FOOTNOTES:

[1] In O. Eng. _heorte_; this is a common Teut. word, cf. Dut.

_hart_, Ger. _Herz_, Goth. _hairto_; related by root are Lat. _cor_

and Gr. [Greek: kardia]: the ultimate root is _kard_-, to quiver,

shake.

[2] This is often called bulbus arteriosus, but it will be seen that

the term is used rather differently in comparative anatomy.

HEART-BURIAL, the burial of the heart apart from the body. This is a

very ancient practice, the special reverence shown towards the heart

being doubtless due to its early association with the soul of man, his

affections, courage and conscience. In medieval Europe heart-burial was

fairly common. Some of the more notable cases are those of Richard I.,

whose heart, preserved in a casket, was placed in Rouen cathedral; Henry

III., buried in Normandy; Eleanor, queen of Edward I., at Lincoln;

Edward I., at Jerusalem; Louis IX., Philip III., Louis XIII. and Louis

XIV., in Paris. Since the 17th century the hearts of deceased members of

the house of Habsburg have been buried apart from the body in the

Loretto chapel in the Augustiner Kirche, Vienna. The most romantic story

of heart-burial is that of Robert Bruce. He wished his heart to rest at

Jerusalem in the church of the Holy Sepulchre, and on his deathbed

entrusted the fulfilment of his wish to Douglas. The latter broke his

journey to join the Spaniards in their war with the Moorish king of

Granada, and was killed in battle, the heart of Bruce enclosed in a

silver casket hanging round his neck. Subsequently the heart was buried

at Melrose Abbey. The heart of James, marquess of Montrose, executed by

the Scottish Covenanters in 1650, was recovered from his body, which had

been buried by the roadside outside Edinburgh, and, enclosed in a steel

box, was sent to the duke of Montrose, then in exile. It was lost on its

journey, and years afterwards was discovered in a curiosity shop in

Flanders. Taken by a member of the Montrose family to India, it was

stolen as an amulet by a native chief, was once more regained, and

finally lost in France during the Revolution. Of notable 17th-century

cases there is that of James II., whose heart was buried in the church

of the convent of the Visitation at Chaillot near Paris, and that of Sir

William Temple, at Moor Park, Farnham. The last ceremonial burial of a

heart in England was that of Paul Whitehead, secretary to the Monks of

Medmenham club, in 1775, the interment taking place in the Le Despenser

mausoleum at High Wycombe, Bucks. Of later cases the most notable are

those of Daniel O'Connell, whose heart is at Rome, Shelley at

Bournemouth, Louis XVII. at Venice, Kosciusko at the Polish museum at

Rapperschwyll, Lake Zurich, and the marquess of Bute, taken by his widow

to Jerusalem for burial in 1900. Sometimes other parts of the body,

removed in the process of embalming, are given separate and solemn

burial. Thus the viscera of the popes from Sixtus V. (1590) onward have

been preserved in the parish church of the Quirinal. The custom of

heart-burial was forbidden by Pope Boniface VIII. (1294-1303), but

Benedict XI. withdrew the prohibition.

See Pettigrew, _Chronicles of the Tombs_ (1857).

HEARTH (a word which appears in various forms in several Teutonic

languages, cf. Dutch _haard_, German _Herd_, in the sense of "floor"),

the part of a room where a fire is made, usually constructed of stone,

bricks, tiles or earth, beaten hard and having a chimney above; the fire

being lighted either on the hearth itself, or in a receptacle placed

there for the purpose. Like the Latin _focus_, especially in the phrase

for "hearth and home" answering to _pro aris et focis_, the word is used

as equivalent to the home or household. The word is also applied to the

fire and cooking apparatus on board ship; the floor of a smith's forge;

the floor of a reverberatory furnace on which the ore is exposed to the

flame; the lower part of a blast furnace through which the metal goes

down into the crucible; in soldering, a portable brazier or chafing

dish, and an iron box sunk in the middle of a flat iron plate or table.

An "open-hearth furnace" is a regenerative furnace of the reverberatory

type used in making steel, hence "open-hearth steel" (see IRON AND

STEEL).

Hearth-money, hearth tax or chimney-money, was a tax imposed in England

on all houses except cottages at a rate of two shillings for every

hearth. It was first levied in 1662, but owing to its unpopularity,

chiefly caused by the domiciliary visits of the collectors, it was

repealed in 1689, although it was producing L170,000 a year. The

principle of the tax was not new in the history of taxation, for in

Anglo-Saxon times the king derived a part of his revenue from a _fumage_

or tax of smoke farthings levied on all hearths except those of the

poor. It appears also in the hearth-penny or tax of a penny on every

hearth, which as early as the 10th century was paid annually to the pope

(see PETER'S PENCE).

HEARTS, a game of cards of recent origin, though founded upon the same

principle as many old games, such as _Slobberhannes_, _Four Jacks_ and

_Enfle_, namely, that of losing instead of winning as many tricks as

possible. Hearts is played with a full pack, ace counting highest and

deuce lowest. In the four-handed game, which is usually played, the

entire pack is dealt out as at whist (but without turning up the last

card, since there are no trumps), and the player at the dealer's left

begins by leading any card he chooses, the trick being taken by the

highest card of the suit led. Each player must follow suit if he can; if

he has no cards of the suit led he is privileged to throw away any card

he likes, thus having an opportunity of getting rid of his hearts, which

is the object of the game. When all thirteen tricks have been played

each player counts the hearts he has taken in and pays into the pool a

certain number of counters for them, according to an arrangement made

before beginning play. In the four-handed, or sweepstake, game the

method of settling called "Howell's," from the name of the inventor, has

been generally adopted, according to which each player begins with an

equal number of chips, say 100, and, after the hand has been played,

pays into the pool as many chips for each heart he had taken as there

are players besides himself. Then each player takes out of the pool one

chip for every heart he did not win. The pool is thus exhausted with

every deal. Hearts may be played by two, three, four or even more

players, each playing for himself.

_Spot Hearts._--In this variation the hearts count according to the

number of spots on the cards, excepting that the ace counts 14, the

king 13, queen 12 and knave 11, the combined score of the thirteen

hearts being thus 104.

_Auction Hearts._--In this the eldest hand examines his hand and bids

a certain number of counters for the privilege of naming the suit to

be got rid of, but without naming the suit. The other players in

succession have the privilege of outbidding him, and whoever bids most

declares the suit and pays the amount of his bid into the pool, the

winner taking it.

_Joker Hearts._--Here the deuce of hearts is discarded, and an extra

card, called the joker, takes its place, ranking in value between ten

and knave. It cannot be thrown away, excepting when hearts are led and

an ace or court card is played, though if an opponent discards the ace

or a court card of hearts, then the holder of the joker may discard

it. The joker is usually considered worth five chips, which are either

paid into the pool or to the player who succeeds in discarding the

joker.

_Heartsette._--In this variation the deuce of spades is deleted and

the three cards left after dealing twelve cards to each player are

called the _widow_ (or _kitty_), and are left face downward on the

table. The winner of the first trick must take the widow without

showing it to his opponents.

_Slobberhannes._--The object of this older form of Hearts is to avoid

taking either the first or last trick or a trick containing the queen

of clubs. A euchre pack (thirty two-cards, lacking all below the 7) is

used, and each player is given 10 counters, one being forfeited to the

pool if a player takes the first or last trick, or that containing the

club queen. If he takes all three he forfeits four points.

_Four Jacks (Polignac or Quatre-Valets)_ is usually played with a

piquet pack, the cards ranking in France as at ecarte, but in Great

Britain and America as at piquet. There is no trump suit. Counters are

used, and the object of the game is to avoid taking any trick

containing a knave, especially the knave of spades, called _Polignac_.

The player taking such a trick forfeits one counter to the pool.

_Enfle_ (or _Schwellen_) is usually played by four persons with a

piquet pack and for a pool. The cards rank as at Hearts, and there is

no trump suit. A player must follow suit if he can, but if he cannot

he may not discard, but must take up all tricks already won and add

them to his hand. Play is continued until one player gets rid of all

his cards and thus wins.

