THERMODYNAMICS AND ABSOLUTE THERMOMETRY
The first statement of the true dynamical theory of heat, based on the
fundamental idea that the work done in a Carnot cycle is to be accounted
for by an excess of the heat received from the source over the heat
delivered to the refrigerator, was given by Clausius in a paper which
appeared in _Poggendorff's Annalen_ in March and April 1850, and in the
_Philosophical Magazine_ for July 1850, under a title which is a German
translation of that of Carnot's essay. In that paper the First Law of
Thermodynamics is explicitly stated as follows: "In all cases in which
work is produced by the agency of heat, a quantity of heat proportional
to the amount of work produced is expended, and, inversely, by the
expenditure of that amount of work exactly the same amount of heat is
generated." Modern thermodynamics is based on this principle and on the
so-called Second Law of Thermodynamics; which is, however, variously
stated by different authors. According to Clausius, who used in his
paper an argument like that of Carnot based on the transference of heat
from the source to the refrigerator, the foundation of the second law
was the fact that heat tends to pass from hotter to colder bodies. In
1854 (_Pogg. Ann._, Dec. 1854) he stated his fundamental principle
explicitly in the form: "Heat can never pass from a colder to a hotter
body, unless some other change, connected therewith, take place at the
same time," and gives in a note the shorter statement, which he regards
as equivalent: "Heat cannot of itself pass from a colder to a hotter
body."
We shall not here discuss the manner in which Clausius applied this
principle: but he arrived at and described in his paper many important
results, of which he must therefore be regarded as the primary
discoverer. His theory as originally set forth was lacking in clearness
and simplicity, and was much improved by additions made to it on its
republication, in 1864, with other memoirs on the Theory of Heat.
In the _Transactions R.S.E._, for March 1851, Thomson published his
great paper, "On the Dynamical Theory of Heat." The object of the paper
was stated to be threefold: (1) To show what modifications of Carnot's
conclusions are required, when the dynamical theory is adopted: (2) To
indicate the significance in this theory of the numerical results
deduced from Regnault's observations on steam: (3) To point out certain
remarkable relations connecting the physical properties of all
substances established by reasoning analogous to that of Carnot, but
founded on the dynamical theory.
This paper, though subsequent to that of Clausius, is very different in
character. Many of the results are identical with those previously
obtained by Clausius, but they are reached by a process which is
preceded by a clear statement of fundamental principles. These
principles have since been the subject of discussion, and are not free
from difficulty even now; but a great step in advance was made by their
careful formulation in Thomson's paper, as a preliminary to the
erection of the theory and the deduction of its consequences. Two
propositions are stated which may be taken as the First and Second Laws
of Thermodynamics. One is equivalent to the First Law as stated in p.
116, the other enunciates the principle of Reversibility as a criterion
of "perfection" of a heat engine. We quote these propositions.
"Prop. I (Joule).--When equal quantities of mechanical effect are
produced by any means whatever from purely thermal sources, or lost in
purely thermal effects, equal quantities of heat are put out of
existence or are generated."
"Prop. II (Carnot and Clausius).--If an engine be such that when worked
backwards, the physical and mechanical agencies in every part of its
motions are all reversed, it produces as much mechanical effect as can
be produced by any thermodynamic engine, with the same temperatures of
source and refrigerator, from a given quantity of heat."
Prop. I was proved by assuming that heat is a form of energy and
considering always the work effected by causing a working substance to
pass through a closed cycle of changes, so that there was no change of
internal energy to be reckoned with.
Prop. II was proved by the following "axiom": "It is impossible, by
means of inanimate material agency, to derive mechanical effect from any
portion of matter by cooling it below the temperature of the coldest of
the surrounding objects." This is rather a postulate than an axiom; for
it can hardly be contended that it commands assent as soon as it is
stated, even from a mind which is conversant with thermal phenomena. It
sets forth clearly, however, and with sufficient guardedness of
statement, a principle which, when the process by which work is done is
always a cyclical one, is not found contradicted by experience, and one,
moreover, which can be at once explicitly applied to demonstrate that no
engine can be more efficient than a reversible one, and that therefore
the efficiency of a reversible engine is independent of the nature of
the working substance.
It has been suggested by Clerk Maxwell that this "axiom" is contradicted
by the behaviour of a gas. According to the kinetic theory of gases an
elevation of temperature consists in an increase of the kinetic energy
of the translatory motion of the gaseous particles; and no doubt there
actually is, from time to time, a passage of some more quickly moving
particles from a portion of a gas in which the average kinetic energy is
low, to a region in which the average kinetic energy is high, and thus a
transference of heat from a region of low temperature to one of higher
temperature. Maxwell imagined a space filled with gas to be divided into
two compartments A and B by a partition in which were small massless
trapdoors, to open and shut which required no expenditure of energy. At
each of these doors was stationed a "sorting demon," whose duty it was
to allow every particle having a velocity greater than the average to
pass through from A to B, and to stop all those of smaller velocity than
the average. Similarly, the demons were to prevent all quickly moving
particles from going across from B to A, and to pass all slowly moving
particles. In this way, without the expenditure of work, all the quickly
moving particles could be assembled in one compartment, and all the
slowly moving particles in the other; and thus a difference of
temperatures between the two compartments could be brought about, or a
previously existing one increased by transference of heat from a colder
to a hotter mass of gas.
Contrary to a not uncommon belief, this process does not invalidate
Thomson's axiom as he intended it to be understood. For the gas referred
to here is what he would have regarded as the working substance of the
engine, by the cycles of which all the mechanical effect was derived;
and it is not, at the end of the process, in the state as regards
average kinetic energy of the particles in which it was at first. That
this was his answer to the implied criticism of his axiom contained in
Maxwell's illustration, those who have heard him refer to the matter in
his lectures are well aware. But of course it is to be understood that
the substance returns to the same state only in a statistical sense.
Thomson's demonstration that a reversible engine is the most efficient
is well known, and need not here be repeated in detail. The reversible
engine may be worked backwards, and the working substance will take in
heat where in the direct action it gave it out, and _vice versa_: the
substance will do work against external forces where in the direct
action it had work done upon it, and _vice versa_: in short, all the
physical and mechanical changes will be of the same amount, but merely
reversed, at every stage of the backward process. Thus if an engine A be
more efficient than a reversible one B, it will convert a larger
percentage of an amount of heat H taken in at the source into work than
would the reversible one working between the same temperatures. Thus if
h be the heat given to the refrigerator by A, and h' that given by B
when both work directly and take in H; h must be less than h'. Then
couple the engines together so that B works backwards while A works
directly. A will take in H and deliver h, and do work equivalent to
H － h. B will take h' from the refrigerator and deliver H to the source,
and have work equivalent to H － h' spent upon it. There will be no heat
on the whole given to or taken from the source; but heat h' － h will be
taken from the refrigerator, and work equivalent to this will be done.
Thus _by a cyclical process_, which leaves the working substance as it
was, work is done at the expense of heat taken from the refrigerator,
which Thomson's postulate affirms to be impossible. Therefore the
assumption that an engine more efficient than the reversible engine
exists must be abandoned; and we have the conclusion that all reversible
engines are equally efficient.
Thomson acknowledged in his paper the priority of Clausius in his proof
of this proposition, but stated that this demonstration had occurred to
him before he was aware that Clausius had dealt with the matter. He now
cited, as examples of the First Law of Thermodynamics, the results of
Joule's experiments regarding the heat produced in the circuits of
magneto-electric machines, and the fact that when an electric current
produced by a thermal agency or by a battery drives a motor, the heat
evolved in the circuit by the passage of the current is lessened by the
equivalent of the work done on the motor.
[Illustration: FIG. 12.]
In the Carnot cycle, the first operation is an isothermal expansion (AB
in Fig. 12), in which the substance increases in volume by dv, and takes
in from the source heat of amount Mdv. The second operation is an
adiabatic expansion, BC, in which the volume is further increased and
the temperature sinks by dt to the temperature of the refrigerator. The
third operation is an isothermal compression, CD, until the volume and
pressure are such that an adiabatic compression DA will just bring the
substance back to the original state. If ∂p⧸∂t be the rate of
increase of pressure with temperature when the volume is constant, the
step of pressure from one isothermal to the other is ∂p⧸∂t.dt; and
thus the area of the closed cycle in the diagram which measures the
external work done in the succession of changes is ∂p⧸∂t.dtdv. Now,
by the second law, the work done must be a certain fraction of the
work-equivalent of the heat, Mdv, taken in from the source. This
fraction is independent of the nature of the working substance, but
varies with the temperature, and is therefore a function of the
temperature. Its ratio to the difference of temperature dt between
source and refrigerator was called "Carnot's function," and the
determination of this function by experiment was at first perhaps the
most important problem of thermodynamics. Denoting it by μ, we have
the equation
∂p⧸∂t = μM ... (A)
which may be taken as expressing in mathematical language the second law
of thermodynamics. M is here so chosen that Mdv is the heat expressed in
units of work, so that μ does not involve Joule's equivalent of heat.
This equation was given by Carnot: it is here obtained by the dynamical
theory which regards the work done as accounted for by disappearance,
not transference merely, of heat.
The work done in the cycle becomes now μMdtdv, or if H denote Mdv, it
is μHdt. The fraction of the heat utilised is thus μdt. This is
called the efficiency of the engine for the cycle.
From the first law Thomson obtained another fundamental equation. For
every substance there is a relation connecting the pressure p (or more
generally the stress of some type), the volume v (or the configuration
according to the specified stress), and the temperature. We may
therefore take arbitrary changes of any two of these quantities: the
relation referred to will give the corresponding change of the third.
Thomson chose v and t as the quantities to be varied, and supposed them
to sustain arbitrary small changes dv and dt in consequence of the
passage of heat to the substance from without. The amount of heat taken
in is Mdv + Ndt, where Mdv and Ndt are heats required for the changes
taken separately. But the substance expanding through dv does external
work pdv. Thus the net amount of energy given to the substance from
without is Mdv + Ndt － pdv or (M － p)dv + Ndt; and if the substance
is made to pass through a cycle of changes so that it returns to the
physical state from which it started, the whole energy received in the
cycle must be zero. From this it follows that the rate of variation of
M － p when the temperature but not the volume varies, is equal to the
rate of variation of N when the volume but not the temperature varies.
To see that this relation holds, the reader unacquainted with the
properties of perfect differentials may proceed thus. Let the substance
be subjected to the infinitesimal closed cycle of changes defined by (1)
a variation consisting of the simultaneous changes dv, dt of volume and
temperature, (2) a variation －dv of volume only, (3) a variation －dt of
temperature only. M － p and N vary so as to have definite values for
the beginning and end of each step, and the proper mean values can
be written down for each step at once, and therefore the value of
(M － p)dv + Ndt obtained. Adding together these values for the three
steps we get the integral for the cycle. The condition that this should
vanish is at once seen to be the relation stated above.
This result combined with the equation A derived from the second law,
gives an important expression for Carnot's function.
We shall not pursue this discussion further: so much is given to make
clear how certain results as to the physical properties of substances
were obtained, and to explain Thomson's scale of absolute thermodynamic
temperature, which is by far the most important discovery within the
range of theoretical thermodynamics.
