THEORY OF SUCCESSIVE CHANGES.
=193. Introduction.= We have seen in previous chapters that the
radio-activity of the radio-elements is always accompanied by the
production of a series of new substances with some distinctive physical
and chemical properties. For example, thorium produces from itself an
intensely radio-active substance, Th X, which can be separated from the
thorium in consequence of its solubility in ammonia. In addition,
thorium gives rise to a gaseous product, the thorium emanation, and also
to another substance which is deposited on the surface of bodies in the
neighbourhood of the thorium, where its presence is indicated by the
phenomenon known as “excited activity.”
A close examination of the origin of these products shows that they are
not produced simultaneously, but arise in consequence of a succession of
changes originating in the radio-element. Thorium first of all gives
rise to the product Th X. The Th X produces from itself the thorium
emanation, and this in turn is transformed into a non-volatile
substance. A similar series of changes is observed in radium, with the
exception that there is no product in radium corresponding to the Th X
in the case of thorium. Radium first of all produces an emanation,
which, like thorium, is transformed into a non-volatile substance. In
uranium only one product, Ur X, has been observed, for uranium does not
give off an emanation and in consequence does not produce excited
activity on bodies.
As a typical example of the evidence, from which it is deduced that one
substance is the parent of another, we will consider the connection of
the two products Th X and the thorium emanation. It has been shown
(section 154) that after the separation of Th X from a thorium solution,
by precipitation with ammonia, the precipitated thorium hydroxide has
lost to a large extent its power of emanating. This cannot be ascribed
to a prevention of escape of the emanation produced in it, for very
little emanation is observed when a current of air is drawn through the
hydroxide in a state of solution, when most of the emanation present
would be carried off. On the other hand, the solution containing the Th
X gives off a large quantity of emanation, showing that the power of
giving off an emanation belongs to the product Th X. Now it is found
that the quantity of emanation given off by the separated Th X decreases
according to an exponential law with the time, falling to half value in
four days. The rate of production of emanation thus falls off according
to the same law and at the same rate as the activity of the Th X
measured in the ordinary manner by the α rays. Now this is exactly the
result to be expected if the Th X is the parent of the emanation, for
the activity of Th X at any time is proportional to its rate of change,
_i.e._, to the rate of production of the secondary type of matter by the
emanation in consequence of a change in it. Since the rate of change of
the emanation (half transformed in 1 minute) is very rapid compared with
the rate of change of Th X, the amount of emanation present will be
practically proportional to the activity of the Th X at any instant,
_i.e._, to the amount of unchanged Th X present. The observed fact that
the hydroxide regains its power of emanating in the course of time is
due to the production of fresh Th X by the thorium, which in turn
produces the emanation.
In a similar way, excited activity is produced on bodies over which the
emanation is passed, and in amount proportional to the activity of the
emanation, _i.e._, to the amount of the emanation present. This shows
that the active deposit, which gives rise to the phenomenon of excited
activity, is itself a product of the emanation. The evidence thus seems
to be conclusive that Th X is the parent of the emanation and that the
emanation is the parent of the deposited matter.
=194. Chemical and Physical properties of the active products.= Each of
these radio-active products is marked by some distinctive chemical and
physical properties which differentiate it from the preceding and
succeeding products. For example, Th X behaves as a solid. It is soluble
in ammonia, while thorium is not. The thorium emanation behaves as a
chemically inert gas and condenses at a temperature of −120° C. The
active deposit from the emanation behaves as a solid and is readily
soluble in sulphuric and hydrochloric acids and is only slightly soluble
in ammonia.
The striking dissimilarity which exists in many cases between the
chemical and the physical properties of the parent matter and the
product to which it gives rise is very well illustrated by the case of
radium and the radium emanation. Radium is an element so closely allied
in chemical properties to barium that, apart from a slight difference in
the solubility of the chlorides and bromides, it is difficult to
distinguish chemically between them. It has a definite spectrum of
bright lines similar in many respects to the spectra of the alkaline
earths. Like barium, it is non-volatile at ordinary temperature. On the
other hand, the emanation which is continually produced from radium is a
radio-active and chemically inert gas, which is condensed at a
temperature of −150° C. Both in its spectrum and in the absence of
definite chemical properties, it resembles the argon-helium group of
inert gases, but differs from these gases in certain marked features.