HEAT (O. E. _haetu_, which like "hot," Old Eng. _hat_, is from the

Teutonic type _haita, hit_, to be hot; cf. Ger. _hitze, heiss_; Dutch,

_hitte, heet_, &c.), a general term applied to that branch of physical

science which deals with the effects produced by heat on material

bodies, with the laws of transference of heat, and with the

transformations of heat into other kinds of energy. The object of the

present article is to give a brief sketch of the historical development

of the science of heat, and to indicate the relation of the different

branches of the subject, which are discussed in greater detail with

reference to the latest progress in separate articles.

1. _Meanings of the Term Heat._--The term heat is employed in ordinary

language in a number of different senses. This makes it a convenient

term to employ for the general title of the science, but the different

meanings must be carefully distinguished in scientific reasoning. For

the present purpose, omitting metaphorical significations, we may

distinguish four principal uses of the term: (a) Sensation of heat; (b)

Temperature, or degree of hotness; (c) Quantity of thermal energy; (d)

Radiant heat, or energy of radiation.

(a) From the sense of heat, aided in the case of very hot bodies by

the sense of sight, we obtain our first rough notions of heat as a

physical entity, which alters the state of a body and its condition in

respect of warmth, and is capable of passing from one body to another.

By touching a body we can tell whether it is warmer or colder than the

hand, and, by touching two similar bodies in succession, we can form a

rough estimate, by the acuteness of the sensation experienced, of

their difference in hotness or coldness over a limited range. If a hot

iron is placed on a cold iron plate, we may observe that the plate is

heated and the iron cooled until both attain appreciably the same

degree of warmth; and we infer from similar cases that something which

we call "heat" tends to pass from hot to cold bodies, and to attain

finally a state of equable diffusion when all the bodies concerned are

equally warm or cold. Ideas such as these derived entirely from the

sense of heat, are, so to speak, embedded in the language of every

nation from the earliest times.

(b) From the sense of heat, again, we naturally derive the idea of a

continuous scale or order, expressed by such terms as summer heat,

blood heat, fever heat, red heat, white heat, in which all bodies may

be placed with regard to their degrees of hotness, and we speak of the

_temperature_ of a body as denoting its place in the scale, in

contradistinction to the quantity of heat it may contain.

(c) The quantity of heat contained in a body obviously depends on the

size of the body considered. Thus a large kettleful of boiling water

will evidently contain more heat than a teacupful, though both may be

at the same temperature. The temperature does not depend on the size

of the body, but on the degree of concentration of the heat in it,

i.e. on the quantity of heat per unit mass, other things being equal.

We may regard it as axiomatic that a given body (say a pound of water)

in a given state (say boiling under a given pressure) must always

contain the same quantity of heat, and conversely that, if it contains

a given quantity of heat, and if it is under conditions in other

respects, it must be at a definite temperature, which will always be

the same for the same given conditions.

(d) It is a matter of common observation that rays of the sun or of a

fire falling on a body warm it, and it was in the first instance

natural to suppose that heat itself somehow travelled across the

intervening space from the sun or fire to the body warmed, in much the

same way as heat may be carried by a current of hot air or water. But

we now know that energy of radiation is not the same thing as heat,

though it is converted into heat when the rays strike an absorbing

substance. The term "radiant heat," however, is generally retained,

because radiation is commonly measured in terms of the heat it

produces, and because the transference of energy by radiation and

absorption is the most important agency in the diffusion of heat.

2. _Evolution of the Thermometer._--The first step in the development of

the science of heat was necessarily the invention of a thermometer, an

instrument for indicating temperature and measuring its changes. The

first requisite in the case of such an instrument is that it should

always give, at least approximately the same indication at the same

temperature. The air-thermoscope of Galileo, illustrated in fig. 1,

which consisted of a glass bulb containing air, connected to a glass

tube of small bore dipping into a coloured liquid, though very sensitive

to variations of temperature, was not satisfactory as a measuring

instrument, because it was also affected by variations of atmospheric

pressure. The invention of the type of thermometer familiar at the

present day, containing a liquid hermetically sealed in a glass bulb

with a fine tube attached, is also generally attributed to Galileo at a

slightly later date, about 1612. Alcohol was the liquid first employed,

and the degrees, intended to represent thousandths of the volume of the

bulb, were marked with small beads of enamel fused on the stem, as shown

in fig. 2. In order to render the readings of such instruments

comparable with each other, it was necessary to select a fixed point or

standard temperature as the zero or starting-point of the graduations.

Instead of making each degree a given fraction of the volume of the

bulb, which would be difficult in practice, and would give different

values for the degree with different liquids, it was soon found to be

preferable to take _two fixed points_, and to divide the interval

between them into the same number of degrees. It was natural in the

first instance to take the temperature of the human body as one of the

fixed points. In 1701 Sir Isaac Newton proposed a scale in which the

freezing-point of water was taken as zero, and the temperature of the

human body as 12 deg. About the same date (1714) Gabriel Daniel

Fahrenheit proposed to take as zero the lowest temperature obtainable

with a freezing mixture of ice and salt, and to divide the interval

between this temperature and that of the human body into 12 deg. To

obtain finer graduations the number was subsequently increased to 96

deg. The freezing-point of water was at that time supposed to be

somewhat variable, because as a matter of fact it is possible to cool

water several degrees below its freezing-point in the absence of ice.

Fahrenheit showed, however, that as soon as ice began to form the

temperature always rose to the same point, and that a mixture of ice or

snow with pure water always gave the same temperature. At a later period

he also showed that the temperature of boiling water varied with the

barometric pressure, but that it was always the same at the same

pressure, and might therefore be used as the second fixed point (as

Edmund Halley and others had suggested) provided that a definite

pressure, such as the average atmospheric pressure, were specified. The

freezing and boiling-points on one of his thermometers, graduated as

already explained, with the temperature of the body as 96 deg., came out

in the neighbourhood of 32 deg. and 212 deg. respectively, giving an

interval of 180 deg. between these points. Shortly after Fahrenheit's

death (1736) the freezing and boiling-points of water were generally

recognized as the most convenient fixed points to adopt, but different

systems of subdivision were employed. Fahrenheit's scale, with its small

degrees and its zero below the freezing-point, possesses undoubted

advantages for meteorological work, and is still retained in most

English-speaking countries. But for general scientific purposes, the

centigrade system, in which the freezing-point is marked 0 deg. and the

boiling-point 100 deg., is now almost universally employed, on account

of its greater simplicity from an arithmetical point of view. For work

of precision the fixed points have been more exactly defined (see

THERMOMETRY), but no change has been made in the fundamental principle

of graduation.

[Illustration: FIG. 1. FIG. 2.]

3. _Comparison of Scales based on Expansion._--Thermometers constructed

in the manner already described will give strictly comparable readings,

provided that the tubes be of uniform bore, and that the same liquid and

glass be employed in their construction. But they possess one obvious

defect from a theoretical point of view, namely, that the subdivision of

the temperature scale depends on the expansion of the particular liquid

selected as the standard. A liquid such as water, which, when

continuously heated at a uniform rate from its freezing-point, first

contracts and then expands, at a rapidly increasing rate, would

obviously be unsuitable. But there is no a priori reason why other

liquids should not behave to some extent in a similar way. As a matter

of fact, it was soon observed that thermometers carefully constructed

with different liquids, such as alcohol, oil and mercury, did not agree

precisely in their indications at points of the scale intermediate

between the fixed points, and diverged even more widely outside these

limits. Another possible method, proposed in 1694 by Carlo Renaldeni

(1615-1698), professor of mathematics and philosophy at Pisa, would be

to determine the intermediate points of the scale by observing the

temperatures of mixtures of ice-cold and boiling water in varying

proportions. On this method, the temperature of 50 deg. C. would be

defined as that obtained by mixing equal weights of water at 0 deg. C.

and 100 deg. C.; 20 deg. C., that obtained by mixing 80 parts of water

at 0 deg. C. with 20 parts of water at 100 deg. C. and so on. Each

degree rise of temperature in a mass of water would then represent the

addition of the same quantity of heat. The scale thus obtained would, as

a matter of fact, agree very closely with that of a mercury thermometer,

but the method would be very difficult to put in practice, and would

still have the disadvantage of depending on the properties of a

particular liquid, namely, water, which is known to behave in an

anomalous manner in other respects. At a later date, the researches of

Gay-Lussac (1802) and Regnault (1847) showed that the laws of the

expansion of gases are much simpler than those of liquids. Whereas the

expansion of alcohol between 0 deg. C. and 100 deg. C. is nearly seven

times as great as that of mercury, all gases (excluding easily

condensible vapours) expand equally, or so nearly equally that the

differences between them cannot be detected without the most refined

observations. This equality of expansion affords a strong a priori

argument for selecting the scale given by the expansion of a gas as the

standard scale of temperature, but there are still stronger theoretical

grounds for this choice, which will be indicated in discussing the

absolute scale (S 21). Among liquids mercury is found to agree most

nearly with the gas scale, and is generally employed in thermometers for

scientific purposes on account of its high boiling-point and for other

reasons. The differences of the mercurial scale from the gas scale

having been carefully determined, the mercury thermometer can be used as

a secondary standard to replace the gas thermometer within certain

limits, as the gas thermometer would be very troublesome to employ

directly in ordinary investigations. For certain purposes, and

especially at temperatures beyond the range of mercury thermometers,

electrical thermometers, also standardized by reference to the gas

thermometer, have been very generally employed in recent years, while

for still higher temperatures beyond the range of the gas thermometer,

thermometers based on the recently established laws of radiation are the

only instruments available. For a further discussion of the theory and

practice of the measurement of temperature, the reader is referred to

the article THERMOMETRY.