There are several scales of temperature: in point of fact the scale of a
mercury-in-glass thermometer is defined by the process of graduation,
and therefore there are as many such scales as there are thermometers,
since no two specimens of glass expand in precisely the same way. Equal
differences of temperature do not correspond to equal increments of
volume of the mercury: for the glass envelope expands also and in its
own way. On the scale of a constant pressure gas thermometer changes of
temperature are measured by variations of volume of the gas, while the
pressure is maintained constant; on a constant volume gas thermometer
changes of temperature are measured by alterations of pressure while the
volume of the gas is kept constant. Each scale has its own independent
definition, thus if the pressure of the gas be kept constant, and the
volume at temperature 0° C. be v₀ and that at any other temperature be
v₁ we define the numerical value t, this latter temperature, by the
equation v = v₀(1 + Et), where E is 1⧸100 of the increase of volume
sustained by the gas in being raised from 0° C. to 100° C. These are
the temperatures of reference on an ordinary centigrade thermometer,
that is, the temperature of melting ice and of saturated steam
under standard atmospheric pressure, respectively. Thus t has
the value (v⧸v₀ － 1)⧸E, and is the temperature (on the constant
pressure scale of the gas thermometer) corresponding to the volume v.
Equal differences of temperature are such as correspond to equal
increments of the volume at 0° C.
Similarly, on the constant volume scale we obtain a definition of
temperature from the pressure p, by the equation t = (p⧸p₀ － 1)⧸E',
where p₀ is the pressure at 0° C., and E' is 1⧸100 of the change of
pressure produced by raising the temperature from 0° C. to 100° C.
For air E is approximately 1⧸273, and thus t = 273(v － v₀)⧸v₀.
If we take the case of v = 0, we get t = －273. Now, although this
temperature may be inaccessible, we may take it as zero, and the
temperature denoted by t is, when reckoned from this zero, 273 + t.
This zero is called the absolute zero on the constant pressure air
thermometer. The value of E' is very nearly the same as that of E; and
we get in a similar manner an absolute zero for the constant volume
scale. If the gas obeyed Boyle's law exactly at all temperatures, E
would not differ from E'.
It was suggested to Thomson by Joule, in a letter dated December 9,
1848, that the value of μ might be given by the equation
μ = JE⧸(1 + Et). Here we take heat in dynamical units, and therefore
the factor J is not required. With these units Joule's suggestion is
that μ = E⧸(1 + Et), or with E = 1⧸273 μ = 1⧸(273 + t), that is,
μ = 1⧸T where T is the temperature reckoned in centigrade degrees from
the absolute zero of the constant pressure air thermometer.
The possibility of adopting this value of μ was shown by Thomson to
depend on whether or not the heat absorbed by a given mass of gas in
expanding without alteration of temperature is the equivalent of the
work done by the expanding gas against external pressure. The heat H
absorbed by the air in expanding from volume V to another volume V' at
constant temperature is the integral of Mdv taken from the former volume
to the latter. But by the value of M given on p. 121, if W be the
integral of pdv, that is the work done by the air in the expansion,
∂W⧸∂t = μH. The equation fulfilled by the gas at constant pressure
(the defining equation for t), v = v₀(1 + Et), gives for the integral
of pdv, that is W, the equation W = pv₀(1 + Et)log(V'⧸V), so that
∂W⧸∂t = EW⧸(1 + Et). Thus μH = EW⧸(1 + Et).
Hence it follows that if μ = E⧸(1 + Et), the value of H will be simply
W. Thus Joule's suggested value of μ is only admissible if the work
done by the gas in expanding from a given volume to any other is the
equivalent of the heat absorbed; or, which is the same thing, if the
external work done in compressing the gas from one volume to another is
the equivalent of the heat developed.
This result naturally suggests the formation of a new scale of
thermometry by the adoption of the defining relation T = 1⧸μ, where T
denotes temperature. A scale of temperature thus defined is proposed
in the paper by Joule and Thomson, "On the Thermal Effects of Fluids
in Motion," Part II, which was published in the _Philosophical
Transactions_ for June 1854, and is what is now universally known as
Thomson's scale of absolute thermodynamic temperature. It can, of
course, be made to give 100 as the numerical value of the temperature
difference between 0° C. and 100° C. by properly fixing the unit of T.
This scale was the natural successor, in the dynamical theory, of one
which Thomson had suggested in 1848, and which was founded, according
to Carnot's idea, on the condition that a unit of heat should do the
same amount of work in descending through each degree. This, as he
pointed out, might justly be called an absolute scale, since it would be
independent of the physical properties of any substance. In the same
sense the scale defined by T = 1⧸μ is truly an absolute scale.
The new scale gives a simple expression for the efficiency of a perfect
engine working between two physically given temperatures, and assigns
the numerical values of these temperatures; for the heat H taken in from
the source in the isothermal expansion which forms the first operation
of the cycle (p. 120) is Mdv, and, as we have seen, the work done in the
cycle is ∂p⧸∂t.dtdv, or μHdt. If we adopt the expression 1⧸T for
μ, we may put dT for dt; and we obtain for the work done the
expression HdT⧸T. The work done is thus the fraction dT⧸T of the heat
taken in, and this is what is properly called the efficiency of the
engine for the cycle.
If we suppose the difference of temperatures between source and
refrigerator to be finite, T － T', say, then since T is the temperature
of the source, we have for the efficiency (T － T')⧸T. If the heat taken
in be H, the heat rejected is HT'⧸T, so that the heat received by the
engine is to the heat rejected by it in the ratio of T' to T. Thus, as
was done by Thomson, we may define the temperatures of the source and
refrigerator as proportional to the heat taken in from the source and
the heat rejected to the refrigerator by a perfect engine, working
between those temperatures. The scale may be made to have 100 degrees
between the temperature of melting ice and the boiling point, as
already explained. We shall return to the comparison of this scale with
that of the air thermometer. At present we consider some of the
thermodynamic relations of the properties of bodies arrived at by
Thomson.
First we take the working substance of the engine as consisting of
matter in two states or phases; for example, ice and water, or water and
saturated steam. Let us apply equation (A) to this case. If v₁, v₂ be
the volume of unit of mass in the first and second states respectively,
the isothermal expansion of the first part of the cycle will take place
in consequence of the conversion of a mass dm from the first state to
the second. Thus dv, the change of volume, is dm(v₂ － v₁). Also if L
be the latent heat of the substance in the second state, _e.g._ the
latent heat of water, Mdv = Ldm; so that M(v₂ － v₁) = L. If dp be the
step of pressure corresponding to the step dT of temperature, equation
(A) becomes
dT⧸T = dp(v₂ － v₁)⧸L ... (B)
In the case of coexistence of the liquid and solid phases, this gives us
the very remarkable result that a change of pressure dp will raise or
lower the temperature of coexistence of the two phases, that is, the
melting point of the solid, by the difference of temperature, dT,
according as v₂ is greater or less than v₁ Thus a substance like
water, which expands in freezing, so that v₂ － v₁ is negative, has
its freezing point lowered by increase of pressure and raised by
diminution of pressure. This is the result predicted by Professor James
Thomson and verified experimentally by his brother (p. 113 above).
On the other hand, a substance like paraffin wax, which contracts in
solidifying, would have its melting point raised by increase of pressure
and lowered by a diminution of pressure.
The same conclusions would be applicable when the phases are liquid and
vapour of the same substance, if there were any case in which v₂ － v₁
is negative. As it is we see, what is well known to be the case, that
the temperature of equilibrium of a liquid with its vapour is raised by
increase of pressure.
Another important result of equation (B), as applied to the liquid and
vapour phases of a substance, is the information which it gives as to
the density of the saturated vapour. When the two phases coexist the
pressure is a function of the temperature only. Hence if the relation of
pressure to temperature is known, dp⧸dT can be calculated, or obtained
graphically from a curve; and the volume v₂ per unit mass of the vapour
will be given in terms of dp⧸dT, the temperature T, and the volume v per
unit mass of the liquid. The density of saturated steam at different
temperatures is very difficult to measure experimentally with any
approach of accuracy: but so far as experiment goes equation (B) is
confirmed. The theory here given is fully confirmed by other results,
and equation (B) is available for the calculation of v₂ for any
substance for which the relation between p and T is known. It is thus
that the density of saturated steam can best be found.
We can obtain another important result for the case of the working
substance in two phases from equation (B). The relation is
∂L⧸∂T + c － h = L⧸T ... (C)
where c and h are the specific heats of the substance in the two phases
respectively, and L is the latent heat of the second phase at absolute
temperature T.
We shall obtain the relation in another way, which will illustrate
another mode of dealing with a cycle of operations which Thomson
employed. Any small step of change of a substance may be regarded as
made up of a step of volume, say, followed by a step of temperature,
that is, by an isothermal step followed by an adiabatic step. In this
way any cycle of operations whatever may be regarded as made up of a
series of Carnot cycles. But without regarding any cycle of a more
general kind than Carnot's as thus compounded, we can draw conclusions
from it by the dynamical theory provided only it is reversible. Suppose
a gramme, say, of the substance to be taken at a specified temperature T
in the lower phase, and to be changed to the other phase at that
temperature. The heat taken in will be L and the expansion will be
v₂ － v₁. Next, keeping the substance in the second phase, and in
equilibrium with the first phase (that is, for example, if the second
phase is saturated vapour, the saturation is to continue in the further
change), let the substance be lowered in temperature by dT. The heat
given out by the substance will be hdT, where h is the specific heat of
the substance in the second phase. Now at the new temperature T － dT let
the substance be wholly brought back to the second phase; the heat given
out will be L － ∂L⧸∂T.dT. Finally, let the substance, now again
all in the first phase, be brought to the original temperature: the heat
taken in will be cdt, where c is the specific heat in the first phase.
Thus the net excess of heat taken in over heat given out in the cycle is
(∂L⧸∂T + c － h)dT. This must, in the indicator diagram for the
changes specified, be the area of the cycle or (v₂ － v₁)∂p⧸∂T.dT.
But by equation (B) L⧸T(v₂ － v₁) = ∂p⧸∂T, and the area of the cycle is
(L⧸T)dT. Equating the two expressions thus found for the area we get
equation (C).
This relation was arrived at by Clausius in his paper referred to above,
and the priority of publication is his: it is here given in the form
which it takes when Thomson's scale of absolute temperature is used.
Regnault's experimental results for the heat required to raise unit mass
of water from the temperature of melting ice to any higher temperature
and evaporate it at that temperature enable the values of L⧸T and
∂L⧸∂T to be calculated, and therefore that of h to be found. It
appears that h is negative for all the temperatures to which Regnault's
experimental results can be held to apply. This, as was pointed out by
Thomson, means that if a mass of saturated vapour is made to expand so
as at the same time to fall in temperature, it must have heat given to
it, otherwise it will be partly condensed into liquid; and, on the other
hand, if the vapour be compressed and made to rise in temperature while
at the same time it is kept saturated, heat must be taken from it,
otherwise the vapour will become superheated and so cease to be
saturated.
It is convenient to notice here the article on Heat which Thomson wrote
for the ninth edition of the _Encyclopædia Britannica_. In that article
he gave a valuable discussion of ordinary thermometry, of thermometry by
means of the pressures of saturated vapour of different
substances--steam-pressure thermometers, he called them--of absolute
thermodynamic thermometry, all enriched with new experimental and
theoretical investigations, and appended to the whole a valuable
synopsis, with additions of his own, of the Fourier mathematics of heat
conduction.
First dealing with temperature as measured by the expansion of a liquid
in a less expansible vessel, he showed how it is in reality numerically
reckoned. This amounted to a discussion of the scale of an ordinary
mercury-in-glass thermometer, a subject concerning which erroneous
statements are not infrequently made in text-books. A sketch of
Thomson's treatment of it is given here.