The emanation must be considered to be an unstable gas which breaks down
into a non-volatile type of matter, the disintegration being accompanied
by the expulsion of heavy atoms of matter (α particles) projected with
great velocity. This rate of breaking up is not affected by temperature
over the considerable range which has been examined. After a month’s
interval, the volume of the emanation has shrunk to a small portion of
its initial value. But the most striking property of the emanation,
which, as we shall see later (chapter XII), is a direct consequence of
its radio-activity, is the enormous amount of energy emitted from it.
The emanation in breaking up through its successive stages emits about 3
million times as much energy as is given out by the explosion of an
equal volume of hydrogen and oxygen, mixed in the proper proportions to
form water; and yet, in this latter chemical reaction more heat is
emitted than in any other known chemical change.
We have seen that the two emanations and the products Ur X, Th X lose
their activity with the time according to a simple exponential law, and
at a rate that is independent—as far as observation has gone—of the
chemical and physical agents at our disposal. The time taken for each of
these products to fall to half its value is thus a definite physical
constant which serves to distinguish it from all other products.
On the other hand, the variation of the excited activity produced by
these emanations does not even approximately obey such a law. The rate
of decay depends not only on the time of exposure to the respective
emanations, but also, in the case of radium, on the type of radiation
which is used as a means of comparative measurement. It will be shown,
in succeeding chapters, that the complexity of the decay is due to the
fact that the matter in the active deposits undergoes several successive
transformations, and that the peculiarities of the curves of decay,
obtained under different conditions, can be explained completely on the
assumption that two changes occur in the active deposit from both
thorium and actinium and six in the active deposit from radium.
=195. Nomenclature.= The nomenclature to be applied to the numerous
radio-active products is a question of great importance and also one of
considerable difficulty. Since there are at least seven distinct
substances produced from radium, and probably five from thorium and
actinium, it is neither advisable nor convenient to give each a special
name such as is applied to the parent elements. At the same time, it is
becoming more and more necessary that each product should be labelled in
such a way as to indicate its place in the succession of changes. This
difficulty is especially felt in discussing the numerous changes in the
active deposits from the different emanations. Many of the names
attached to the products were given at the time of their discovery,
before their position in the scheme of changes was understood. In this
way the names Ur X, Th X were applied to the active residues obtained by
chemical treatment of uranium and thorium. Since, in all probability,
these substances are the first products of the two elements, it may be
advisable to retain these names, which certainly have the advantage of
brevity. The name “emanation” was originally given to the radio-active
gas from thorium, and has since been applied to the similar gaseous
products of radium and actinium.
Finding the name “radium emanation” somewhat long and clumsy, Sir
William Ramsay[294] has recently suggested “ex-radio” as an equivalent.
This name is certainly brief and is also suggestive of its origin; but
at least six other ex-radios, whose parentage is as certain as that of
the emanation, remain unnamed. A difficulty arises in applying the
corresponding names ex-thorio, ex-actinio to the other gaseous products,
for, unlike radium, the emanations of thorium and actinium are probably
the second, not the first, disintegration product of the radio-elements
in question. Another name thus has to be applied to the first product in
these cases. It may be advisable to give a special name to the
emanation, since it has been the product most investigated and was the
first to be isolated chemically; but, on the other hand, the name
“radium emanation” is historically interesting, and suggests a type of
volatile or gaseous matter. Since the term “excited” or “induced”
activity refers only to the radiations from the active body, a name is
required for the radiating matter itself. The writer in the first
edition of this book suggested the name “emanation X.”[295] This title
was given from analogy to the names Ur X and Th X, to indicate that the
active matter was product of the emanation. The name, however, is not
very suitable, and, in addition, can only be applied to the initial
product deposited, and not to the further products of its decomposition.
It is very convenient in discussing mathematically the theory of
successive changes to suppose that the deposited matter called _A_ is
changed into _B_, _B_ into _C_, _C_ into _D_, and so on. I have
therefore discarded the name emanation _X_, and have used the terms
radium _A_, radium _B_, and so on, to signify the successive products of
the decomposition of the emanation of radium. A similar nomenclature is
applied to thorium and actinium. This system of notation is elastic and
simple, and I have found it of great convenience in the discussion of
successive products. In speaking generally of the active matter, which
causes excited activity, without regard to its constituents, I have used
the term “active deposit.” The scheme of nomenclature employed in this
book is clearly shown below:—
Radium Thorium Uranium Actinium
Radium Th X Ur X Actinium X
emanation
Radium _A_ Thorium Final Actinium
(Active) emanation product emanation
Radium _B_ Thorium Actinium
(Active) _A_ _A_
(Active) (Active)
Radium _C_ Thorium Actinium
(Active) _B_ _B_
(Active) (Active)
Radium _D_ Thorium Actinium
(Active) _C_ _C_
(final) (final)
&c.