_4. Change of State._--Among the most important effects of heat is that

of changing the state of a substance from solid to liquid, or from

liquid to vapour. With very few exceptions, all substances, whether

simple or compound, are known to be capable of existing in each of the

three states under suitable conditions of temperature and pressure. The

transition of any substance, from the state of liquid to that of solid

or vapour under the ordinary atmospheric pressure, takes place at fixed

temperatures, the freezing and boiling-points, which are very sharply

defined for pure crystalline substances, and serve in fact as fixed

points of the thermometric scale. A change of state cannot, however, be

effected in any case without the addition or subtraction of a certain

definite quantity of heat. If a piece of ice below the freezing-point is

gradually heated at a uniform rate, its temperature may be observed to

rise regularly till the freezing-point is reached. At this point it

begins to melt, and its temperature ceases to rise. The melting takes a

considerable time, during the whole of which heat is being continuously

supplied without producing any rise of temperature, although if the same

quantity of heat were supplied to an equal mass of water, the

temperature of the water would be raised nearly 80 deg. C. Heat thus

absorbed in producing a change of state without rise of temperature is

called "Latent Heat," a term introduced by Joseph Black, who was one of

the first to study the subject of change of state from the point of view

of heat absorbed, and who in many cases actually adopted the

comparatively rough method described above of estimating quantities of

heat by observing the time required to produce a given change when the

substance was receiving heat at a steady rate from its surroundings. For

every change of state a definite quantity of heat is required, without

which the change cannot take place. Heat must be added to melt a solid,

or to vaporize a solid or a liquid, and conversely, heat must be

subtracted to reverse the change, i.e. to condense a vapour or freeze a

liquid. The quantity required for any given change depends on the nature

of the substance and the change considered, and varies to some extent

with the conditions (as to pressure, &c.) under which the change is

made, but is always the same for the same change under the same

conditions. A rough measurement of the latent heat of steam was made as

early as 1764 by James Watt, who found that steam at 212 deg. F., when

passed from a kettle into a jar of cold water, was capable of raising

nearly six times its weight of water to the boiling point. He gives the

volume of the steam as about 1800 times that of an equal weight of

water.

The phenomena which accompany change of state, and the physical laws

by which such changes are governed, are discussed in a series of

special articles dealing with particular cases. The articles on FUSION

and ALLOYS deal with the change from the solid to the liquid state,

and the analogous case of solution is discussed in the article on

SOLUTION. The articles on CONDENSATION OF GASES, LIQUID GASES and

VAPORIZATION deal with the theory of the change of state from liquid

to vapour, and with the important applications of liquid gases to

other researches. The methods of measuring the latent heat of fusion

or vaporization are described in the article CALORIMETRY, and need not

be further discussed here except as an introduction to the history of

the evolution of knowledge with regard to the nature of heat.

5. _Calorimetry by Latent Heat._--In principle, the simplest and most

direct method of measuring quantities of heat consists in observing the

effects produced in melting a solid or vaporizing a liquid. It was, in

fact, by the fusion of ice that quantities of heat were first measured.

If a hot body is placed in a cavity in a block of ice at 0 deg. C., and

is covered by a closely fitting slab of ice, the quantity of ice melted

will be directly proportional to the quantity of heat lost by the body

in cooling to 0 deg. C. None of the heat can possibly escape through the

ice, and conversely no heat can possibly get in from outside. The body

must cool exactly to 0 deg. C., and every fraction of the heat it loses

must melt an equivalent quantity of ice. Apart from heat lost in

transferring the heated body to the ice block, the method is

theoretically perfect. The only difficulty consists in the practical

measurement of the quantity of ice melted. Black estimated this quantity

by mopping out the cavity with a sponge before and after the operation.

But there is a variable film of water adhering to the walls of the

cavity, which gives trouble in accurate work. In 1780 Laplace and

Lavoisier used a double-walled metallic vessel containing broken ice,

which was in many respects more convenient than the block, but

aggravated the difficulty of the film of water adhering to the ice. In

spite of this practical difficulty, the quantity of heat required to

melt unit weight of ice was for a long time taken as the unit of heat.

This unit possesses the great advantage that it is independent of the

scale of temperature adopted. At a much later date R. Bunsen (_Phil.

Mag._, 1871), adopting a suggestion of Sir John Herschel's, devised an

ice-calorimeter suitable for measuring small quantities of heat, in

which the difficulty of the water film was overcome by measuring the

change in volume due to the melting of the ice. The volume of unit mass

of ice is approximately 1.0920 times that of unit mass of water, so that

the diminution of volume is 0.092 a cubic centimetre for each gramme of

ice melted. The method requires careful attention to details of

manipulation, which are more fully discussed in the article on

CALORIMETRY.

For measuring large quantities of heat, such as those produced by the

combustion of fuel in a boiler, the most convenient method is the

evaporation of water, which is commonly employed by engineers for the

purpose. The natural unit in this case is the quantity of heat required

to evaporate unit mass of water at the boiling point under atmospheric

pressure. In boilers working at a higher pressure, or supplied with

water at a lower temperature, appropriate corrections are applied to

deduce the quantity evaporated in terms of this unit.

For laboratory work on a small scale the converse method of condensation

has been successfully applied by John Joly, in whose steam-calorimeter

the quantity of heat required to raise the temperature of a body from

the atmospheric temperature to that of steam condensing at atmospheric

pressure is observed by weighing the mass of steam condensed on it. (See

CALORIMETRY.)

6. _Thermometric Calorimetry._--For the majority of purposes the most

convenient and the most readily applicable method of measuring

quantities of heat, is to observe the rise of temperature produced in a

known mass of water contained in a suitable vessel or calorimeter. This

method was employed from a very early date by Count Rumford and other

investigators, and was brought to a high pitch of perfection by Regnault

in his extensive calorimetric researches (_Memoires de l'Institut de

Paris_, 1847); but it is only within comparatively recent years that it

has really been placed on a satisfactory basis by the accurate

definition of the units involved. The theoretical objections to the

method, as compared with latent heat calorimetry, are that some heat is

necessarily lost by the calorimeter when its temperature is raised above

that of the surroundings, and that some heat is used in heating the

vessel containing the water. These are small corrections, which can be

estimated with considerable accuracy in practice. A more serious

difficulty, which has impaired the value of much careful work by this

method, is that the quantity of heat required to raise the temperature

of a given mass of water 1 deg. C. depends on the temperature at which

the water is taken, and also on the scale of the thermometer employed.

It is for this reason, in many cases, impossible to say, at the present

time, what was the precise value, within 1/2 or even 1% of the heat

unit, in terms of which many of the older results, such as those of

Regnault, were expressed. For many purposes this would not be a serious

matter, but for work of scientific precision such a limitation of

accuracy would constitute a very serious bar to progress. The unit

generally adopted for scientific purposes is the quantity of heat

required to raise 1 gram (or kilogram) of water 1 deg. C., and is called

the calorie (or kilo-calorie). English engineers usually state results

in terms of the British Thermal Unit (B.Th.U.), which is the quantity of

heat required to raise 1 lb. of water 1 deg. F.