Considering this thermometer as a vessel consisting of a glass bulb and
a long glass stem of fine and uniform bore, hermetically sealed and
containing only mercury and mercury vapour, he explained the numerical
relation between the temperature as shown by the instrument and the
volumes of the mercury and vessel. The scale is really defined by the
method of graduation adopted. Two points of reference are marked on the
stem at which the top of the mercury stands when the vessel is immersed
(1) in melting ice, (2) in saturated steam under standard atmospheric
pressure. The stem is divided into parts of equal volume of bore between
these two points and beyond each of them. For a centigrade thermometer
the bore-space between the two points is divided into 100 equal parts,
and the lower point of reference is marked 0 and the upper 100, and the
other dividing marks are numbered in accordance with this along the
stem. Each of these parts of the bore may be called a degree-space.
Now let the instrument contain in its bulb and stem, up to the mark 0, N
degree-spaces, and let v be the volume of a degree-space at that
temperature. The volume up to the mark 0 will be Nv, at that
temperature; and if the substance of the vessel be quite uniform in
quality and free from stress, N will be the same for all temperatures.
If v₀ be the volume of a degree-space at the temperature of melting ice
the volume of the mercury at that temperature will be Nv₀. If G be the
expansion of the glass when the volume of a degree-space is increased
from v₀ to v by the rise of temperature, then v = v₀(1 + G). The
volume of the mercury has been increased therefore to (N + n)v₀(1 + G)
by the same rise of temperature, if the top of the column is thereby
made to rise from the mark 0 so as to occupy n degree-spaces more than
before. But if E be the expansion of the mercury between the temperature
of melting ice and that which has now been attained, the volume of the
mercury is also Nv₀(1 + E). Hence N(1 + E) = (N + n)(1 + G). This gives
n = N(E － G)⧸(1 + G).
If we take, as is usual, n as measuring the temperature, and substitute
for it the symbol t, we have, since N = 100(1 + G₁₀₀)⧸(E₁₀₀ － G₁₀₀),
t = 100 {(1 + G₁₀₀)⧸(1 + G)} {(E － G)⧸(E₁₀₀ － G₁₀₀)} ... (D)
In this reckoning the definition of any temperature, let us say 37° C.,
is the temperature of the vessel and its contents when the top of the
mercury column stands at the mark 37 above 0, on the scale defined by
the graduation of the instrument; but the numerical signification with
relation to the volumes is given by equation (D). This shows that the
numerical measure of any temperature involves both the expansion of the
vessel and that of the glass vessel between the temperature of melting
ice and the temperature in question. This result may be contrasted with
the erroneous statement frequently made that equal increments of
temperature correspond to equal increments of the volume of the
thermometric substance. It also shows that different mercury-in-glass
thermometers, however accurately made and graduated, need not agree when
placed in a bath at any other temperature than 0° C. or 100° C. This
fact, and the results of the comparison of thermometers made with
different kinds of glass with the normal air thermometer, which was
carried out by Regnault, were always insisted on by Thomson in his
teaching when he dealt with the subject of heat. The scale of a
mercury-in-glass thermometer is too often in text-books, and even in
Acts of Parliament regarded as a perfectly definite thing, and the
expansion of a gas is not infrequently defined by this indefinite scale,
instead of being used as it ought to be, as the basis of definition of
the scale of the gas thermometer. The whole treatment of the so-called
gaseous laws is too often, from a logical point of view, a mass of
confusion.
In his article on Heat Thomson gave two definitions of the scale of
absolute temperature. One is that stated on p. 126 above, namely, that
the temperature of the source and refrigerator are in the ratio of the
heat taken in from the source to the heat given to the refrigerator,
when the engine describes a Carnot cycle consisting of two isothermal
and two adiabatic changes.
The other definition is better adapted for general use, as it applies to
any cycle whatever which is reversible. Let the working substance
expand under constant pressure by an amount dv (AB' in Fig. 12), and let
heat H be given to the substance at the same time. The external work
done is pdv. Thomson called pdv⧸H the work ratio. Now let the
temperature be raised by dT without giving heat to the substance or
taking heat from it, and let the corresponding pressure rise be dp; and
call dp⧸p the pressure ratio. The temperature ratio dT⧸T is equal to
the product of the work ratio and the pressure ratio, that is,
dT⧸T = dvdp⧸H
This is clearly true; for dvdp is the area of a cycle like AB'C'D,
represented in Fig. 12, for which an amount of heat H is taken in,
though not in this case strictly at one temperature. And clearly, since
in Fig. 12 the change from B' to B is adiabatic, H is the heat which
would have to be taken in for the isothermal change AB in the Carnot
cycle ABCD, which has the same area as AB'C'D. Thus the efficiency of
the cycle is dvdp⧸H, and this by the former definition is dT⧸T.
Or we may regard the matter thus:--The amount of heat H which
corresponds to an infinitesimal expansion dv may be used in equation (A)
whether the expansion is isothermal or not, if we take T as the average
temperature of the expansion. Hence we have dp⧸dT = H⧸(dv.T), that is,
dT⧸T = dpdv⧸H. The theorem on p. 128 is obtained by what is virtually
this process.
COMPARISON OF ABSOLUTE SCALE WITH SCALE OF AIR THERMOMETER
The comparison which Joule and Thomson carried out of the absolute
thermodynamic scale with the scale of the constant pressure gas
thermometer has already been referred to, and it has been shown that the
two scales would exactly agree, that is, absolute temperature would be
simply proportional to the volume of the gas in a gas thermometer kept
at the temperature to be measured, if the internal energy of the gas
were not altered by an alteration of volume without alteration of
temperature, that is, if the de － ∂e of p. 107 above were zero. Joule
tested whether this was the case by immersing two vessels, connected by
a tube which could be opened or closed by a stopcock, in the water of a
calorimeter, ascertaining the temperature with a very sensitive
thermometer, and then allowing air which had already been compressed
into one of the vessels to flow into the other, which was initially
empty. It was found that no alteration of temperature of the water of
the calorimeter that could be observed was produced. But the volume of
the air had been doubled by the process, and if any sensible alteration
of internal energy had taken place it would have shown itself by an
elevation or a lowering of the temperature of the water, according as
the energy had been diminished or increased.
Thomson suggested that the gas to be examined should be forced through a
pipe ending in a fine nozzle, or, preferably, through a plug of porous
material placed in a pipe along which the gas was forced by a pump, and
observations made of the temperature in the steady stream on both sides
of the plug. The experiments were carried out with a plug of compressed
cotton-wool held between two metal disks pierced with holes, in a tube
of boxwood surrounded also by cotton-wool, and placed in a bath of water
closely surrounding the supply pipe. This was of metal, and formed the
end of a long spiral all immersed in the bath. Thus the temperature of
the gas approaching the plug was kept at a uniform temperature
determined by a delicate thermometer; another thermometer gave the
temperature in the steady stream beyond the plug.
In the case of hydrogen the experiments showed a slight heating effect
of passage through the plug; air, oxygen, nitrogen and carbonic acid
were cooled by the passage.
The theory of the matter is set forth in the original papers, and in a
very elegant manner in the article on Heat. The result of the analysis
shows that if ∂w be the positive or negative work-value of the heat
which will convert one gramme of the gas after passage to its original
temperature; and T be absolute temperature, and v volume of a gramme of
the gas at pressure p, and the difference of pressure on the two sides
of the plug be dp, the equation which holds is
(1⧸T) (∂T⧸∂v) = 1⧸{v + (∂w⧸dp)} ... (E)[18]
It was found by Joule and Thomson that ∂w was proportional to dp for
values of dp up to five or six atmospheres. At different temperatures,
however, in the case of hydrogen the heating effect was found to
diminish with rise of temperature, being .100 of a degree centigrade at
4° or 5° centigrade, and .155 at temperatures of from 89° to 93°
centigrade for a difference of pressure due to 100 inches of mercury.
If there is neither heating nor cooling ∂w = 0, and we obtain by
integration T = Cv, where C is a constant.
Elaborate discussions of the theory of this experiment will be found in
modern treatises on thermodynamics, and in various recent memoirs, and
the differential equation has been modified in various ways, and
integrated on various suppositions, which it would be out of place to
discuss here.
The cooling effect of passing a gas such as air or oxygen through a
narrow orifice has been used to liquefy the gas. The stream of gas is
pumped along a pipe towards the opening, and that which has passed the
orifice and been slightly cooled is led on its way back to the pump
along the outside of the pipe by which more gas is approaching the
orifice, and so cools slightly the advancing current. The gas which
emerges later is thus cooler than that which emerged before, and the
process goes on until the issuing gas is liquefied and falls down into
the lower part of the pipe surrounding the orifice, whence it can be
drawn off into vessels constructed to receive and preserve it.
It is possible thus to liquefy hydrogen, which shows that at the low
temperature at which the process is usually started (an initial cooling
is applied) the passage through the orifice has a cooling effect as in
the other cases.
Another idea, that of _thermodynamic motivity_, on which Thomson
suggested might be founded a fruitful presentation of the subject of
thermodynamics, may be mentioned here. It was set forth in a letter
written to Professor Tait in May 1879. If a system of bodies be given,
all at different temperatures, it is possible to reduce them to a common
temperature, and by doing so to extract a certain amount of mechanical
energy from them. The temperatures must for this purpose be equalised
by perfect thermodynamic engines working between the final temperature
T₀, say, and the temperatures of the different parts of the system.
This process is one of the levelling up and the levelling down of
temperature; and the temperature T₀ is such that exactly the heat given
out at T₀ by certain engines, receiving heat from bodies of higher
temperature than T₀, is supplied to the engines which work between T₀
and bodies at lower temperatures. The whole useful work obtained in this
way was called by Thomson the motivity of the system. Of course
equalisation of temperature may be obtained by conduction, and in this
case the energy which might be utilised is lost. With two equal and
similar bodies at absolute temperatures T, T' the temperature to which
they are reduced when their motivity is extracted is √(TT'). If the
temperatures are equalised by conduction the resulting temperature is
higher, being ½(T + T'). Thus, if only the two bodies are available
for engines to work between, the motivity is the measure of the energy
lost when conduction brings about equalisation of temperature.
A very suggestive paper on the subject was published by Lord Kelvin in
the _Trans. R.S.E._, vol. 28, 1877-8.
DISSIPATION OF ENERGY
In connection with the theory of heat must be mentioned Thomson's great
generalisation, the theory of the dissipation of energy.[19] Most people
have some notion of the meaning of the physical doctrine of
conservation of energy, though in popular discourses it is usually
misstated. What is meant is that in a finite material system, which is
isolated in the sense that it is not acted on by force from without, the
total amount of energy--that is, energy of motion and energy of relative
position (including energy of chemical affinity) of the parts--remains
constant. The usual misstatement is that the energy of the universe is
constant. This may be true if the universe is finite; if the universe is
infinite in extent the statement has no meaning. In any case, we know
nothing about the universe as a whole, and therefore make no statements
regarding it.
But while there is thus conservation or constancy of amount of energy in
an isolated and finite material system, this energy may to residents on
the system become unavailable. For useful work within such a system is
done by conversion of energy from one form to another and the total
amount remains unchanged. But if this conversion is prevented all
processes which involve such conversion must cease, and among these are
vital processes.
The unavailable form which the energy of the system with which we are
directly and at present concerned, whatever may become of us ultimately,
is taking, according to Thomson's theory, is universally diffused heat.
How this comes about may be seen as follows. Even a perfect engine, if
the refrigerator be at the lowest available temperature, rejects a
quantity of heat which cannot be utilised for the performance of the
work. This heat is diffused by conduction and radiation to surrounding
bodies, and so to bodies more remote, and the general temperature of
the system is raised. Moreover, as heat engines are imperfect there is
heat rejected to the surroundings by conduction, and produced by work
done against friction, so that the heat thrown on the unavailable or
waste heap is still further increased.