Each product on this scheme is the parent of the product below it. Since
only two products have been observed in the active deposit of thorium
and actinium, thorium _C_ and actinium _C_ respectively refer to their
final inactive products. It will be shown in the next chapter that, as
in the case of thorium, an intermediate product exists between actinium
and its emanation. From analogy to the products Th X and Ur X, this
substance is termed “actinium X.”
=196. Theory of Successive Changes.= Before considering the evidence
from which these changes are deduced, the general theory of successive
changes of radio-active matter will be considered. It is supposed that
the matter _A_ changes into _B_, _B_ into _C_, _C_ into _D_, and so on.
Each of these changes is supposed to take place according to the same
law as a monomolecular change in chemistry, _i.e._, the number _N_ of
particles unchanged after a time _t_ is given by
$$ M = N₀ e^{–λt} $$,
where _N₀_ is the initial number and λ the constant of the change.
Since _dN_/_dt_ = -λ_N_, the rate of change at any time is always
proportional to the amount of matter unchanged. It has previously been
pointed out that this law of decay of the activity of the radio-active
products is an expression of the fact that the change is of the same
type as a monomolecular chemical change.
Suppose that _P_, _Q_, _R_ represent the number of particles of the
matter _A_, _B_, and _C_ respectively at any time _t_. Let λ₁, λ₂, λ₃ be
the constants of change of the matter _A_, _B_, and _C_ respectively.
Each atom of the matter _A_ is supposed to give rise to one atom of the
matter _B_, one atom of _B_ to one of _C_, and so on.
The expelled “rays” or particles are non-radio-active, and so do not
enter into the theory.
It is not difficult to deduce mathematically the number of atoms of _P_,
_Q_, _R_, ... of the matter _A_, _B_, _C_, ... existing at any time _t_
after this matter is set aside, if the initial values of _P_, _Q_, _R_, ...are given. In practice, however, it is generally only necessary to
employ three special cases of the theory which correspond, for example,
to the changes in the active deposit, produced on a wire exposed to a
constant amount of radium emanation and then removed, (1) when the time
of exposure is extremely short compared with the period of the changes,
(2) when the time of exposure is so long that the amount of each of the
products has reached a steady limiting value, and (3) for any time of
exposure.
There is also another case of importance which is practically a converse
of Case 3, viz. when the matter _A_ is supplied at a constant rate from
a primary source and the amounts of _A_, _B_, _C_ are required at any
subsequent time. The solution of this can, however, be deduced
immediately from Case 3 without analysis.
=197.= CASE 1. _Suppose that the matter initially considered is all of
one kind A. It is required to find the number of particles P, Q, R of
the matter A, B, C respectively present after any time t._
Then
$$ P = ne^{–λ_1 t} $$,
if _n_ is the number of particles of _A_ initially present. Now _dQ_,
the increase of the number of particles of the matter _B_ per unit time,
is the number supplied by the change in the matter _A_, less the number
due to the change of _B_ into _C_, thus
_dQ_/_dt_ = λ₁_P_ − λ₂_Q_ (1).
Similarly _dR_/_dt_ = λ₂_Q_ − λ₃_R_ (2).
Substituting in (1) the value of _P_ in terms of _n_,
$$ \frac {dQ} {dt} = λ_1 ne^{–λ_1 t} − λ_2 Q $$ .
The solution of this equation is of the form
$$ Q = n (ae^{–λ_1 t} + be^{–λ_2 t}) $$ ......(3).
By substitution it is found that _a_ = λ₁/(λ₂ − λ₁).
Since _Q_ = 0 when _t_ = 0, _b_ = –λ₁(λ₂ − λ₁).
Thus
$$ Q = \frac {nλ_1} {λ_1 − λ_2} (e^{–λ_2 t} -
e^{–λ_1 t}) $$ .... (4).