7. _Watt's Indicator Diagram; Work of Expansion._--The rapid development

of the steam-engine (q.v.) in England during the latter part of the 18th

century had a marked effect on the progress of the science of heat. In

the first steam-engines the working cylinder served both as boiler and

condenser, a very wasteful method, as most of the heat was transferred

directly from the fire to the condensing water without useful effect.

The first improvement (about 1700) was to use a separate boiler, but the

greater part of the steam supplied was still wasted in reheating the

cylinder, which had been cooled by the injection of cold water to

condense the steam after the previous stroke. In 1769 James Watt showed

how to avoid this waste by using a separate condenser and keeping the

cylinder as hot as possible. In his earlier engines the steam at full

boiler pressure was allowed to raise the piston through nearly the whole

of its stroke. Connexion with the boiler was then cut off, and the steam

at full pressure was discharged into the condenser. Here again there was

unnecessary waste, as the steam was still capable of doing useful work.

He subsequently introduced "expansive working," which effected still

further economy. The connexion with the boiler was cut off when a

fraction only, say 1/4, of the stroke had been completed, the remainder

of the stroke being effected by the expansion of the steam already in

the cylinder with continually diminishing pressure. By the end of the

stroke, when connexion was made to the condenser, the pressure was so

reduced that there was comparatively little waste from this cause. Watt

also devised an instrument called an _indicator_ (see STEAM ENGINE), in

which a pencil, moved up and down vertically by the steam pressure,

recorded the pressure in the cylinder at every point of the stroke on a

sheet of paper moving horizontally in time with the stroke of the

piston. The diagram thus obtained made it possible to study what was

happening inside the cylinder, and to deduce the work done by the steam

in each stroke. The method of the indicator diagram has since proved of

great utility in physics in studying the properties of gases and

vapours. The work done, or the useful effect obtained from an engine or

any kind of machine, is measured by the product of the resistance

overcome and the distance through which it is overcome. The result is

generally expressed in terms of the equivalent weight raised through a

certain height against the force of gravity.[1] If, for instance, the

pressure on a piston is 50 lb. per sq. in., and the area of the piston

is 100 sq. in., the force on the piston is 5000 lb. weight. If the

stroke of the piston is 1 ft., the work done per stroke is capable of

raising a weight of 5000 lb. through a height of 1 ft., or 50 lb.

through a height of 100 ft. and so on.

[Illustration: FIG. 3.--Watt's Indicator Diagram. Patent of 1782.]

Fig. 3 represents an imaginary indicator diagram for a steam-engine,

taken from one of Watt's patents. Steam is admitted to the cylinder

when the piston is at the beginning of its stroke, at S. ST represents

the length of the stroke or the limit of horizontal movement of the

paper on which the diagram is drawn. The indicating pencil rises to

the point A, representing the absolute pressure of 60 lb. per sq. in.

As the piston moves outwards the pencil traces the horizontal line AB,

the pressure remaining constant till the point B is reached, at which

connexion to the boiler is cut off. The work done so far is

represented by the area of the rectangle ABSF, namely AS X SF,

multiplied by the area of the piston in sq. in. The result is in

foot-pounds if the fraction of the stroke SF is taken in feet. After

cut-off at B the steam expands under diminishing pressure, and the

pencil falls gradually from B to C, following the steam pressure until

the exhaust valve opens at the end of the stroke. The pressure then

falls rapidly to that of the condenser, which for an ideal case may be

taken as zero, following Watt. The work done during expansion is found

by dividing the remainder of the stroke FT into a number of equal

parts (say 8, Watt takes 20) and measuring the pressure at the points

1, 2, 3, 4, &c., corresponding to the middle of each. We thus obtain a

number of small rectangles, the sum of which is evidently very nearly

equal to the whole area BCTF under the expansion curve, or to the

remainder of the stroke FT multiplied by the average or mean value of

the pressure. The whole work done in the forward stroke is represented

by the area ABCTSA, or by the average value of the pressure P over the

whole stroke multiplied by the stroke L. This area must be multiplied

by the area of the piston A in sq. in. as before, to get the work done

per stroke in foot-pounds, which is PLA. If the engine repeats this

cycle N times per minute, the work done per minute is PLAN

foot-pounds, which is reduced to horse-power by dividing by 33,000. If

the steam is ejected by the piston at atmospheric pressure (15 lb. per

sq. in.) instead of being condensed at zero pressure, the area CDST

under the atmospheric line CD, representing work done against

back-pressure on the return stroke must be subtracted. If the engine

repeats the same cycle or series of operations continuously, the

indicator diagram will be a closed curve, and the nett work done per

cycle will be represented by the included area, whatever the form of

the curve.

8. _Thermal Efficiency._--The thermal efficiency of an engine is the

ratio of the work done by the engine to the heat supplied to it.

According to Watt's observations, confirmed later by Clement and

Desormes, the total heat required to produce 1 lb. of saturated steam at

any temperature from water at 0 deg. C. was approximately 650 times the

quantity of heat required to raise 1 lb. of water 1 deg. C. Since 1 lb.

of steam represented on this assumption a certain quantity of heat, the

efficiency could be measured naturally in foot-pounds of work obtainable

per lb. of steam, or conversely in pounds of steam consumed per

horse-power-hour.

In his patent of 1782 Watt gives the following example of the

improvement in thermal efficiency obtained by expansive working. Taking

the diagram already given, if the quantity of steam represented by AB,

or 300 cub. in. at 60 lb. pressure, were employed without expansion, the

work realized, represented by the area ABSF, would be 6000/4 = 1500

foot-pounds. With expansion to 4 times its original volume, as shown in

the diagram by the whole area ABCTSA, the mean pressure (as calculated

by Watt, assuming Boyle's law) would be 0.58 of the original pressure,

and the work done would be 6000 X 0.58 = 3480 foot-pounds for the same

quantity of steam, or the thermal efficiency would be 2.32 times

greater. The advantage actually obtained would not be so great as this,

on account of losses by condensation, back-pressure, &c., which are

neglected in Watt's calculation, but the margin would still be very

considerable. Three hundred cub. in. of steam at 60 lb. pressure would

represent about .0245 of 1 lb. of steam, or 28.7 B.Th.U., so that,

neglecting all losses, the possible thermal efficiency attainable with

steam at this pressure and four expansions (1/4 cut-off) would be

3480/28.7, or 121 foot-pounds per B.Th.U. At a later date, about 1820,

it was usual to include the efficiency of the boiler with that of the

engine, and to reckon the efficiency or "duty" in foot-pounds per bushel

or cwt. of coal. The best Cornish pumping-engines of that date achieved

about 70 million foot-pounds per cwt., or consumed about 3.2 lb. per

horse-power-hour, which is roughly equivalent to 43 foot-pounds per

B.Th.U. The efficiency gradually increased as higher pressures were

used, with more complete expansion, but the conditions upon which the

efficiency depended were not fully worked out till a much later date.

Much additional knowledge with regard to the nature of heat, and the

properties of gases and vapours, was required before the problem could

be attacked theoretically.

9. _Of the Nature of Heat._--In the early days of the science it was

natural to ascribe the manifestations of heat to the action of a subtle

imponderable fluid called "caloric," with the power of penetrating,

expanding and dissolving bodies, or dissipating them in vapour. The

fluid was imponderable, because the most careful experiments failed to

show that heat produced any increase in weight. The opposite property of

levitation was often ascribed to heat, but it was shown by more cautious

investigators that the apparent loss of weight due to heating was to be

attributed to evaporation or to upward air currents. The fundamental

idea of an imaginary fluid to represent heat was useful as helping the

mind to a conception of something remaining invariable in quantity

through many transformations, but in some respects the analogy was

misleading, and tended greatly to retard the progress of science. The

caloric theory was very simple in its application to the majority of

calorimetric experiments, and gave a fair account of the elementary

phenomena of change of state, but it encountered serious difficulties in

explaining the production of heat by friction, or the changes of

temperature accompanying the compression or expansion of a gas. The

explanation which the calorists offered of the production of heat by

friction or compression was that some of the latent caloric was squeezed

or ground out of the bodies concerned and became "sensible." In the case

of heat developed by friction, they supposed that the abraded portions

of the material were capable of holding a smaller quantity of heat, or

had less "capacity for heat," than the original material. From a logical

point of view, this was a perfectly tenable hypothesis, and one

difficult to refute. It was easy to account in this way for the heat

produced in boring cannon and similar operations, where the amount of

abraded material was large. To refute this explanation, Rumford (_Phil.