Conduction of heat is the great agency by which energy is more and more
dispersed in this unavailable form throughout the totality of material
bodies. As has been seen, available motivity is continually wasted
through its agency; and in the flow of heat in the earth and in the sun
and other unequally heated bodies of our system the waste of energy is
prodigious. Aided by convection currents in the air and in the ocean it
continually equalises temperatures, but does so at an immense cost of
useful energy.
Then in our insanely wasteful methods of heating our houses by open
fires, of half burning the coal used in boiler furnaces, and allowing
unconsumed carbon to escape into the atmosphere in enormous quantities,
while a very large portion of the heat actually generated is allowed to
escape up chimneys with heated gases, the store of unavailable heat is
being added to at a rate which will entail great distress, if not ruin,
on humanity at no indefinitely distant future. It will be the height of
imprudence to trust to the prospect, not infrequently referred to at the
present time, of drawing on the energy locked up in the atomic structure
of matter. He would be a foolish man who would wastefully squander the
wealth he possesses, in the belief that he can recoup himself from mines
which all experience so far shows require an expenditure to work them
far beyond any return that has as yet been obtained.
It is not apart from our present theme to urge that it is high time the
question of the national economy of fuel, and the desirability of
utilising by afforestation the solar energy continually going to waste
on the surface of the earth, were dealt with by statesmen. If statesmen
would but make themselves acquainted with the results of physical
science in this magnificent region of cosmic economics there would be
some hope, but, alas! as a rule their education is one which inevitably
leads to neglect, if not to disdain of physical teaching.
From the causes which have been referred to, energy is continually being
dissipated, not destroyed, but locked up in greater and greater quantity
in the general heat of bodies. There is always friction, always heat
conduction and convection, so that as our stores of motional or
positional energy, whether of chemical substances uncombined, the
earth's motion, or what not, are drawn upon, the inevitable fraction,
too often a large proportion, is shed off and the general temperature
raised. After a large part of the whole existent energy has gone thus to
raise the dead level of things, no difference of temperature adequate
for heat engines to work between will be possible, and the inevitable
death of all things will approach with headlong rapidity.
THERMOELASTICITY AND THERMOELECTRICITY
In the second definition of the scale of absolute temperature just
discussed, stress of any type may be substituted for pressure, and the
corresponding displacement s for the change of volume. Thus for a piece
of elastic material put through a cycle of changes we may substitute dS
for dp and Ads for dv; where A is such a factor that AdSds is the work
done in the displacement ds by the stress dS. As an example consider a
wire subjected to simple longitudinal stress S. Longitudinal extension
is produced, but this is not the only change; there is at the same time
lateral contraction. However, s within certain limits is proportional to
S.
Let heat dH in dynamical measure be given to the wire while the stress S
is maintained constant, and let the extension increase from s to s + ds.
The stress S will do work ASds _on the wire_, and the work ratio will be
-ASds⧸dH. Now let the stress be increased to S + dS while the extension
is kept constant, and the absolute temperature raised from T to T + dT.
The stress ratio (as we may call it) is dS⧸S and the temperature ratio
dT⧸T. Thus we obtain (p. 134 above)
－(dS⧸dT) = (1⧸TA) (dH⧸ds)
In his Heat article Thomson used the alteration e of strain under
constant stress (that is ds⧸l, where l is the length of the wire)
corresponding to an amount of heat sufficient to raise the temperature
under constant stress by 1°. Hence if K be the specific heat under
constant stress, and le be put for ds in the sense just stated, we have
dT = －(TedS⧸Kρ) ... (F)
where ρ is the density, since dH = KρlA.
The ratio of dH to the increase ds of the extension is positive or
negative, that is, the substance absorbs or evolves heat, when strained
under the condition of constant stress, according as dS⧸dT is negative
or positive. Or we may put the same thing in another way which is
frequently useful. If a wire subjected to constant stress has heat given
to it, ds is negative or positive, in other words the wire shortens or
lengthens, according as dS⧸dT is positive or negative, that is,
according as the stress for a given strain is increased or diminished by
increase of temperature.
It is known from experiment that a metal wire expands under constant
stress when heat is given to it, and thus we learn from the equation (F)
that the stress required for a given strain is diminished when the
temperature of the wire is raised. Again, a strip of india-rubber
stretched by a weight is shortened if its temperature is raised,
consequently the stress required for a given strain is increased by
rise of temperature.
These results, from a qualitative point of view, are self-evident. But
from what has been set forth it will be obvious that an equation exactly
similar to (F) holds whether the change ds of s is taken as before under
constant stress, or at uniform temperature, or whether the change dS of
S is effected adiabatically or at constant strain.
In all these cases the same equation
dT = －T (edS⧸Kρ) ... (G)
applies, with the change of meaning of dT involved.
This equation differs from that of Thomson as given in various places
(_e.g._ in the _Encyclopædia Britannica_ article on Elasticity which he
also wrote) in the negative sign on the right-hand side, but the
difference is only apparent. According to his specification a pressure
would be a positive stress, and an expansion a positive displacement,
and in applying the equation to numerical examples this must be borne in
mind so that the proper signs may be given to each numerical magnitude.
As an example of adiabatic change, a sudden extension of the wire
already referred to by an increase of stress dS may be considered. If
there is not time for the passage of heat from or to the surroundings of
the wire, the change of temperature will be given by equation (G).
This equation was applied by Thomson (article Elasticity) to find the
relation between what he called the kinetic modulus of elasticity and
the static modulus, that is, between the modulus for adiabatic strain
and the modulus for isothermal strain.
The augmentation of the strain produced by raising the temperature 1°
is e, and therefore edT, that is, －Te²dS⧸Kρ, is the increase of strain
due to the sudden rise of temperature dT. This added to the isothermal
strain produced by dS will give the whole adiabatic strain. Thus
if M be the static or isothermal modulus, the adiabatic strain
is dS⧸M － Te²dS⧸Kρ. If M' denote the kinetic or adiabatic modulus
its value is dS divided by the whole adiabatic strain, that is,
M' = M⧸(1 － MTe²⧸Kρ) and the ratio M'⧸M = 1⧸(1 － MTe²⧸Kρ).
It is well known and easy to prove, without the use of any theorem which
can be properly called thermodynamic, that this ratio of moduli is equal
to the ratio of the specific heat K of the substance, under the
condition of constant stress, to the specific heat N under the condition
of constant strain of the corresponding type. This, indeed, is
self-evident if two changes of stress, one isothermal the other
adiabatic, _which produce the same steps of displacement ds_, be
considered, and it be remembered that the step ∂T of temperature which
accompanies the adiabatic change may be regarded as made up of a step
－dT of temperature, accompanying a displacement ds effected at constant
stress, and then two successive steps dT and ∂T effected, at constant
strain, along with the steps of stress dS. The ratio M'⧸M is easily seen
to have the value (∂T + dT)⧸dt, and since －KdT + N(∂T + dT) = 0, by
the adiabatic condition, the theorem is proved.
Laplace's celebrated result for air, according to which the adiabatic
bulk-modulus is equal to the static bulk-modulus multiplied by the ratio
of the specific heat of air pressure constant to the specific heat of
air volume constant, is a particular example of this theory.
Thomson showed in the Elasticity article how, by the value of M'⧸M,
derived as above from thermodynamic theory, the value of K⧸N could be
obtained for different substances and for different types of stress, and
gave very interesting tables of results for solids, liquids, and gases
subjected to pressure-stress (bulk-modulus) and for solids subjected to
longitudinal stress (Young's modulus).
The discussion as to the relation of the adiabatic and isothermal moduli
of elasticity is part of a very important paper on "Thermoelastic,
Thermomagnetic, and Thermoelectric Properties of Matter," which he
published in the _Philosophical Magazine_ for January 1878. This was in
the main a reprint of an article entitled, "On the Thermoelastic and
Thermomagnetic Properties of Matter, Part I," which appeared in April
1855 in the first number of the _Quarterly Journal of Mathematics_. Only
thermoelasticity was considered in this article; the thermomagnetic
results had, however, been indicated in an article on "Thermomagnetism"
in the second edition of the _Cyclopædia of Physical Science_, edited
and in great part written by Professor J. P. Nichol, and published in
1860. For the same Cyclopædia Thomson also wrote an article entitled,
"Thermo-electric, Division I.--Pyro-Electricity, or Thermo-Electricity
of Non-conducting Crystals," and the enlarged _Phil. Mag._ article also
contained the application of thermodynamics to this kind of
thermoelectric action.
This great paper cannot be described without a good deal of mathematical
analysis; but the student who has read the earlier thermodynamical
papers of Thomson will have little difficulty in mastering it. It must
suffice to say here that it may be regarded as giving the keynote of
much of the general thermodynamic treatment of physical phenomena, which
forms so large a part of the physical mathematics of the present day,
and which we owe to Willard Gibbs Duhem, and other contemporary writers.
Thomson had, however, previous to the publication of this paper, applied
thermodynamic theory to thermoelectric phenomena. A long series of
papers containing experimental investigations, and entitled,
"Electrodynamic Qualities of Metals," are placed in the second volume of
his _Mathematical and Physical Papers_. This series begins with the
Bakerian Lecture (published in the _Transactions of the Royal Society_
for 1856) which includes an account of the remarkable experimental work
accomplished during the preceding four or five years by the volunteer
laboratory corps in the newly-established physical laboratory in the old
College. The subjects dealt with are the Electric Convection of Heat,
Thermoelectric Inversions, the Effects of Mechanical Strain and of
Magnetisation on the Thermoelectric Qualities of Metals, and the Effects
of Tension and Magnetisation on the Electric Conductivity of Metals. It
is only possible to give here a very short indication of the
thermodynamic treatment, and of the nature of Thomson's remarkable
discovery of the electric convection of heat.
It was found by Seebeck in 1822 that when a circuit is formed of two
different metals (without any cell or battery) a current flows round the
circuit if the two junctions are not at the same temperature. For
example, if the two metals be rods of antimony and bismuth, joined at
their extremities so as to form a complete circuit, and one junction be
warmed while the other is kept at the ordinary temperature, a current
flows across the hot junction in the direction from bismuth to antimony.
Similarly, if a circuit be made of a copper wire and an iron wire, a
current passes across the warmer junction from copper to iron. The
current strength--other things being the same--depends on the metals
used; for example, bismuth and antimony are more effective than other
metals.
It was found by Peltier that when a current, say from a battery, is sent
round such a circuit, that junction is cooled and that junction is
heated by the passage of the current, which, being respectively heated
and cooled, would without the cell have caused a current to flow in the
same direction. Thus the current produced by the difference of
temperature of the junctions causes an absorption of heat from the
warmer junction, and an evolution of heat at the colder junction.
This naturally suggested to Thomson the consideration of a circuit of
two metals, with the junctions at different temperatures, as a heat
engine, of which the hot junction was the source and the cold junction
the refrigerator, while the heat generated in the circuit by the current
and other work performed, if there was any, was the equivalent of the
difference between the heat absorbed and the heat evolved. Of course in
such an arrangement there is always irreversible loss of heat by
conduction; but when such losses are properly allowed for the circuit is
capable of being correctly regarded as a reversible engine.