Substituting this value of _Q_ in (2), it can readily be shown that
$$ R = n (ae^{–λ_1 t} + be^{–λ_2 t} + ce^{–λ_3 t}) $$ ..... (5).
where
$$ a = \frac {λ_1 λ_2} {(λ_1 − λ_2) (λ_1 -
λ_3)} $$,
$$ b = \frac {- λ_1 λ_2} {(λ_1 − λ_2) (λ_2 −
λ_3)} $$,
$$ c = \frac {λ_1 λ_2} {(λ_1 − λ_3) (λ_2 -
λ_3)} $$,
[Illustration: Fig. 72.]
The variation of the values of _P_, _Q_, _R_ with the time _t_, after
removal of the source, is shown graphically in Fig. 72, curves _A_, _B_,
and _C_ respectively. In order to draw the curves for the practical case
which will be considered later corresponding to the first three changes
in radium _A_, the values of λ₁, λ₂, λ₃ were taken as 3·85 × 10⁻³, 5·38
× 10⁻⁴, 4·13 × 10⁻⁴ respectively, _i.e._, the times required for each
successive type of matter to be half transformed are about 3, 21, and 28
minutes respectively.
The ordinates of the curves represent the relative number of atoms of
the matter _A_, _B_, and _C_ existing at any time, and the value of _n_,
the original number of atoms of the matter _A_ deposited, is taken as
100. The amount of matter _B_ is initially zero, and in this particular
case, passes through a maximum about 10 minutes later, and then
diminishes with the time. In a similar way, the amount of _C_ passes
through a maximum about 37 minutes after removal. After an interval of
several hours the amount of both _B_ and _C_ diminishes very
approximately according to an exponential law with the time, falling to
half value after intervals of 21 and 28 minutes respectively.
=198.= CASE 2. _A primary source supplies the matter A at a constant
rate and the process has continued so long that the amount of the
products A, B, C, ... has reached a steady limiting value. The primary
source is then suddenly removed. It is required to find the amounts of
A, B, C, ... remaining at any subsequent time t._
In this case, the number _n₀_ of particles of _A_, deposited per second
from the source, is equal to the number of particles of _A_ which change
into _B_ per second, and of _B_ into _C_, and so on. This requires the
relation
_n₀_ = λ₁_P₀_ = λ₂_Q₀_ = λ₃_R₀_ (6),
where _P₀_, _Q₀_, _R₀_ are the maximum numbers of particles of the
matter _A_, _B_, and _C_ when a steady state is reached.
The values of _P_, _Q_, _R_ at any time _t_ after removal of the source
are given by equations of the same form as (3) and (5) for a short
exposure. Remembering the condition that initially
_P_ = _P₀_ = _n₀_/λ₁,
_Q_ = _Q₀_ = _n₀_/λ₂,
_R_ = _R₀_ = _n₀_/λ₃,
it can readily be shown that
$$ P = \frac {n₀} {λ_1} e^{–λ_1 t} $$ .... (7),
$$ Q = \frac {n₀} {λ_1 − λ_2} (\frac {λ_1}
{λ_2} e^{–λ_2 t} − e^{–λ_1 t}) $$ .... (8),
$$ R = n₀ (ae^{–λ_1 t} + be^{–λ_2 t} + ce^{–λ_3
t}) $$ .... (9),
where
$$ a = \frac {λ_2} {(λ_1 − λ_2) (λ_1 -
λ_3)} $$,
$$ b = \frac {–λ_1} {(λ_1 − λ_2) (λ_2 -
λ_3)} $$,
$$ c = \frac {λ_1 λ_2} {λ_3 (λ_1 − λ_3)
(λ_2 − λ_3)} $$ .
[Illustration: Fig. 73.]
The relative numbers of atoms of _P_, _Q_, _R_ existing at any time are
shown graphically in Fig. 73, curves _A_, _B_, _C_ respectively. The
number of atoms _R₀_ is taken as 100 for comparison, and the values of
λ₁, λ₂, λ₃ are taken corresponding to the 3, 21, and 28-minute changes
in the active deposit of radium. A comparison with Fig. 72 for a short
exposure brings out very clearly the variation in the relative amounts
of _P_, _Q_, _R_ in the two cases. Initially the amount of _R_ decreases
very slowly. This is a result of the fact that the supply of _C_ due to
the breaking up of _B_ at first, nearly compensates for the breaking up
of _C_. The values of _Q_ and _R_ after several hours decrease
exponentially, falling to half value in 28 minutes.
=199.= CASE 3. _Suppose that a primary source has supplied the matter A
at a constant rate for any time T and is then suddenly removed. Required
the amounts of A, B, C at any subsequent time._
Suppose that _n₀_ particles of the matter _A_ are deposited each second.