Trans._, 1798) made his celebrated experiments with a blunt borer, in

one of which he succeeded in boiling by friction 26.5 lb. of cold water

in 2(1/2) hours, with the production of only 4145 grains of metallic

powder. He then showed by experiment that the metallic powder required

the same amount of heat to raise its temperature 1 deg., as an equal

weight of the original metal, or that its "capacity for heat" (in this

sense) was unaltered by reducing it to powder; and he argued that "in

any case so small a quantity of powder could not possibly account for

all the heat generated, that the supply of heat appeared to be

inexhaustible, and that heat could not be a material substance, but must

be something of the nature of motion." Unfortunately Rumford's argument

was not quite conclusive. The supporters of the caloric theory appear,

whether consciously or unconsciously, to have used the phrase "capacity

for heat" in two entirely distinct senses without any clear definition

of the difference. The phrase "capacity for heat" might very naturally

denote the total quantity of heat contained in a body, which we have no

means of measuring, but it was generally used to signify the quantity of

heat required to raise the temperature of a body one degree, which is

quite a different thing, and has no necessary relation to the total

heat. In proving that the powder and the solid metal required the same

quantity of heat to raise the temperature of equal masses of either one

degree, Rumford did not prove that they contained equal quantities of

heat, which was the real point at issue in this instance. The metal tin

actually changes into powder below a certain temperature, and in so

doing evolves a measurable quantity of heat. A mixture of the gases

oxygen and hydrogen, in the proportions in which they combine to form

water, evolves when burnt sufficient heat to raise more than thirty

times its weight of water from the freezing to the boiling point; and

the mixture of gases may, in this sense, be said to contain so much more

heat than the water, although its capacity for heat in the ordinary

sense is only about half that of the water produced. To complete the

refutation of the calorists' explanation of the heat produced by

friction, it would have been necessary for Rumford to show that the

powder when reconverted into the same state as the solid metal did not

absorb a quantity of heat equivalent to that evolved in the grinding; in

other words that the heat produced by friction was not simply that due

to the change of state of the metal from solid to powder.

Shortly afterwards, in 1799, Davy[2] described an experiment in which he

melted ice by rubbing two blocks together. This experiment afforded a

very direct refutation of the calorists' view, because it was a

well-known fact that ice required to have a quantity of heat added to it

to convert it into water, so that the water produced by the friction

contained more heat than the ice. In stating as the conclusion to be

drawn from this experiment that "friction consequently does not diminish

the capacity of bodies for heat," Davy apparently uses the phrase

capacity for heat in the sense of total heat contained in a body,

because in a later section of the same essay he definitely gives the

phrase this meaning, and uses the term "capability of temperature" to

denote what we now term capacity for heat.

The delay in the overthrow of the caloric theory, and in the acceptance of

the view that heat is a mode of motion, was no doubt partly due to some

fundamental confusion of ideas in the use of the term "capacity for heat"

and similar phrases. A still greater obstacle lay in the comparative

vagueness of the motion or vibration theory. Davy speaks of heat as being

"repulsive motion," and distinguishes it from light, which is "projective

motion"; though heat is certainly not a substance--according to Davy in

the essay under discussion--and may not even be treated as an imponderable

fluid, light as certainly is a material substance, and is capable of

forming chemical compounds with ordinary matter, such as oxygen gas, which

is not a simple substance, but a compound, termed phosoxygen, of light and

oxygen. Accepting the conclusions of Davy and Rumford that heat is not a

material substance but a mode of motion, there still remains the question,

what definite conception is to be attached to a quantity of heat? What do

we mean by a quantity of vibratory motion, how is the quantity of motion

to be estimated, and why should it remain invariable in many

transformations? The idea that heat was a "mode of motion" was applicable

as a qualitative explanation of many of the effects of heat, but it lacked

the quantitative precision of a scientific statement, and could not be

applied to the calculation and prediction of definite results. The state

of science at the time of Rumford's and Davy's experiments did not admit

of a more exact generalization. The way was paved in the first instance by

a more complete study of the laws of gases, to which Laplace, Dalton,

Gay-Lussac, Dulong and many others contributed both on the experimental

and theoretical side. Although the development proceeded simultaneously

along many parallel lines, it is interesting and instructive to take the

investigation of the properties of gases, and to endeavour to trace the

steps by which the true theory was finally attained.

10. _Thermal Properties of Gases._--The most characteristic property of

a gaseous or elastic fluid, namely, the elasticity, or resistance to

compression, was first investigated scientifically by Robert Boyle

(1662), who showed that the pressure p of a given mass of gas varied

inversely as the volume v, provided that the temperature remained

constant. This is generally expressed by the formula pv = C, where C is

a constant for any given temperature, and v is taken to represent the

specific volume, or the volume of unit mass, of the gas at the given

pressure and temperature. Boyle was well aware of the effect of heat in

expanding a gas, but he was unable to investigate this properly as no

thermometric scale had been defined at that date. According to Boyle's

law, when a mass of gas is compressed by a small amount at constant

temperature, the percentage increase of pressure is equal to the

percentage diminution of volume (if the compression is v/100, the

increase of pressure is very nearly p/100). Adopting this law, Newton

showed, by a most ingenious piece of reasoning (_Principia_, ii., sect.

8), that the velocity of sound in air should be equal to the velocity

acquired by a body falling under gravity through a distance equal to

half the height of the atmosphere, considered as being of uniform

density equal to that at the surface of the earth. This gave the result

918 ft. per sec. (280 metres per sec.) for the velocity at the freezing

point. Newton was aware that the actual velocity of sound was somewhat

greater than this, but supposed that the difference might be due in some

way to the size of the air particles, of which no account could be taken

in the calculation. The first accurate measurement of the velocity of

sound by the French Academie des Sciences in 1738 gave the value 332

metres per sec. as the velocity at 0 deg. C. The true explanation of the

discrepancy was not discovered till nearly 100 years later.

The law of expansion of gases with change of temperature was

investigated by Dalton and Gay-Lussac (1802), who found that the volume

of a gas under constant pressure increased by 1/267th part of its volume

at 0 deg. C. for each 1 deg. C. rise in temperature. This value was

generally assumed in all calculations for nearly 50 years. More exact

researches, especially those of Regnault, at a later date, showed that

the law was very nearly correct for all permanent gases, but that the

value of the coefficient should be 1/273rd. According to this law the

volume of a gas at any temperature t deg. C. should be proportional to

273 + t, i.e. to the temperature reckoned from a zero 273 deg. below

that of the Centigrade scale, which was called the absolute zero of the

gas thermometer. If T = 273 + t, denotes the temperature measured from

this zero, the law of expansion of a gas may be combined with Boyle's

law in the simple formula

pv = RT (1)

which is generally taken as the expression of the gaseous laws. If equal

volumes of different gases are taken at the same temperature and

pressure, it follows that the constant R is the same for all gases. If

equal masses are taken, the value of the constant R for different gases

varies inversely as the molecular weight or as the density relative to

hydrogen.

Dalton also investigated the laws of vapours, and of mixtures of gases

and vapours. He found that condensible vapours approximately followed

Boyle's law when compressed, until the condensation pressure was

reached, at which the vapour liquefied without further increase of

pressure. He found that when a liquid was introduced into a closed

space, and allowed to evaporate until the space was saturated with the

vapour and evaporation ceased, the increase of pressure in the space was

equal to the condensation pressure of the vapour, and did not depend on

the volume of the space or the presence of any other gas or vapour

provided that there was no solution or chemical action. He showed that

the condensation or saturation-pressure of a vapour depended only on the

temperature, and increased by nearly the same fraction of itself per

degree rise of temperature, and that the pressures of different vapours

were nearly the same at equal distances from their boiling points. The

increase of pressure per degree C. at the boiling point was about 1/28th

of 760 mm. or 27.2 mm., but increased in geometrical progression with

rise of temperature. These results of Dalton's were confirmed, and in

part corrected, as regards increase of vapour-pressure, by Gay-Lussac,

Dulong, Regnault and other investigators, but were found to be as close

an approximation to the truth as could be obtained with such simple

expressions. More accurate empirical expressions for the increase of

vapour-pressure of a liquid with temperature were soon obtained by

Thomas Young, J. P. L. A. Roche and others, but the explanation of the

relation was not arrived at until a much later date (see VAPORIZATION).