Shortly after Seebeck's discovery it was found by Cumming that when the
hot junction was increased in temperature the electromotive force
increased more and more slowly, at a certain temperature of the hot
junction took its maximum value, and then as the temperature of the hot
junction was further increased began to diminish, and ultimately, at a
sufficiently high temperature, in most instances changed sign. The
temperature of maximum electromotive force was found to be independent
of the temperature of the colder junction. It is called the temperature
of the neutral point, from the fact that if the two junctions of a
thermoelectric circuit be kept at a constant small difference of
temperature, and be both raised in temperature until one is at a higher
temperature than the neutral point, and the other is at a lower, the
electromotive force will fall off, until finally, when this point is
reached, it has become zero.
Thus it was found that for every pair of metals there was at least one
such temperature of the hot junction, and it was assumed, with
consequences in agreement with experimental results, that when the
temperature was the neutral temperature there was neither absorption nor
evolution of heat at the junction. But then the source provided by the
thermodynamic view just stated had ceased to exist. The current still
flowed, there was evolution of heat at the cold junction, and likewise
Joulean evolution of heat in the wires of the circuit in consequence of
their resistance. Hence it was clear that energy must be obtained
elsewhere than at the junctions. Thomson solved the problem by showing
that (besides the Joulean evolution of heat) there is absorption (or
evolution) of heat when a current flows in a conductor along which there
is a gradient of temperature. For example, when an electric current
flows along an unequally heated copper wire, heat is evolved where the
current flows from the hot parts to the cold, and heat is absorbed where
the flow is from cold to hot. When the hot junction is at the
temperature of zero absorption or evolution of heat--the so-called
neutral temperature--the heat absorbed in the flow of the circuit along
the unequally heated conductors is greater than that evolved on the
whole, by an amount which is the equivalent of the energy electrically
expended in the circuit in the same time.
It was found, moreover, that the amount of heat absorbed by a given
current in ascending or descending through a given difference of
temperature is different in different metals. When the current was unit
current and the temperature difference also unity, Thomson called the
heat absorbed or evolved in a metal the specific heat of electricity in
the metal, a name which is convenient in some ways, but misleading in
others. The term rather conveys the notion that electricity has a
material existence. A substance such as copper, lead, water, or mercury
has a specific heat in a perfectly understood sense; electricity is not
a substance, hence there cannot be in the same proper sense a specific
heat of electricity.
However, this absorption and evolution of heat was investigated
experimentally and mathematically by Thomson, and is generally now
referred to in thermoelectric discussions as the "Thomson effect."
Part VI (_Trans. R.S._, 1875) of the investigations of the
electrodynamic qualities of metals dealt with the effects of stretching
and compressing force, and of torsion, on the magnetisation of iron and
steel and of nickel and cobalt.
One of the principal results was the discovery that the effect of
longitudinal pull is to increase the inductive magnetisation of soft
iron, and of transverse thrust to diminish it, so long as the
magnetising field does not exceed a certain value. When this value,
which depends on the specimen, is exceeded, the effect of stress is
reversed. The field-intensity at which the effect is reversed is called
the Villari critical intensity, from the fact, afterwards ascertained,
that the result had previously been established by Villari in Italy. No
such critical value of the field was found to exist for steel, or
nickel, or cobalt.
In some of the experiments the specimen was put through a cycle of
magnetic changes, and the results recorded by curves. These proved that
in going from one state to another and returning the material lagged in
its return path behind the corresponding states in the outward path.
This is the phenomenon called later "hysteresis," and studied in minute
detail by Ewing and others. Thomson's magnetic work was thus the
starting point of many more recent researches.
CHAPTER IX
HYDRODYNAMICS--DYNAMICAL THEOREM OF MINIMUM ENERGY--VORTEX MOTION
Thomson devoted great attention from time to time to the science of
hydrodynamics. This is perhaps the most abstruse subject in the domain
of applied mathematics, and when viscosity (the frictional resistance to
the relative motion of particles of the fluid) is taken into account,
passes beyond the resources of mathematical science in its present state
of development. But leaving viscosity entirely aside, and dealing only
with so-called perfect fluids, the difficulties are often overwhelming.
For a long time the only kind of fluid motion considered was, with the
exception of a few simple cases, that which is called irrotational
motion. This motion is characterised by the analytical peculiarity, that
the velocity of an element of the fluid in any direction is the rate of
variation per unit distance in that direction of a function of the
coordinates (the distances which specify the position) of the particle.
This condition very much simplifies the analysis; but when it does not
hold we have much more serious difficulties to overcome. Then the
elements of the fluid have what is generally, but quite improperly,
called molecular rotation. For we know little of the molecules of a
fluid; even when we deal with infinitesimal elements, in the analysis of
fluid motion, we are considering the fluid in mass. But what is meant
is elemental rotation, a rotation of the infinitesimal elements as they
move. We have an example of such motion in the air when a ring of smoke
escapes from the funnel of a locomotive or the lips of a tobacco-smoker,
in the motion of part of the liquid when a cup of tea is stirred by
drawing the spoon from one side to the other, or when the blade of an
oar is moving through the water. In these last two cases the depressions
seen in the surface are the ends of a vortex which extends between them
and terminates on the surface. In all these examples what have been
called vortices are formed, and hence the name vortex motion has been
given to all those cases in which the condition of irrotationality is
not satisfied.
The first great paper on vortex motion was published by von Helmholtz in
1858, and ten years later a memoir on the same subject by Thomson was
published in the _Transactions of the Royal Society of Edinburgh_. In
that memoir are given very much simpler proofs of von Helmholtz's main
theorems, and, moreover, some new theorems of wide application to the
motion of fluids. One of these is so comprehensive that it may be said
with truth to contain the whole of the dynamics of a perfect fluid. We
go on to indicate the contents of the principal papers, as far as that
can be done without the introduction of analysis of a difficult
description.
In Chapter VI reference has been made to the "Notes on Hydrodynamics"
published by Thomson in the _Cambridge and Dublin Mathematical Journal_
for 1848 and 1849. These Notes were not intended to be entirely
original, but were composed for the use of students, like Airy's Tracts
of fifteen years before.
The first Note dealt with the equation of continuity, that is to say,
the mathematical expression of the obvious fact that if any region of
space in a moving fluid be considered, the excess of rate of flow into
the space across the bounding surface, above the rate of flow out, is
equal to the rate of growth of the quantity of fluid within the space.
The proof given is that now usually repeated in text-books of
hydrodynamics.
The second Note discussed the condition fulfilled at the bounding
surface of a moving fluid. The chief mathematical result is the equation
which expresses the fact, also obvious without analysis, that there is
no flow of the fluid across the surface. In other words, the component
of the motion of a fluid particle in the immediate neighbourhood of the
surface at any instant, taken in the direction perpendicular to the
surface, must be equal to the motion of the surface in that direction at
the same instant.
The third Note, published a year later (February 1849), is of
considerable scientific importance. It is entitled, "On the Vis Viva of
a Liquid in Motion." What used to be called the "vis viva" of a body is
double what is now called the energy of motion, or kinetic energy, of
the body. The term liquid is merely a brief expression for a fluid, the
mass of which per unit volume is the same throughout, and suffers no
variation. The fluid, moreover, is supposed devoid of friction, that is,
the relative motions of its parts are unresisted by tangential force
between them. The chief theorem proved and discussed may be described as
follows.
The liquid is supposed to fill the space within a closed envelope, which
fulfils the condition of being "simply continuous." The condition will
be understood by imagining any two points A, B, within the space, to be
joined by two lines ACB, ADB both lying within the space. These two
lines will form a circuit ACBDA. If now this circuit, however it may be
drawn, can be contracted down to a point, without any part of the
circuit passing out of the space, the condition is fulfilled. Clearly
the space within the surface of an anchor-ring, or a curtain-ring, would
not fulfil this condition, for one part of the circuit might pass from A
to B round the ring one way, and the other from A to B the other way.
The circuit could not then be contracted towards a point without passing
out of the ring.
Now let the liquid given at rest in such a space be set in motion by any
arbitrarily specified variation of position of the envelope. The liquid
within will be set in motion in a manner depending entirely on the
motion of the envelope. It is possible to conceive of other motions of
the liquid than that taken, which all agree in having the specified
motion of the surface. Thomson's theorem asserts that the motion
actually taken has less kinetic energy than that of any of the other
motions which have the same motion of the bounding surface.
The motion produced has the property described by the word
"irrotational," that is, the elements of the fluid have no spinning
motion--they move without rotation. A small portion of a fluid may
describe any path--may go round in a circle, for example--and yet have
no rotation. The reader may imagine a ball carried round in a circle,
but in such a way that no line in the body ever changes its direction.
The body has translation, but no spin.
Irrotationality of a fluid is secured, as stated above, when the
velocity of each element in any direction is the rate of variation per
unit distance in that direction of a certain function of the
coordinates, the distances, taken parallel to three lines perpendicular
to one another and drawn from a point, which specify the position of the
particle. In fact, what is called a velocity-potential exists, similar
to the potential described in Chapter IV above, for an electric field.
This condition, together with the specified motion of the surface,
suffices to determine the motion of the fluid.
Two important particular consequences were pointed out by Thomson: (1)
that the motion of the fluid at any instant depends solely on the form
and motion of the bounding surface, and is therefore independent of the
previous motion; and (2) that if the bounding surface be instantaneously
brought to rest, the liquid throughout the vessel will also be instantly
brought to rest.
This theorem was afterwards generalised by Thomson (_Proc. R.S.E._,
1863), and applied to any material system of connected particles set
into motion by specified velocities simultaneously and suddenly imposed
at selected points of the system. It was already known that the kinetic
energy of a system of bodies connected in any manner, and set in motion
by impulses applied at specified points, was either a maximum or a
minimum, as compared with that for any other motion compatible with
these impulses, and with the connections of the system. This was proved
by Lagrange in the _Mécanique Analytique_ as a generalisation of a
theorem given by Euler for a rigid body set into rotation by an impulse.
Bertrand proved in 1842 that when the impulses applied are given in
amount, and are applied at specified points, the system starts off with
kinetic energy greater than that of any other motion which is consistent
with the given impulses and the connections of the system. This other
motion must be such as could be produced in the system by the given
impulses, together with any other set of impulses capable of doing no
work on the whole.
Thomson's theorem is curiously complementary to Bertrand's. Let the
system be acted on by impulses applied at certain specified points, and
by no other impulses of any kind; and let the impulses be such as to
start those selected points with any prescribed velocities. The system
will start off with kinetic energy which is less than that of any other
motion which the system could have consistently with the prescribed
velocities, and which it could be constrained to take by impulses which
do no work on the whole. In each case the difference of energies is the
energy of the motion which must be compounded with one motion to give
the other which is compared with it.
A simple example, such as might be taken of the particular case
considered by Euler, may help to make these theorems clear. Imagine a
straight uniform rod to lie on a horizontal table, between which and the
rod there is no friction. Let the rod be struck a blow at one end in a
horizontal direction at right angles to the length of the rod. If no
other impulse acts, the end of the rod will move off with a certain
definite velocity, and the other parts of the rod (which is supposed
perfectly unbending) will be started by the connections of the system.
It is obvious that any number of other motions of the rod can be
imagined, all of which give the same motion of the extremity struck. But
the actual motion taken is one of turning about that point of the rod
which is two-thirds of the length from the end struck. If the reader
will consider the kinetic energy for any other horizontal turning motion
consistent with the same motion of the end, he will find that the
kinetic energy is greater than that of the motion just specified. This
motion could be produced by applying at the point about which the rod
turns the impulse required to keep that point at rest. The impulse so
applied would do no work. The actual value is 1⧸8mv², where m denotes
the mass of the rod and v the velocity of the end. If the motion taken
were one of rotation about a point of the rod at distance x from the end
struck, the kinetic energy would be m(4l² － 6lx + 3x²)v²⧸6x², where
2l is the length of the rod, and this has its least value 1⧸8mv² for
x = 4l⧸3. For example, x = 2l gives 1⧸6mv², which is greater than the
value just found.