After a time of exposure _T_, the number of particles _P_{T}_ of the
matter _A_ present is given by
$$ P_T = n₀ \int₀^T e^{–λ_1 t} dt = \frac {n₀} {λ_1} (1 -
e^{–λ_1 T}) $$ .
At any time _t_, after removal of the source, the number of particles
_P_ of the matter _A_ is given by
$$ P = P_T e^{–λ_1 t} = \frac {n₀} {λ_1} (1 − e^{–λ_1
T}) e^{–λ_1 t} $$ .
Consider the number of particles _n₀dt_ of the matter _A_ produced
during the interval _dt_. At any later time _t_, the number of particles
_dQ_ of the matter _B_, which result from the change in _A_, is given
(see equation 4) by
$$ dQ = \frac {n₀ λ_1} {λ_1 − λ_2} (e^{–λ_2 t} −
e^{–λ_1 t}) dt = n₀ f(t) dt $$ .... (10).
After a time of exposure _T_, the number of particles _Q_{T}_ of the
matter _B_ present is readily seen to be given by
$$ Q_T = n₀ (f(T) dt + f(T − dt) dt + ... + f(0) dt) = n₀ \int₀^T
f(t) dt $$ .
If the body is removed from the emanation after an exposure _T_, at any
later time _t_ the number of particles of _B_ is in the same way given
by
$$ Q = n₀ \int_t^{T+t} f(t) dt $$ .
It will be noted that the method of deduction of _Q_{T}_ and _Q_ is
independent of the particular form of the function _f_(_t_).
Substituting the particular value of _f_(_t_) given in equation (10) and
integrating, it can readily be deduced that
$$ \frac {Q} {Q_T} = \frac {ae^{–λ_2 t} − be^{–λ_1 t}} {a -
b} $$ .... (11),
where
$$ a = \frac {(1 − e^{–λ_2 T} {λ_2} $$,
$$ b = \frac {(1 − e^{–λ_1 T} {λ_1} $$,
In a similar way, the number of particles _R_ of the matter _C_ present
at any time can be deduced by substitution of the value of _f_(_t_) in
equation (5). These equations are, however, too complicated in form for
simple application to experiment, and will not be considered here.
=200.= CASE 4. _The matter A is supplied at a constant rate from a
primary source. Required to find the number of particles of A, B, C at
any subsequent time t, when initially A, B, C are absent._
The solution can be simply obtained in the following way. Suppose that
the conditions of Case 2 are fulfilled. The products _A_, _B_, _C_ are
in radio-active equilibrium and let _P₀_, _Q₀_, _R₀_ be the number of
particles of each present. Suppose the source is removed. The values of
_P_, _Q_, _R_ at any subsequent time are given by equations (7), (8) and
(9) respectively. Now suppose the source, which has been removed, still
continues to supply _A_ at the same constant rate and let _P₁_, _Q₁_,
_R₁_ be the number of particles of _A_, _B_, _C_ again present with the
source at any subsequent time. Now we have seen, that the rate of change
of any individual product, considered by itself, is independent of
conditions and is the same whether the matter is mixed with the parent
substance or removed from it. Since the values of _P₀_, _Q₀_, _R₀_
represent a steady state where the rate of supply of each kind of matter
is equal to its rate of change, the sum of the number of particles _A_,
_B_, _C_ present at any time with the source, and in the matter from
which it was removed, must at all times be equal to _P₀_, _Q₀_, _R₀_, ...,
that is
_P₁_ + _P_ = _P₀_,
_Q₁_ + _Q_ = _Q₀_,
_R₁_ + _R_ = _R₀_.
This must obviously be the case, for otherwise there would be a
destruction or creation of matter by the mere process of separation of
the source from its products; but, by hypothesis, neither the rate of
supply from the source, nor the law of change of the products, has been
in any way altered by removal.