11. _Specific Heats of Gases._--In order to estimate the quantities of

heat concerned in experiments with gases, it was necessary in the first

instance to measure their specific heats, which presented formidable

difficulties. The earlier attempts by Lavoisier and others, employing

the ordinary methods of calorimetry, gave very uncertain and discordant

results, which were not regarded with any confidence even by the

experimentalists themselves. Gay-Lussac (_Memoires d'Arcueil_, 1807)

devised an ingenious experiment, which, though misinterpreted at the

time, is very interesting and instructive. With the object of comparing

the specific heats of different gases, he took two equal globes A and B

connected by a tube with a stop-cock. The globe B was exhausted, the

other A being filled with gas. On opening the tap between the vessels,

the gas flowed from A to B and the pressure was rapidly equalized. He

observed that the fall of temperature in A was nearly equal to the rise

of temperature in B, and that for the same initial pressure the change

of temperature was very nearly the same for all the gases he tried,

except hydrogen, which showed greater changes of temperature than other

gases. He concluded from this experiment that equal volumes of gases had

the same capacity for heat, except hydrogen, which he supposed to have a

larger capacity, because it showed a greater effect. The method does not

in reality afford any direct information with regard to the specific

heats, and the conclusion with regard to hydrogen is evidently wrong. At

a later date (_Ann. de Chim._, 1812, 81, p. 98) Gay-Lussac adopted A.

Crawford's method of mixture, allowing two equal streams of different

gases, one heated and the other cooled about 20 deg. C., to mix in a tube

containing a thermometer. The resulting temperature was in all cases

nearly the mean of the two, from which he concluded that equal volumes

of all the gases tried, namely, hydrogen, carbon dioxide, air, oxygen

and nitrogen, had the same thermal capacity. This was correct, except as

regards carbon dioxide, but did not give any information as to the

actual specific heats referred to water or any known substance. About

the same time, F. Delaroche and J. E. Berard (_Ann. de chim._, 1813, 85,

p. 72) made direct determinations of the specific heats of air, oxygen,

hydrogen, carbon monoxide, carbon dioxide, nitrous oxide and ethylene,

by passing a stream of gas heated to nearly 100 deg. C. through a spiral

tube in a calorimeter containing water. Their work was a great advance

on previous attempts, and gave the first trustworthy results. With the

exception of hydrogen, which presents peculiar difficulties, they found

that equal volumes of the permanent gases, air, oxygen and carbon

monoxide, had nearly the same thermal capacity, but that the compound

condensible gases, carbon dioxide, nitrous oxide and ethylene, had

larger thermal capacities in the order given. They were unable to state

whether the specific heats of the gases increased or diminished with

temperature, but from experiments on air at pressures of 740 mm. and

1000 mm., they found the specific heats to be .269 and .245

respectively, and concluded that the specific heat diminished with

increase of pressure. The difference they observed was really due to

errors of experiment, but they regarded it as proving beyond doubt the

truth of the calorists' contention that the heat disengaged on the

compression of a gas was due to the diminution of its thermal capacity.

Dalton and others had endeavoured to measure directly the rise of

temperature produced by the compression of a gas. Dalton had observed a

rise of 50 deg. F. in a gas when suddenly compressed to half its volume,

but no thermometers at that time were sufficiently sensitive to indicate

more than a fraction of the change of temperature. Laplace was the first

to see in this phenomenon the probable explanation of the discrepancy

between Newton's calculation of the velocity of sound and the observed

value. The increase of pressure due to a sudden compression, in which no

heat was allowed to escape, or as we now call it an "adiabatic"

compression, would necessarily be greater than the increase of pressure

in a slow isothermal compression, on account of the rise of temperature.

As the rapid compressions and rarefactions occurring in the propagation

of a sound wave were perfectly adiabatic, it was necessary to take

account of the rise of temperature due to compression in calculating the

velocity. To reconcile the observed and calculated values of the

velocity, the increase of pressure in adiabatic compression must be

1.410 times greater than in isothermal compression. This is the ratio of

the adiabatic elasticity of air to the isothermal elasticity. It was a

long time, however, before Laplace saw his way to any direct

experimental verification of the value of this ratio. At a later date

(_Ann. de chim._, 1816, 3, p. 238) he stated that he had succeeded in

proving that the ratio in question must be the same as the ratio of the

specific heat of air at constant pressure to the specific heat at

constant volume.

In the method of measuring the specific heat adopted by Delaroche and

Berard, the gas under experiment, while passing through a tube at

practically constant pressure, contracts in cooling, as it gives up

its heat to the calorimeter. Part of the heat surrendered to the

calorimeter is due to the contraction of volume. If a gramme of gas at

pressure p, volume v and temperature T abs. is heated 1 deg. C. at

constant pressure p, it absorbs a quantity of heat S = .238 calorie

(according to Regnault) the specific heat at constant pressure. At the

same time the gas expands by a fraction 1/T of v, which is the same as

1/273 of its volume at 0 deg. C. If now the air is suddenly compressed

by an amount v/T, it will be restored to its original volume, and its

temperature will be raised by the liberation of a quantity of heat R',

the latent heat of expansion for an increase of volume v/T. If no heat

has been allowed to escape, the air will now be in the same state as

if a quantity of heat S had been communicated to it at its original

volume v without expansion. The rise of temperature above the original

temperature T will be S/s degrees, where s is the specific heat at

constant volume, which is obviously equal to S - R'. Since p/T is the

increase of pressure for 1 deg. C. rise of temperature at constant

volume, the increase of pressure for a rise of S/s degrees will be

[gamma]p/T, where [gamma] is the ratio S/s. But this is the rise of

pressure produced by a sudden compression v/T, and is seen to be

[gamma] times the rise of pressure p/T produced by the same

compression at constant temperature. The ratio of the adiabatic to the

isothermal elasticity, required for calculating the velocity of sound,

is therefore the same as the ratio of the specific heat at constant

pressure to that at constant volume.

12. _Experimental Verification of the Ratio of Specific Heats._--This

was a most interesting and important theoretical relation to discover,

but unfortunately it did not help much in the determination of the

ratio required, because it was not practically possible at that time

to measure the specific heat of air at constant volume in a closed

vessel. Attempts had been made to do this, but they had signally

failed, on account of the small heat capacity of the gas as compared

with the containing vessel. Laplace endeavoured to extract some

confirmation of his views from the values given by Delaroche and

Berard for the specific heat of air at 1000 and 740 mm. pressure. On

the assumption that the quantities of heat contained in a given mass

of air increased in direct proportion to its volume when heated at

constant pressure, he deduced, by some rather obscure reasoning, that

the ratio of the specific heats S and s should be about 1.5 to 1,

which he regarded as a fairly satisfactory agreement with the value

[gamma] = 1.41 deduced from the velocity of sound.

The ratio of the specific heats could not be directly measured, but a

few years later, Clement and Desormes (_Journ. de Phys._, Nov. 1819)

succeeded in making a direct measurement of the ratio of the

elasticities in a very simple manner. They took a large globe

containing air at atmospheric pressure and temperature, and removed a

small quantity of air. They then observed the defect of pressure p0

when the air had regained its original temperature. By suddenly

opening the globe, and immediately closing it, the pressure was

restored almost instantaneously to the atmospheric, the rise of

pressure p0 corresponding to the sudden compression produced. The air,

having been heated by the compression, was allowed to regain its

original temperature, the tap remaining closed, and the final defect

of pressure p^1 was noted. The change of pressure for the same

compression performed isothermally is then p0 - p^1. The ratio p0/(p0

- p^1) is the ratio of the adiabatic and isothermal elasticities,

provided that p0 is small compared with the whole atmospheric

pressure. In this way they found the ratio 1.354, which is not much

smaller than the value 1.410 required to reconcile the observed and

calculated values of the velocity of sound. Gay-Lussac and J. J.