Bertrand's theorem applied to this case of motion is not quite so easy,
perhaps, to understand. The motion which is said to have maximum energy
is one given by a specified impulse at the end struck, and this, in the
absence of any other impulses, would be a motion of minimum energy. But
let the alternative motion, which is to be compared with that actually
taken, be one constrained by additional impulses such as can together
effect no work, and the existence of the maximum is accounted for.
The kinetic energy produced is one-half the product of the impulse
into the velocity of the point struck, that is ½Iv, and it has just
been seen that this is the product of (1⧸6)mv² by the factor
(4l² － 6lx + 3x²)⧸x². This factor is 3I⧸mv, and is a minimum when
x = 4l⧸3. Thus for a given I, v will have its maximum value when the
factor referred to is least, and ½Iv will then be a maximum.
The bar can be constrained to turn about another point by a fixed pivot
there situated. An impulse will be applied to the rod by the pivot,
simultaneously with the blow; and it is obvious that this impulse does
no work, since there is no displacement of the point to which it is
applied.
The two theorems are consequences of one principle. The constraint in
each case increases what may be called the effective inertia, which may
be taken as I⧸v. Thus when v is given, I is increased by any constraint
compelling the rod to rotate about a particular axis, and so ½Iv, or
the kinetic energy, is increased. On the other hand, when I is given the
same constraint diminishes v, and so ½Iv is diminished.
A short paper published in the B. A. Report for 1852 points out that the
lines of force near a small magnet, placed with its axis along the lines
of force in a uniform magnetic field, as it would rest under the action
of the field, are at corresponding points similar to those of the field
of an insulated spherical conductor, under the inductive influence of a
distant electric change. Further, the fact is noted that, if the magnet
be oppositely directed to the field, the lines of force are curved
outwards, just as the lines of flow of a uniform stream would be by a
spherical obstacle, at the surface of which no eddies were caused. This
is one of those instructive analogies between the theory of fluid motion
and other theories involving perfectly analogous fundamental ideas,
which Thomson was fond of pointing out, and which helped him in his
repeated attempts to imagine mechanical representations of physical
phenomena of different kinds.
With these may be placed another, which in lectures he frequently dwelt
on--a simple doublet, as it is called, consisting of a point-source of
fluid and an equal and closely adjacent point-sink. A short tube in an
infinite mass of liquid, which is continually flowing in at one end and
out at the other, may serve as a realisation of this arrangement. The
lines of flow outside the tube are exactly analogous to the lines of
force of a small magnet; and if at the same time there exist a uniform
flow of the liquid in the direction of the length of the tube, the field
of flow will be an exact picture of the field of force of the small
magnet, when it is placed with its length along the lines of a
previously existing uniform field. The flow in the doublet will be with
or against the general flow according as the magnet is directed with or
against the field.
The paper on vortex-motion has been referred to above, and an indication
given of the nature of the fluid-motion described by this title. There
are, however, two cases of fluid-motion which are referred to as
vortices, though the fundamental criterion of vortex-motion--the
non-existence of a velocity-potential--is satisfied in only one of them.
The exhibition of one of these was a favourite experiment in Thomson's
ordinary lectures, as his old students will remember. If water in a
large bowl is stirred rapidly with a teaspoon carried round and round in
a circle about the axis of the bowl, the surface will become concave,
and the form of the central part will be a paraboloid of revolution
about the vertical through the lowest point, that is to say, any section
of that part of the surface made by a vertical plane containing the axis
will be a parabola symmetrical about the axis. The motion can be better
produced by mounting the vessel on a whirling-table, and rotating it
about the vertical axis coinciding with its axis of figure; but the
phenomenon can be quite well seen without this machinery. In this case
the velocity of each particle of the water is proportional to its
distance from the axis, and the whole mass, when relative equilibrium is
set up, turns, as if it were rigid, about the axis of the vessel. Each
element of the fluid in this "forced vortex," as it is called, is in
rotation, and, like the moon, makes one turn in one revolution about the
centre of its path. This is, therefore, a true, though very simple, case
of vortex-motion.
On the other hand, what may be called a "free vortex" may exist, and is
approximated to sometimes when water in a vessel is allowed to run off
through an escape pipe at the bottom. The velocity of an element in this
"vortex" is inversely proportional to its distance from the centre, and
the form of the free surface is quite different from that in the other
case. The name "free vortex" is often given to this case of motion, but
there is no vortex-motion about it whatever.
Thomson's great paper on vortex-motion was read before the Royal Society
of Edinburgh in 1867, and was recast and augmented in the following
year. It will be possible to give here only a sketch of its scope and
main results.
The fluid is supposed contained in a closed fixed vessel which is either
simply or multiply continuous (see p. 156), and may contain immersed in
it simply or multiply continuous solids. When these solids exist their
surfaces are part of the boundary of the liquid; they are surrounded by
the liquid unless they are anywhere in contact with the containing
vessel, and their density is supposed to be the same as that of the
liquid. They may be acted on by forces from without, and they act on the
liquid with pressure-forces, and either directly or through the liquid
on one another.
The first result obtained is fairly obvious. The centre of mass of the
whole system must remain at rest whatever external forces act on the
solids, since the density is the same everywhere within the vessel, and
the vessel is fixed; that is to say, there is no momentum of the
contents of the vessel in any direction. For whatever motion of the
solids is set up by the external forces, must be accompanied by a motion
of the liquid, equal and opposite in the sense here indicated.
After a discussion of what he calls the impulse of the motion, which is
the system of impulsive forces on the movable solids which would
generate the motion from rest, Thomson proceeds to prove the important
proposition that the rotational motion of every portion of the liquid
mass, if it is zero at any one instant for every portion of the mass,
remains always zero. This is done by considering the angular momentum of
any small spherical portion of the liquid relatively to an axis through
the centre of the sphere, and proving that in order that it may vanish,
for every axis, the component velocities of the fluid at the centre
must be derivable from a velocity-potential. The angular momentum
of a particle about an axis is the product of the component of the
particle's momentum, at right angles to the plane through the particle
and the axis, by the distance of the particle from the axis. The sum of
all such products for the particles making up the body (when proper
account is taken of the signs according to the direction of turning
round the axis) is the angular momentum. The proof of this result
adopted is due to Stokes. The angular velocities of an element of
fluid at a point x, y, z, about the axes of x, y, z are shown to be
½(∂w⧸∂y － ∂v⧸∂z), etc.
The condition was therefore shown to be necessary; it remained to prove
that it was sufficient. This is obvious at once from the definition of
the velocity-potential, which must now be supposed to exist in order
that its sufficiency may be proved. If any diameter of the spherical
portion be taken as the axis, and any plane through that axis be
considered, the velocity of a particle at right angles to that plane can
be at once expressed as the rate at which the velocity-potential varies
per unit distance along the circle, symmetrical about the axis, on which
the particle lies. The integral of the velocity-potential round this
circle vanishes, and so the angular momentum for any thin uniform ring
of particles about the axis also vanishes, and as the sphere is made up
of such rings, the whole angular momentum is zero. Thus the condition is
sufficient.
Thomson then proves that if the angular momentum thus considered be
zero for every portion of the liquid at any one instant, it remains zero
at every subsequent instant; that is, no physical action whatsoever
could set up angular momentum within the fluid, which, it is to be
remembered, is supposed to be frictionless. The proof here given cannot
be sketched because it depends on the differential equation of
continuity satisfied by the velocity-potential throughout the fluid (the
same differential equation, in fact, that is satisfied by the
distribution of temperature in a uniform conducting medium in the
stationary state), and the consequent expression of this function for
any spherical space in the fluid as a series of spherical harmonic
functions. To a reader to whom the properties of these functions are
known the process can present no difficulty.
An entirely different proof of this proposition is given subsequently in
the paper, and depends on a new and very general theorem, which has been
described as containing almost the whole theory of the motion of a
fluid. This depends on what Thomson called the flow along any path
joining any two points P, Q in the fluid. Let q be the velocity of the
fluid at any element of length ds of such a path, and θ be the
angle between the direction of ds (taken positive in the sense from P
to Q) and the direction of q: q cos θ.ds is the flow along ds. If u,
v, w be the components of q at ds, parallel to the axes, and dx, dy, dz
be the projections of ds on the axes, udx + vdy + wdz is the same thing
as q cos θ.ds. The sum of the values of either of these expressions
for all the elements of the path between P and Q is the flow along the
path. The statement that u, v, w are the space-rates of variation of a
function φ (of x, y, z) parallel to the axes, or that q cos θ is
the space-rate of variation of φ along ds, merely means that this
sum is the same for whatever path may be drawn from P to Q. This,
however, is only the case when the paths are so taken that in each case
the value of φ returns after variation along a closed path to the
value which it had at the starting point, that is, the closed path must
be capable of being contracted to a point without passing out of space
occupied by irrotationally moving fluid.
Since the flow from P to Q is the same for any two paths which fulfil
this condition, the flow from P to Q by any one path and from Q to P by
any other must be zero. The flow round such a closed path is not zero if
the condition is not fulfilled, and its value was called by Thomson the
circulation round the path.
The general theorem which he established may now be stated. Consider
any path joining PQ, and moving with the fluid, so that the line
contains always the same fluid particles. Let u̇, v̇, ẇ be the
time-rates of change of u, v, w at an element ds of the path, at
any instant, and du, dv, dw the excesses of the values of u, v, w
at the terminal extremity of ds above the values at the other
extremity; then the time-rate of variation of udx + vdy + wdz
is u̇dx + v̇dy + ẇdz + udu + vdv + wdw or u̇dx + v̇dy + ẇdz + qdq,
where q has the meaning specified above. Thus if S be the flow for
the whole path PQ, and Ṡ its time-rate of variation, S' denote the
sum of u̇dx + v̇dy + ẇdz along the path from P to Q, and q₁, q₀ the
resultant fluid velocities at Q and P, we get Ṡ = S' + ½(q₁² － q₀²).
This is Thomson's theorem. If the curve be closed, that is, if P and Q
be coincident, q₁ = q₀ and Ṡ = S'. But in certain circumstances S' is
zero, and so therefore is also Ṡ. Thus in the circumstances referred to,
as the closed path moves with the fluid Ṡ is continually zero, and it
follows that if Ṡ is zero at any instant it remains zero ever after. But
Ṡ is only zero if u, v, w are derivable from a potential, single valued
in the space in which the closed path is drawn, so that the path could
be shrunk down to a point without ever passing out of such space. In a
perfect fluid if this condition is once fulfilled for a closed curve
moving with the fluid, it is fulfilled for this curve ever after.
The circumstances in which S' is zero are these:--the external force,
per unit mass, acting on the fluid at any point is to be derivable from
a potential-function, and the density of the fluid is to be a function
of the pressure (also a function of the coordinates); and these
functions must be such as to render S' always zero for the closed path.
This condition is manifestly fulfilled in many important cases; for
example, the forces are derivable from a potential due to actions, such
as gravity, the origin of which is external to the fluid; and the
density is a function of the pressure (in the present case it is a
constant), such that the part of S' which depends on pressure and
density vanishes for the circuit.