Substituting the values of _P_, _Q_, _R_ from equations (7), (8), and
(9), we obtain
$$ \frac {P_1} {P₀} = 1 − e^{–λ_1 t} $$,
$$ \frac {Q_1} {Q₀} = 1 − \frac {(λ_1 e^{–λ_2 t} -
λ_2e^{–λ_1 t})} {λ_1 − λ_2} $$,
$$ \frac {R_1} {R₀} = 1 − λ_3 (ae^{–λ_1 t} + be^{–λ_2
t} + ce^{–λ_3 t}) $$,
where _a_, _b_, and _c_ have the values given after equation (9). The
curves representing the increase of _P_, _Q_, _R_, are thus, in all
cases, complementary to the curves shown in Fig. 73. The sum of the
ordinates of the two curves of rise and decay at any time is equal to
100. We have already seen examples of this in the case of the decay and
recovery curves of Ur X and Th X.
=201. Activity of a mixture of products.= In the previous calculations
we have seen how the number of particles of each of the successive
products varies with the time under different conditions. It is now
necessary to consider how this number is connected with the activity of
the mixture of products.
If _N_ is the number of particles of a product, the number of particles
breaking up per second is λ_N_, where λ is the constant of change. If
each particle of each product, in breaking up, emits one α particle, we
see that the number of α particles expelled per second from the mixture
of products at any time is equal to λ₁_P_ + λ₂_Q_ + λ₃_R_ + ...,
where _P_, _Q_, _R_, ... are the numbers of particles of the successive
products _A_, _B_, _C_, .... Substituting the values of _P_, _Q_, _R_
already found from any one of the four cases previously considered, the
variation of the number of α particles expelled per second with the time
can be determined.
The ideal method of measuring the activity of any mixture of
radio-active products would be to determine the number of α or β
particles expelled from it per second. In practice, however, this is
inconvenient and also very difficult experimentally.
Certain practical difficulties arise in endeavouring to compare the
activity of one product with another. We shall see later that, in many
cases, all of the successive products do not emit α rays. Some give out
β and γ rays alone, while there are several “rayless” products, that is,
products which do not emit either α, β, or γ rays. In the case of
radium, for example, radium _A_ gives out only α rays, radium _B_ no
rays at all, while radium _C_ gives out α, β, and γ rays.
In practice, the relative activity of any individual product at any time
is usually determined by relative measurements of the saturation
ionization current produced between the electrodes of a suitable testing
vessel.
Let us consider, for example, the case of a product which gives out only
α rays. The passage of the α particles through the gas produces a large
number of ions in its path. Since the α particles from any individual
product are projected with the same average velocity under all
conditions, the relative amount of the ionization produced per second in
the testing vessel serves as an accurate means of determining the
variation of its activity. No two products, however, emit α particles
with the same average velocity. We have seen that the rays from some
products are more readily stopped in the gas than others. Thus the
relative saturation current, due to two different products in a testing
vessel, does not serve as an accurate method of comparing the relative
number of α particles expelled per second. The ratio of the currents
will in general depend upon the distance between the plates of the
testing vessel, and, unless the relative ionization due to the average α
particle from the two products is known from other data, the comparison
of the currents can, at best, be only an approximate guide to the
relative number of α particles escaping into the gas.
=202.= Some examples will now be considered to show how the factors,
above considered, influence the character of the curves of activity
obtained under different experimental conditions. For the purpose of
illustration, we shall consider the variation after removal of the
excited activity on a body exposed for different times to a constant
supply of the radium emanation. The active deposit on removal consists
in general of a mixture of the products radium _A_, _B_, and _C_. The
nature of the rays from each product, the time for each product to be
transformed, and the value of λ are tabulated below for convenience:—
Product Rays T. λ (sec⁻¹)
Radium _A_ α rays 3 min. 3·85 ×
10⁻³
Radium _B_ no rays 21 min. 5·38 ×
10⁻⁴
Radium _C_ α, β, γ 28 min. 4·13 ×
rays 10⁻⁴
Since only the product _C_ gives rise to β and γ rays, the activity
measured by either of these types of rays will be proportional to the
amount of _C_ present at any time, _i.e._ to the value of _R_ at any
time. For a long exposure, the variation of activity with time measured
by the β and γ rays will thus be represented by the upper curve _CC_ of
Fig. 73, where the ordinates represent activity. This curve will be seen
to be very similar in shape to the experimental curve for a long
exposure which is given in Fig. 68.
Since radium _B_ does not give out rays, the number of α particles
expelled from the active deposit per second is proportional to λ₁_P_ +
λ₃_R_. The activity measured by the α rays, using the electrical method,
is thus proportional at any time to λ₁_P_ + _K_λ₃_R_, where _K_ is a
constant which represents the ratio of the number of ions, produced in
the testing vessel, by an α particle from _C_ compared with that from an
α particle emitted by _A_.