Welter (_Ann. de chim._, 1822) repeated the experiment with slight

improvements, using expansion instead of compression, and found the

ratio 1.375. The experiment has often been repeated since that time,

and there is no doubt that the value of the ratio deduced from the

velocity of sound is correct, the defect of the value obtained by

direct experiment being due to the fact that the compression or

expansion is not perfectly adiabatic. Gay-Lussac and Welter found the

ratio practically constant for a range of pressure 144 to 1460 mm.,

and for a range of temperature from -20 deg. to +40 deg. C. The

velocity of sound at Quito, at a pressure of 544 mm. was found to be

the same as at Paris at 760 mm. at the same temperature. Assuming on

this evidence the constancy of the ratio of the specific heats of air,

Laplace (_Mecanique celeste_, v. 143) showed that, if the specific

heat at constant pressure was independent of the temperature, the

specific heat per unit volume at a pressure p must vary as

p^(1/[gamma]), according to the caloric theory. The specific heat per

unit mass must then vary as p^(1/[gamma]-1) which he found agreed

precisely with the experiment of Delaroche and Berard already cited.

This was undoubtedly a strong confirmation of the caloric theory.

Poisson by the same assumptions (_Ann. de chim._, 1823, 23, p. 337)

obtained the same results, and also showed that the relation between

the pressure and the volume of a gas in adiabatic compression or

expansion must be of the form pv^[gamma] = constant.

P. L. Dulong (_Ann. de chim._, 1829, 41, p. 156), adopting a method

due to E. F. F. Chladni, compared the velocities of sound in different

gases by observing the pitch of the note given by the same tube when

filled with the gases in question. He thus obtained the values of the

ratios of the elasticities or of the specific heats for the gases

employed. For oxygen, hydrogen and carbonic oxide, these ratios were

the same as for air. But for carbonic acid, nitrous oxide and olefiant

gas, the values were much smaller, showing that these gases

experienced a smaller change of temperature in compression. On

comparing his results with the values of the specific heats for the

same gases found by Delaroche and Berard, Dulong observed that the

changes of temperature for the same compression were in the inverse

ratio of the specific heats at constant volume, and deduced the

important conclusion that "_Equal volumes of all gases under the same

conditions evolve on compression the same quantity of heat_." This is

equivalent to the statement that the difference of the specific heats,

or the latent heat of expansion R' per 1 deg., is the same for all

gases if equal volumes are taken. Assuming the ratio [gamma] = 1.410,

and taking Delaroche and Berard's value for the specific heat of air

at constant pressure S = .267, we have s = S/1.41 = .189, and the

difference of the specific heats per unit mass of air S - s = R' =

.078. Adopting Regnault's value of the specific heat of air, namely, S

= .238, we should have S - s = .069. This quantity represents the heat

absorbed by unit mass of air in expanding at constant temperature T by

a fraction 1/T of its volume v, or by 1/273rd of its volume 0 deg. C.

If, instead of taking unit mass, we take a volume v0 = 22.30 litres at

0 deg. C. and 760 mm. being the volume of the molecular weight of the

gas in grammes, the quantity of heat evolved by a compression equal to

v/T will be approximately 2 calories, and is the same for all gases.

The work done in this compression is pv/T = R, and is also the same

for all gases, namely, 8.3 joules. Dulong's experimental result,

therefore, shows that the heat evolved in the compression of a gas is

proportional to the work done. This result had previously been deduced

theoretically by Carnot (1824). At a later date it was assumed by

Mayer, Clausius and others, on the evidence of these experiments, that

the heat evolved was not merely proportional to the work done, but was

equivalent to it. The further experimental evidence required to

justify this assumption was first supplied by Joule.

Latent heat of expansion R' = .069 calorie per gramme of air, per 1

deg. C.

= 2.0 calories per gramme-molecule of any

gas.

Work done in expansion R = .287 joule per gramme of air per 1 deg. C.

= 8.3 joules per gramme-molecule of any

gas.

13. _Carnot: On the Motive Power of Heat._--A practical and theoretical

question of the greatest importance was first answered by Sadi Carnot

about this time in his _Reflections on the Motive Power of Heat_ (1824).

How much motive power (defined by Carnot as weight lifted through a

certain height) can be obtained from heat alone by means of an engine

repeating a regular succession or "cycle" of operations continuously? Is

the efficiency limited, and, if so, how is it limited? Are other agents

preferable to steam for developing motive power from heat? In discussing

this problem, we cannot do better than follow Carnot's reasoning which,

in its main features could hardly be improved at the present day.

Carnot points out that in order to obtain an answer to this question, it

is necessary to consider the essential conditions of the process, apart

from the mechanism of the engine and the working substance or agent

employed. Work cannot be said to be produced _from heat alone_ unless

nothing but heat is supplied, and the working substance and all parts of

the engine are at the end of the process in precisely the same state as

at the beginning.[3]

_Carnot's Axiom._--Carnot here, and throughout his reasoning, makes a

fundamental assumption, which he states as follows: "When a body has

undergone any changes and after a certain number of transformations is

brought back identically to its original state, considered relatively to

density, temperature and mode of aggregation, it must contain the same

quantity of heat as it contained originally."[4]

Heat, according to Carnot, in the type of engine we are considering, can

evidently be a cause of motive power only by virtue of changes of volume

or form produced by alternate heating and cooling. This involves the

existence of cold and hot bodies to act as boiler and condenser, or

source and sink of heat, respectively. Wherever there exists a

difference of temperature, it is possible to have the production of

motive power from heat; and conversely, production of motive power, from

heat alone, is impossible without difference of temperature. In other

words the production of motive power from heat is not merely a question

of the consumption of heat, but always requires transference of heat

from hot to cold. What then are the conditions which enable the

difference of temperature to be most advantageously employed in the

production of motive power, and how much motive power can be obtained

with a given difference of temperature from a given quantity of heat?

_Carnot's Rule for Maximum Effect._--In order to realize the maximum

effect, it is necessary that, in the process employed, there should not

be any direct interchange of heat between bodies at different

temperatures. Direct transference of heat by conduction or radiation

between bodies at different temperatures is equivalent to wasting a

difference of temperature which might have been utilized to produce

motive power. The working substance must throughout every stage of the

process be in equilibrium with itself (i.e. at uniform temperature and

pressure) and also with external bodies, such as the boiler and

condenser, at such times as it is put in communication with them. In the

actual engine there is always some interchange of heat between the steam

and the cylinder, and some loss of heat to external bodies. There may

also be some difference of temperature between the boiler steam and the

cylinder on admission, or between the waste steam and the condenser at

release. These differences represent losses of efficiency which may be

reduced indefinitely, at least in imagination, by suitable means, and

designers had even at that date been very successful in reducing them.

All such losses are supposed to be absent in deducing the ideal limit of

efficiency, beyond which it would be impossible to go.

14. _Carnot's Description of his Ideal Cycle._--Carnot first gives a

rough illustration of an incomplete cycle, using steam much in the same

way as it is employed in an ordinary steam-engine. After expansion down

to condenser pressure the steam is completely condensed to water, and is

then returned as cold water to the hot boiler. He points out that the

last step does not conform exactly to the condition he laid down,

because although the water is restored to its initial state, there is

direct passage of heat from a hot body to a cold body in the last

process. He points out that this difficulty might be overcome by

supposing the difference of temperature small, and by employing a series

of engines, each working through a small range, to cover a finite

interval of temperature. Having established the general notions of a

perfect cycle, he proceeds to give a more exact illustration, employing

a gas as the working substance. He takes as the basis of his

demonstration the well-established experimental fact that a gas is

heated by rapid compression and cooled by rapid expansion, and that if

compressed or expanded slowly in contact with conducting bodies, the gas

will give out heat in compression or absorb heat in expansion while its

temperature remains constant. He then goes on to say:--

"This preliminary notion being settled, let us imagine an elastic

fluid, atmospheric air for example, enclosed in a cylinder _abcd_,

fig. 4, fitted with a movable diaphragm or piston cd. Let there also

be two bodies A, B, each maintained at a constant temperature, that of

A being more elevated than that of B. Let us now suppose the following

series of operations to be performed:

[Illustration: FIG 4. Carnot's Cylinder.]

"1. Contact of the body A with the air contained in the space _abcd_,

or with the bottom of the cylinder, which we will suppose to transmit

heat easily. The air is now at the temperature of the body A, and _cd_

is the actual position of the piston.

"2. The piston is gradually raised, and takes the position _ef_. The

air remains in contact with the body A, and is thereby maintained at a

constant temperature during the expansion. The body A furnishes the

heat necessary to maintain the constancy of temperature.