It is to be clearly understood that the motion of a fluid may be
irrotational although the value of S does not vanish for every closed
path that can be drawn in it. The fluid may occupy multiply continuous
space, and the path may or may not be drawn so that S shall be zero; but
what is necessary for irrotational motion within any space is that S
should vanish for all paths which are capable of being shrunk down to
zero without passing out of that space. S need not vanish for a path
which cannot be so shrunk down, but it must, if the condition just
stated is fulfilled, have the same value for any two paths, one of which
can be made to pass into the other by change of position without ever
passing in whole or in part out of the space. The potential is always
single valued in fluid filling a singly continuous space such as that
within a spherical shell, or between two concentric shells; within a
hollow anchor-ring the potential, though it exist, and the motion be
irrotational, is not single valued. In the latter case the motion is
said to be cyclic, in the former acyclic.
A number of consequences are deduced from this theorem; and from these
the properties of vortices, which had previously been discovered by von
Helmholtz, immediately follow. First take any surface whatever which has
for bounding edge a closed curve drawn in the fluid, and draw from any
element of this surface, of area dS, a line perpendicular to the surface
towards the side chosen as the positive side, and calculate the angular
velocity ω, say, of the fluid about that normal from the components of
angular velocity determined in the manner explained at p. 164. This
Thomson called the rotation of the element. Now take the product ωdS for
the surface element. It is easy to see that this is equal to half the
circulation round the bounding edge of the element. As the fluid
composing the element moves the area dS may change, but the circulation
round its edge by Thomson's theorem remains unaltered. Thus ω alters in
the inverse ratio of dS, and the line drawn at right angles to the
surface at dS, if kept of length proportional to ω, will lengthen or
shorten as dS contracts or expands.
Now sum the values of ωdS for the finite surface enclosed by the
bounding curve. It follows from the fact that ωdS is equal to half the
circulation round the edge of dS, that this sum, which is usually
denoted by ΣωdS, is equal to half the circulation round the closed
curve which forms the edge of the surface. Also as the fluid moves the
circulation round the edge remains unaltered, and therefore so does also
ΣωdS for the elements enclosed by it. It is important to notice
that this sum being determined by the circulation in the bounding curve
is the same for all surfaces which have the same boundary.
The equality of 2ΣωdS for the surface to the circulation round its
edge was expressed by Thomson as an analytical theorem of integration,
which was first given by Stokes in a Smith's Prize paper set in 1854. It
is here stated, apparently by an oversight, that it was first given in
Thomson and Tait's _Natural Philosophy_, § 190. In the second edition of
the _Natural Philosophy_ the theorem is attributed to Stokes. It is now
well known as Stokes's theorem connecting a certain surface integral
with a line integral, and has many applications both in physics and in
geometry.
Now consider the resultant angular velocity at any point of the fluid,
and draw a short line through that point in the direction of the axis of
rotation. That line may be continued from point to point, and will
coincide at every one of its points with the direction of the axis of
rotation there. Such an axial curve, as it may be called, it is clear
moves with the fluid. For take any infinitesimal area containing an
element of the line; the circulation round the edge of this area is
zero, since there is no rotation about a line perpendicular to the area.
Hence the circulation along the axial curve is zero, and the axial
curves move with the fluid.
Take now any small plane area dS moving with the fluid, and draw axial
lines through every point of its boundary. These will form an axial tube
enclosing dS. If θ be the angle between the direction of resultant
rotation and a perpendicular to dS, the cross-section of the tube at
right angles to the normal, and to the axial lines which bound it, is
dS.cosθ. Let these axial lines be continued in both directions from the
element dS. They will enclose a tube of varying normal cross-section;
but the product of rotation and area of normal cross-section has
everywhere the same value. A vortex-tube with the fluid within it is
called a vortex-filament.
It will be seen that this vortex-tube must be endless, that is, it must
either return into itself, or be infinitely long in one or both
directions. For if it were terminated anywhere within the fluid, it
would be possible to form a surface, starting from a closed circuit
round the tube, continued along the surface of the tube to the
termination, and then closed by a cap situated beyond the termination.
At no part of this surface would there be any rotation, and ΣωdS,
which is equal to the circulation, would be zero for it; and of course
this cannot be the case. Thus the tube cannot terminate within the
fluid. It can, however, have both of its ends on the surface, or one on
the bounding surface and the other at infinity, if the fluid is
infinitely extended in one direction, but in that case the termination
is only apparent. The section is widened out at the surface; some of the
bounding lines pass across to the other apparent termination, when it
also lies on the surface, while the other lines pass off to infinity
along the surface, and correspond to other lines coming in from
infinity to the other termination. Whether the surface is infinite or
not, the vortex is spread out into what is called a vortex-sheet, that
is, in a surface on the two sides of which the fluid moves with
different tangential velocities.
Through a vortex-ring or tube, the fluid circulates in closed lines of
flow, each one of which is laced through the tube. The circulation along
every line of flow which encloses the same system of vortex-tubes has
the same value.
If any surface be drawn cutting a vortex-tube, it is clear from the
definition of the tube that the value of ΣωdS for every such
surface must be the same. This Thomson calls the "rotation of the tube."
As was pointed out first by von Helmholtz, vortex-filaments correspond
to circuits carrying currents and the velocity in the surrounding fluid
to magnetic field-intensity. The "rotation of the tube" corresponds to
the strength of the current, and sources and sinks to positive and
negative magnetic poles. Thomson made great use of this analogy in his
papers on electromagnetism.
Examples of vortex-tubes are indicated on p. 154; and the reader may
experiment with vortices in liquids with water in a tea-cup, or in a
river or pond, at pleasure. Air vortices may be experimentally studied
by means of a simple apparatus devised by Professor Tait, which may be
constructed by anyone.
In one end of a packing-box, about 2ft. long by 18in. wide and 18in.
deep, a circular hole is cut, and the edges of the hole are thinned down
to a blunt edge. This can be closed at pleasure by a piece of board. The
opposite end is removed, and a sheet of canvas stretched tightly in its
place, and tacked to the ends of the sides. Through two holes bored in
one of the sides the mouths of two flasks with bent necks protrude into
the box. One of these flasks contains ammonia, the other hydrochloric
acid. When the hole at one end is closed up by a slip of tinplate, and
the liquids are heated with a spirit-lamp, the vapours form a cloud of
sal-ammoniac within the box, which is retained during its formation. The
hole is then opened, and the canvas struck smartly with the palm of the
open hand. Immediately a beautiful ring of smoke emerges, clear-cut and
definite as a solid, and moves across the room. (See Fig. 13.) Of
course, it is a ring of air, made visible by the smoke carried with it.
By varying the shape of the aperture--for example, by using instead of
the hole cut in the wood, a slide of tinplate with an elliptic hole cut
in it--the vortex-rings can be set in vibration as they are created, and
the vibrations studied as the vortex moves.
[Illustration: FIG. 13.]
Still more beautiful vortices can be formed in water by using a long
tank of clear water to replace the air in which the vortex moves, and a
compartment at one end filled with water coloured with aniline, instead
of the smoke-box. A hole in the dividing partition enables the vortex to
be formed, and a piston arrangement fitted to the opposite side enables
the impulse to the water to be given from without.
From the account of the nature of vortex-motion given above, it will be
clear that vortices in a perfect fluid once existent must be ever
existent. To create a vortex within a mass of irrotationally moving
perfect fluid is physically impossible. It occurred to Thomson,
therefore, that ordinary matter might be portions of a perfect fluid,
filling all space, differentiated from the surrounding fluid by the
rotation which they possess. Such matter would fulfil the law of
conservation, as it could neither be created nor destroyed by any
physical act.
The results of such experiments led Thomson to frame his famous
vortex-atom theory of matter, a theory, however, which he felt
ultimately was beset with so many difficulties as to be unworkable.
The paper on vortex-motion also deals with the modification of Green's
celebrated theorem of analysis, which, it was pointed out by Helmholtz,
was necessary to adapt it to a space which is multiply continuous. The
theorem connects a certain volume-integral taken throughout a closed
space with an integral taken over the bounding surface of the space.
This arises from the fact noticed above that in multiply continuous
space (for example, the space within an endless tube) the functions
which are the subject of integration may not be single valued. Such a
function would be the velocity-potential for fluid circulating round the
tube--cyclic motion, as it was called by Thomson. If a closed path of
any form be drawn in such a tube, starting from a point P, and doubling
back so as to return to P without making the circuit of the tube, the
velocity-potential will vary along the tube, but will finally return to
its original value when the starting point is reached. And the
circulation round this circuit will be zero. But if the closed path make
the circuit of the tube, the velocity-potential will continuously vary
along the path, until finally, when P is reached again, the value of
the function is greater (or less) than the value assumed for the
starting point, by a certain definite amount which is the same for every
circuit of the space. If the path be carried twice round in the same
direction, the change of the function will be twice this amount, and so
on. The space within a single endless tube such as an anchor-ring is
doubly continuous; but much more complicated cases can be imagined. For
example, an anchor-ring with a cross-connecting tube from one side to
the other would be triply continuous.
Thomson showed that the proper modification of the theorem is obtained
by imagining diaphragms placed across the space, which are not to be
crossed by any closed path drawn within the space, and the two surfaces
of each of which are to be reckoned as part of the bounding surface of
the space. One such diaphragm is sufficient to convert a hollow
anchor-ring into a singly continuous space, two would be required for
the hollow anchor-ring with cross-connection, and so on. The number of
diaphragms required is always one less than the degree of multiplicity
of the continuity.
The paper also deals with the motion of solids in the fluid and the
analogous motions of vortex-rings and their attraction by ordinary
matter. These can be studied with vortex-rings in air produced by the
apparatus described above. Such a ring made to pass the re-entrant
corner of a wall--the edge of a window recess, for example--will appear
to be attracted. A large sphere such as a large terrestrial globe serves
also very well as an attracting body.
Two vortex-rings projected one after the other also act on one another
in a very curious manner. Their planes are perpendicular to the
direction of motion, and the fluid is moving round the circular core of
the ring. There is irrotational cyclic motion of the fluid through the
ring in one direction and back outside, as shown in Fig. 13, which can
be detected by placing a candle flame in the path of the centre. The
first ring, in consequence of the existence of that which follows it,
moves more slowly, and opens out more widely, the following ring hastens
its motion and diminishes in diameter, until finally it overtakes the
former and penetrates it. As soon as it has passed through it moves
ahead more and more slowly, until the one which has been left behind
begins to catch it up, and the changes which took place before are
repeated. The one penetrating becomes in its turn the penetrated, and so
on in alternation. Great care and skill are, however, necessary to make
this interesting experiment succeed.
We have not space to deal here with other hydrodynamical investigations,
such as the contributions which Thomson made to the discussion of the
many difficult problems of the motion of solids through a liquid, or to
his very numerous and important contributions to the theory of waves.
The number and importance of his hydrodynamical papers may be judged
from the fact that there are no less than fifty-two references to his
papers, and thirty-five to Thomson and Tait's _Natural Philosophy_ in
the latest edition of Lamb's Hydrodynamics, and that many of these are
concerned with general theorems and results of great value.
CHAPTER X
THE ENERGY THEORY OF ELECTROLYSIS--ELECTRICAL UNITS--ELECTRICAL
OSCILLATIONS
ELECTROLYSIS AND ELECTRICAL UNITS
In December 1851 Thomson communicated an important paper to the
_Philosophical Magazine_ on "The Mechanical Theory of Electrolysis," and
"Applications of Mechanical Effect to the Measurement of Electromotive
Forces, and of Galvanic Resistances, in Absolute Units."