It will be seen later that, for this particular case, _K_ is nearly
unity. Taking _K_ = 1, the activity at any time after removal is
proportional to λ₁_P_ + λ₃_R_.
CASE 1. We shall first consider the activity curve for a short exposure
to the radium emanation. The relative values of _P_, _Q_, and _R_ at any
time corresponding to this case are graphically shown in Fig. 74. The
activity measured by the α rays at any time will be the sum of the
activities due to _A_ and _C_ separately.
Let curve _AA_ (Fig. 74) represent the activity due to _A_. This
decreases exponentially, falling to half value in 3 minutes. In order to
show the small activity due to _C_ clearly in the Figure, the activity
due to _A_ is plotted after an interval of 6 minutes, when the activity
has been reduced to 25 per cent. of its maximum value. The activity due
to _C_ is proportional to λ₃_R_, and in order to represent the activity
due to _C_ to the same scale as _A_, it is necessary to reduce the scale
of the ordinates of curve _CC_ in Fig. 72 in the ratio λ₃/λ₁.
[Illustration: Fig. 74.]
The activity due to _C_ is thus represented by the curve _CCC_, Fig. 74.
The total activity is thus represented by a curve _A_ + _C_ whose
ordinates are the sum of the ordinates of _A_ and _C_.
This theoretical activity curve is seen to be very similar in its
general features to the experimental curve shown in Fig. 66, where the
activity from a very short exposure is measured by the α rays.
CASE 2. The activity curve for a long exposure to the emanation will now
be considered. The activity after removal of _A_ and _C_ is proportional
to λ₁_P_ + λ₃_R_, where the values of _P_ and _R_ are graphically shown
in Fig. 75 by the curves _AA_, _CC_. Initially after removal, λ₁_P₀_ =
λ₃_R₀_, since _A_ and _C_ are in radio-active equilibrium, and the same
number of particles of each product break up per second. The activity
due to _A_ alone is shown in curve _AA_, Fig. 75. The activity decreases
exponentially, falling to half value in 3 minutes. The activity due to
_C_ at any time is proportional to _R_, and is initially equal to that
of _A_. The activity curve due to _C_ is thus represented by the curve
_CC_, which is the same curve as the upper curve _CC_ of Fig. 73. The
activity of _A_ and _C_ together is represented by the upper curve _A_ +
_C_ (Fig. 75), where the ordinates are equal to the sum of the ordinates
of the curves _A_ and _C_. This theoretical curve is seen to be very
similar in shape to the experimental curve (Fig. 67) showing the decay
of activity of the active deposit from a long exposure measured by the α
rays.
[Illustration: Fig. 75.]
=203. Effect of a rayless change on the activity curves.= Certain
important cases occur in the analysis of radio-active changes, when one
of the products does not give rise to rays and so cannot be detected
directly. The presence of this rayless change can, however, be readily
observed by the variations which occur in the activity of the succeeding
product.
Let us consider, for example, the case where the inactive matter _A_,
initially all of one kind, changes into the matter _B_ which gives out
rays. The inactive matter _A_ is supposed to be transformed according to
the same law as the radio-active products. Let λ₁, λ₂ be the constants
of the change of _A_ and _B_ respectively. If _n_ is the number of
particles of _A_, initially present, we see from the equation (4),
section 197, that the number of particles of the matter _B_ present at
any time is given by
$$ Q = \frac {nλ_1} {λ_1 − λ_2} (e^{–λ_2 t} -
e^{λ_1 t}) $$ .
Differentiating and equating to zero, it is seen that the value of _Q_
passes through a maximum at a time _T_ given by the equation
$$ λ_2 e^{–λ_2 T} = λ_1 e^{–λ_1 T} $$ .
For the sake of illustration, we shall consider the variation of the
activity of the active deposit of thorium, due to a very short exposure
to the emanation. Thorium _A_ gives out no rays, and thorium _B_ gives
out α, β, and γ rays, while thorium _C_ is inactive.
The matter _A_ is half transformed in 11 hours, and _B_ is half
transformed in 55 minutes. The value of λ₁ = 1·75 x 10⁻⁵(sec.)⁻¹ and λ₂
= 2·08 x 10⁻⁴(sec.)⁻¹.
The activity of the mixture of products _A_ + _B_ is due to _B_ alone,
and will, in consequence, be always proportional to the amount of _B_
present, that is, to the value of _Q_.