In the first of these he supposed a machine of the kind imagined by
Faraday, consisting of a metal disk, rotating uniformly with its plane
at right angles to the lines of force of a uniform magnetic field, and
touched at its centre and its circumference by fixed wires, to send a
current through an electrochemical apparatus, to which the wires are
connected. A certain amount of work W was supposed to be spent in a
given time, during which a quantity of heat H was evolved in the
circuit, and a certain amount of work M spent in the chemical apparatus
in effecting chemical change. If H be taken in dynamical units, W = H +
M.
The work done in driving the disk, if the intensity of the field is I,
the current produced c, the radius of the disc r, and the angular
velocity of turning w, is ½Ir²cw.
Thomson assumed that the work done in the electrochemical apparatus was
equal to the heat of chemical combination of the substance or
substances which underwent the chemical action, taken with the proper
sign according to the change, if more compound substances than one were
acted on. Hence M represented this resultant heat of combination.
The electrochemical apparatus was a voltameter containing a definite
compound to be electrolysed, or a voltaic cell or battery. And by
Faraday's experiments on electrolysis it was known that the amount of
chemical action was proportional to the whole quantity of electricity
passed through the cell in a given time, so that the rate at which
energy was being spent in the cell was at any instant proportional to
the current at that instant.
The chemical change could be measured by considering only one of the
elements set free, or made to combine, by the passage of the current,
and considering the quantity of heat θ, say, for the whole chemical
change in the cell corresponding to the action on unit mass of that
element. Thus if E denote the whole quantity of that element operated on
the heat of combination in the vessel was θE. If E be taken for unit of
time, and ε denote the quantity set free by the passage of unit quantity
of electricity, then E = εc, since a current conveys c units of
electricity in one second. The number ε is a definite quantity of the
element, and is called its electrochemical equivalent. Again, from
Joule's experiments, H = Rc², if R denote the resistance of the current,
and so
½Ir²cw = Rc² + θεc
and
c = (½Ir²w － θε)⧸R
The quantity ½Ir²w is the electromotive force due to the disk.
Thus c was positive or negative according as ½Ir²w was greater or less
than θε, and was zero when ½Ir²w = θε. Thus the electromotive force
of the disk was opposed by a back electromotive force θε due to the
chemical action in the voltameter or battery, to which the wires from
the disk were connected.
The conclusion arrived at therefore was that the electromotive force
(or, as it was then termed, the intensity) of the electrochemical action
was equal to the dynamical value of the whole chemical change effected
by a current of unit strength in unit of time.
From this result Thomson proceeded to calculate the electromotive forces
required to effect chemical changes of different kinds, and those of
various types of voltaic cell. Supposing a unit of electricity to be
carried by the current through the cell, he considered the chemical
changes which accompanied its passage, and from the known values of
heats of combination calculated their energy values. In some parts the
change was one of chemical combination, in others one of decomposition
of the materials, and regard had to be paid to the sign of the
heat-equivalent. By properly summing up the whole heat-equivalents a net
total was obtained which, according to Thomson, was the energy consumed
in the passage of unit current, and was therefore the electromotive
force. The theory was incomplete, and required to be supplemented by
thermodynamic theory, which shows that besides the electromotive force
there must be included in the quantity set against the sum of heats a
term represented by the product of the absolute temperature multiplied
by the rate of variation of electromotive force with alteration of
temperature. Thus the theory is only applicable when the electromotive
force is not affected by variation of temperature. The necessary
addition here indicated was made by Helmholtz.
In the next paper, which appeared in the same number (December 1851) of
the _Philosophical Magazine_, the principle of work is applied to the
measurement of electromotive forces and resistances in absolute units.
The advantages of such units are obvious. Nearly the whole of the
quantitative work of the older experimenters was useless except for
those who had actually made the observations: it was hardly possible for
one man to advance his researches by employing data obtained by others.
For the results were expressed by reference to apparatus and materials
in the possession of the observers, and to these others could obtain
access only with great difficulty and at great expense--to say nothing
of the uncertainty of comparisons made to enable the results of one man
to be linked on to those made elsewhere, and with other apparatus, by
another. It was imperative, therefore, to obtain absolute units--units
independent of accidents of place and apparatus--for the expression of
currents, electromotive forces, and resistances, so as to enable the
results of the work of experiments all over the world to be made
available to every one who read the published record. (See Chap. XIII.)
The magneto-electric machine imagined in the former paper gave a means
of estimating the electromotive force of a cell or battery in absolute
units. The same kind of machine is used here, in the simpler form of a
sliding conductor connecting a pair of insulated rails laid with their
plane perpendicular to the lines of force of a uniform magnetic field.
If the rails be connected by a wire, and the slider be moved so as to
cut across the lines of force, a current will be produced in the
circuit. The current can be measured in terms of the already known unit
of current, that current which flowing in a circle of radius unity
produces a magnetic field at the centre of 2π units. This current, c,
say, in strength, flowing in the circuit, renders a dynamical force cIl
necessary to move the slider of length l across the lines of force of
the field of intensity I, and if the speed of the slider required for
the current c be v, the rate at which work is done in moving the slider
is cIlv. This must be the rate at which work is done in the circuit by
the current, and if the only work done be in the heating of the
conductor, we have cIlv = Rc², or Ilv = Rc, so that Ilv is the
electromotive force. Any electromotive force otherwise produced, which
gave rise to the same current, must obviously be equal to Ilv, so that
the unit of electromotive force can thus be properly defined.
Thomson used a foot-grain-second system of units; but from this
arrangement are now obtained the C.G.S. units of electromotive force and
resistance. If I is one C.G.S. unit, l one centimetre, and v one
centimetre per second, we have unit electromotive force in the C.G.S.
system. Also in one C.G.S. unit of resistance if c be unity as well as
Ilv.
The idea of the determination of a resistance in absolute units on
correct principles was due to W. Weber, who also gave methods of
carrying out the measurement; and the first determination was made by
Kirchhoff in 1849. Thomson appears, however, to have been the first to
discuss the subject of units from the point of view of energy. This mode
of regarding the matter is important, as the absolute units are so
chosen as to enable work done by electric and magnetic forces to be
reckoned in the ordinary dynamical units. A vast amount of experimental
resource and skill has been spent since that time on the determination
of resistance, though not more than the importance of the subject
warranted. We shall have to return to the subject in dealing with the
work of the British Association on Electrical Standards, of which
Thomson was for long an active member.
ELECTRICAL OSCILLATIONS
In his famous tract on the conservation of energy, published in 1847,
von Helmholtz discussed some puzzling results obtained by Riess in the
magnetisation of iron wires by the current of a Leyden jar discharge
flowing in a coil surrounding them, and by the fact, observed by
Wollaston, that when water was decomposed by Leyden jar discharges a
mixture of oxygen and hydrogen appeared at each electrode, and suggested
that possibly the discharge was oscillatory in character.
In 1853 the subject was discussed mathematically by Thomson, in a paper
which was to prove fruitful in our own time in a manner then little
anticipated. The jar is given, let us say, with the interior coating
charged positively, and the exterior coating charged negatively. A coil
or helix of wire has its ends connected to the two coatings, and a
current immediately begins in the wire, and gradually (not slowly)
increases in strength. Accompanying the creation of the current is the
production of a magnetic field, that is, the surrounding space is made
the seat of magnetic action. The magnetic field, as we shall see from
another investigation of Thomson's, almost certainly involves motion in
or of a medium--the ether--filling the space where the magnetic action
is found to exist. The charge of the jar consists of a state of intense
and peculiar strain in the glass plate between the coatings. When the
plates are connected by the coil, this state of strain breaks down and
motion in the medium ensues, not merely between the plates, but also in
the surrounding space--in fact, in the whole field. This motion--which
is not to be confused with bodily displacement of finite parts of the
medium--is opposed by something akin to inertia of the medium (the
property that confers energy on matter when in motion), so that when the
motion is started it persists, until it is finally wiped out by
resistance of the nature of friction. The inertia here referred to
depends on the mode in which the coil is wound, or whether it contains
or not an iron core.
If the work done in charging a Leyden jar or electric condenser, by
bringing the charge to the condenser in successive small portions, is
considered, it is at once clear that it must be proportional to the
square of the whole quantity of electricity brought up. For whatever the
charge may be, let it be brought up from a great distance in a large
number N of equal instalments. The larger the whole amount the larger
must each instalment be, and therefore the greater the amount
accumulated on the condenser when any given number of instalments have
been deposited. But the greater any charge that is being brought up, and
also the greater the charge that has already arrived, the greater is
the repulsion that must be overcome in bringing up that instalment, in
simple proportion in each case, and therefore the greater the work done.
Thus the whole work done in bringing up the charge must be proportional
to Q². We suppose it to be ½Q²⧸C, where C is a constant depending on
the condenser and called its capacity.
The idea of the charge as a quantity of some kind of matter, brought up
and placed on the insulated plate of the condenser, has only a
correspondence to the fact, which is that the medium between the plates
is the seat, when the condenser is charged, of a store of energy, which
can only be made available by connecting the plates of the condenser by
a wire or other conductor. The charge is only a surface aspect of the
state of the medium, apparently a state of strain, to which the energy
belongs.
When a wire is used to connect the plates the state of strain
disappears; the energy comes out from the medium between the plates by
motion sideways of the tubes of strain (so that the insulating medium is
under longitudinal tension and lateral pressure) which, according to
Faraday's conception of lines of electric force connecting the charge on
a body with the opposite charges on other bodies, run from plate to
plate, when the condenser is in equilibrium in the changed state. These
tubes move out with their ends on the wire, carrying the energy with
them, and the ends run towards one another along the wire; the tube
shortens in the process, and energy is lost in the wire. The ends of a
tube thus moving represent portions of the charges which were on the
plates, and the oppositely-directed motions of the opposite charges
represent a current along the wire from one conductor to the other. The
motion of the tubes is accompanied by the development of a magnetic
field, the lines of force of which are endless, and the direction of
which at every point is perpendicular at once to the length of the tube
and to the direction in which it is there moving. In certain
circumstances the tube, by the time its ends have met, will have wholly
disappeared in the wire, and the whole energy will have gone to heat the
wire: in other circumstances the ends will meet before the tube has
disappeared, the ends will cross, and the tube will be carried back to
the condenser and reinserted in the opposite direction. At a certain
time this will have happened to all the tubes, though they will have
lost some of their energy in the process; and the condenser will again
be charged, though in the opposite way to that in which it was at first.
Then the tubes will move out again, and the same process will be
repeated: once more the condenser will be charged, but in the same
direction as at first, and once more with a certain loss of energy.
Again the process of discharge and charge will take place, and so on,
again and again, until the whole energy has disappeared. This process
represents, according to the modern theory of the flow of energy in the
electromagnetic field, with more or less accuracy, what takes place in
the oscillatory discharge of a condenser.
The motion of the tubes with their ends on the wire represents a certain
amount of energy, commonly regarded as kinetic, and styled
electrokinetic energy. If c denote the current, that is, the rate,
-dQ⧸dt, at which the charge of the condenser is being changed, and L a
quantity called self-inductance, depending mainly on the arrangement of
the connecting wire--whether it is wound in a coil or helix, with or
without an iron core, or not--the electrokinetic energy will be ½Lc².
This is analogous to the kinetic energy ½mv² of a body (say a pendulum
bob) of mass m and velocity v, so that L represents a quantity for the
conducting arrangement analogous to inertia, and c is the analogue of
the velocity of the body. The whole energy at any instant is thus
½Q²⧸C + ½Lc², or ½Q²⧸C + ½L(dQ⧸dt)².