[Illustration: Fig. 76.]
The variation of activity with time is shown graphically in Fig. 76. The
activity rises from zero to a maximum in 220 minutes and then decays,
finally decreasing, according to an exponential law, with the time,
falling to half value in 11 hours.
This theoretical curve is seen to agree closely in shape with the
experimental curve (Fig. 65), which shows the variation of the activity
of the active deposit of thorium, produced by a short exposure in
presence of the emanation.
There are several points of interest in connection with an activity
curve of this character. The activity, some hours after removal, decays
according to an exponential law, not at the rate of the product _B_,
from which the activity rises, but at the same rate as the first rayless
transformation. This will also be the case if the rayless product has a
slower rate of change than the succeeding active product. Given an
activity curve of the character of Fig. 76, we can deduce from it that
the first change is not accompanied by rays and also the period of the
two changes in question. We are, however, unable to determine from the
curve which of the periods of change refers to the rayless product. It
is seen that the activity curve is unaltered if the values of λ₁, λ₂,
that is, if the periods of the products are interchanged, for the
equation is symmetrical in λ₁, λ₂. For example, in the case of the
active deposit of thorium, without further data it is impossible to
decide whether the period of the first change has a value of 55 minutes
or 11 hours. In such cases the question can only be settled by using
some physical or chemical means in order to separate the product _A_
from _B_, and then testing the rate of decay of their activity
separately. In practice, this can often be effected by electrolysis or
by utilizing the difference in volatility of the two products. If now a
product is separated from the mixture of _A_ and _B_ which loses its
activity according to an exponential law, falling to half value in 55
minutes (and such is experimentally observed), we can at once conclude
that the active product _B_ has the period of 55 minutes.
The characteristic features of the activity curve shown in Fig. 76
becomes less marked with increase of the time of exposure of a body to
the emanation, that is, when more and more of _B_ is mixed with _A_ at
the time of removal. For a long time of exposure, when the products _A_
and _B_ are in radio-active equilibrium, the activity after removal is
proportional to _Q_, where
$$ Q = \frac {n₀} {λ_1 − λ_2} (\frac {λ_1}
{λ_2} e^{–λ_2 t} − e^{λ_1 t}) $$,
(see equation 8, section 198). The value of _Q_, in this case, does not
increase after removal, but at once commences to diminish. The activity,
in consequence, decreases from the moment of removal, but more slowly
than would be given by an exponential law. The activity finally decays
exponentially, as in the previous case, falling to half value in 11
hours.
In the previous case we have discussed the activity curve obtained when
both the active and inactive product have comparatively rapid rates of
transformation. In certain cases which arise in the analysis of the
changes in actinium and radium, the rayless product has a rate of change
extremely slow compared with that of the active product. This
corresponds to the case where the active matter _B_ is supplied from _A_
at a constant rate. The activity curve will thus be identical in form
with the recovery curves of Th X and Ur X, that is, the activity _I_ at
any time _t_ will be represented by the equation
$$ \frac {I_t} {I₀} = 1 − e^{–λ_2 t} $$,
where _I₀_ is the maximum value of the activity and λ₂ the constant of
change of _B_.
=204.= In this chapter we have considered the variation with time, under
different conditions, of the number of atoms of the successive products,
when the period and number of the changes are given. It has been seen
that the activity curves to be expected under various conditions can be
readily deduced from the simple theory. In practice, however, the
investigator has been faced with the much more difficult inverse problem
of deducing the period, number, and character of the products, by
analysis of the activity curves obtained under various conditions.
In the case of radium, where at least seven distinct changes occur, the
problem has been one of considerable difficulty, and a solution has only
been possible by devising special physical and chemical methods of
isolation of some of the products.
We shall see later that two rayless changes occur in radium and actinium
and one in thorium. It is at first sight a very striking fact that the
presence of a substance which does not emit rays can be detected, and
its properties investigated. This is only possible when the rayless
product is transformed into another substance which emits rays; for the
variation of the activity of the latter may be such as to determine not
only the period but also the physical and chemical properties of the
parent product. In the two following chapters the application of the
theory of successive changes will be shown to account satisfactorily for
the complicated processes occurring in the radio-elements.
Footnote 294:
Ramsay, _Proc. Roy. Soc._ p. 470, June, 1904; _C. R._ 138, June 6,
1904.
Footnote 295:
_Phil. Mag._ February, 1904.