"3. The body A is removed, and the air no longer being in contact with

any body capable of giving it heat, the piston continues nevertheless

to rise, and passes from the position _ef_ to _gh_. The air expands

without receiving heat and its temperature falls. Let us imagine that

it falls until it is just equal to that of the body B. At this moment

the piston is stopped and occupies the position _gh_.

"4. The air is placed in contact with the body B; it is compressed by

the return of the piston, which is brought from the position _gh_ to

the position _cd_. The air remains meanwhile at a constant

temperature, because of its contact with the body B to which it gives

up its heat.

"5. The body B is removed, and the compression of the air is

continued. The air being now isolated, rises in temperature. The

compression is continued until the air has acquired the temperature of

the body A. The piston passes meanwhile from the position _cd_ to the

position _ik_.

"6. The air is replaced in contact with the body A, and the piston

returns from the position _ik_ to the position _ef_, the temperature

remaining invariable.

"7. The period described under (3) is repeated, then successively the

periods (4), (5), (6); (3), (4), (5), (6); (3), (4), (5), (6); and so

on.

"During these operations the air enclosed in the cylinder exerts an

effort more or less great on the piston. The pressure of the air

varies both on account of changes of volume and on account of changes

of temperature; but it should be observed that for equal volumes, that

is to say, for like positions of the piston, the temperature is higher

during the dilatation than during the compression. Since the pressure

is greater during the expansion, the quantity of motive power produced

by the dilatation is greater than that consumed by the compression. We

shall thus obtain a balance of motive power, which may be employed for

any purpose. The air has served as working substance in a heat-engine;

it has also been employed in the most advantageous manner possible,

since no useless re-establishment of the equilibrium of heat has been

allowed to occur.

"All the operations above described may be executed in the reverse

order and direction. Let us imagine that after the sixth period, that

is to say, when the piston has reached the position _ef_, we make it

return to the position _ik_, and that at the same time we keep the air

in contact with the hot body A; the heat furnished by this body during

the sixth period will return to its source, that is, to the body A,

and everything will be as it was at the end of the fifth period. If

now we remove the body A, and if we make the piston move from _ik_ to

_cd_, the temperature of the air will decrease by just as many degrees

as it increased during the fifth period, and will become that of the

body B. We can evidently continue in this way a series of operations

the exact reverse of those which were previously described; it

suffices to place oneself in the same circumstances and to execute for

each period a movement of expansion in place of a movement of

compression, and vice versa.

"The result of the first series of operations was the production of a

certain quantity of motive power, and the transport of heat from the

body A to the body B; the result of the reverse operations is the

consumption of the motive power produced in the first case, and the

return of heat from the body B to the body A, in such sort that these

two series of operations annul and neutralize each other.

"The impossibility of producing by the agency of heat alone a quantity

of motive power greater than that which we have obtained in our first

series of operations is now easy to prove. It is demonstrated by

reasoning exactly similar to that which we have already given. The

reasoning will have in this case a greater degree of exactitude; the

air of which we made use to develop the motive power is brought back

at the end of each cycle of operations precisely to its initial state,

whereas this was not quite exactly the case for the vapour of water,

as we have already remarked."

15. _Proof of Carnot's Principle._--Carnot considered the proof too

obvious to be worth repeating, but, unfortunately, his previous

demonstration, referring to an incomplete cycle, is not so exactly

worded that exception cannot be taken to it. We will therefore repeat

his proof in a slightly more definite and exact form. Suppose that a

reversible engine R, working in the cycle above described, takes a

quantity of heat H from the source in each cycle, and performs a

quantity of useful work W_r. If it were possible for any other engine S,

working with the same two bodies A and B as source and refrigerator, to

perform a greater amount of useful work W_s per cycle for the same

quantity of heat H taken from the source, it would suffice to take a

portion W_r of this motive power (since W_s is by hypothesis greater

than W_r) to drive the engine R backwards, and return a quantity of heat

H to the source in each cycle. The process might be repeated

indefinitely, and we should obtain at each repetition a balance of

useful work W_s - W_r, _without taking any heat from the source_, which

is contrary to experience. Whether the quantity of heat taken from the

condenser by R is equal to that given to the condenser by S is

immaterial. The hot body A might be a comparatively small boiler, since

no heat is taken from it. The cold body B might be the ocean, or the

whole earth. We might thus obtain without any consumption of fuel a

practically unlimited supply of motive power. Which is absurd.

_Carnot's Statement of his Principle._[5]--If the above reasoning be

admitted, we must conclude with Carnot that _the motive power obtainable

from heat is independent of the agents employed to realize it_. _The

efficiency is fixed solely by the temperatures of the bodies between

which, in the last resort, the transfer of heat is effected._ "We must

understand here that each of the methods of developing motive power

attains the perfection of which it is susceptible. This condition is

fulfilled if, according to our rule, there is produced in the body no

change of temperature that is not due to change of volume, or in other

words, if there is no direct interchange of heat between bodies of

sensibly different temperatures."

It is characteristic of a state of frictionless mechanical equilibrium

that an indefinitely small difference of pressure suffices to upset the

equilibrium and reverse the motion. Similarly in thermal equilibrium

between bodies at the same temperature, an indefinitely small difference

of temperature suffices to reverse the transfer of heat. Carnot's rule

is therefore the criterion of the reversibility of a cycle of operations

as regards transfer of heat. It is assumed that the ideal engine is

mechanically reversible, that there is not, for instance, any

communication between reservoirs of gas or vapour at sensibly different

pressures, and that there is no waste of power in friction. If there is

equilibrium both mechanical and thermal at every stage of the cycle, the

ideal engine will be perfectly reversible. That is to say, all its

operations will be exactly reversed as regards transfer of heat and

work, when the operations are performed in the reverse order and

direction. On this understanding Carnot's principle may be put in a

different way, which is often adopted, but is really only the same thing

put in different words: _The efficiency of a perfectly reversible engine

is the maximum possible, and is a function solely of the limits of

temperature between which it works_. This result depends essentially on

the existence of a state of thermal equilibrium defined by equality of

temperature, and independent, in the majority of cases, of the state of

a body in other respects. In order to apply the principle to the

calculation and prediction of results, it is sufficient to determine the

manner in which the efficiency depends on the temperature for one

particular case, since the efficiency must be the same for all

reversible engines.

16. _Experimental Verification of Carnot's Principle._--Carnot

endeavoured to test his result by the following simple calculations.

Suppose that we have a cylinder fitted with a frictionless piston,

containing 1 gram of water at 100 deg. C., and that the pressure of

the steam, namely 760 mm., is in equilibrium with the external

pressure on the piston at this temperature. Place the cylinder in

connexion with a boiler or hot body at 101 deg. C. The water will then

acquire the temperature of 101 deg. C., and will absorb 1 gram-calorie

of heat. Some waste of motive power occurs here because heat is

allowed to pass from one body to another at a different temperature,

but the waste in this case is so small as to be immaterial. Keep the

cylinder in contact with the hot body at 101 deg. C. and allow the

piston to rise. It may be made to perform useful work as the pressure

is now 27.7 mm. (or 37.7 grams per sq. cm.) in excess of the external

pressure. Continue the process till all the water is converted into

steam. The heat absorbed from the hot body will be nearly 540

gram-calories, the latent heat of steam at this temperature. The

increase of volume will be approximately 1620 c.c., the volume of 1

gram of steam at this pressure and temperature. The work done by the

excess pressure will be 37.7 X 1620 = 61,000 gram-centimetres or 0.61

of a kilogrammetre. Remove the hot body, and allow the steam to expand

further till its pressure is 760 mm. and its temperature has fallen to

100 deg. C. The work which might be done in this expansion is less

than 1/1000th part of a kilogrammetre, and may be neglected for the

present purpose. Place the cylinder in contact with the cold body at

100 deg. C., and allow the steam to condense at this temperature. No

work is done on the piston, because there is equilibrium of pressure,

but a quantity of heat equal to the latent heat of steam at 100 deg.

C. is given to the cold body. The water is now in its initial

condition, and the result of the process has been to gain 0.61 of a

kilogrammetre of work by allowing 540 gram-calories of heat to pass

from a body at 101 deg. C. to a body at 100 deg. C. by means of an

ideally simple steam-engine. The work obtainable in this way from 1000

gram-calories of heat, or 1 kilo-calorie, would evidently be 1.13

kilogrammetre (= 0.61 X 1000/540).

Taking the same range of temperature, namely 101 deg. to 100 deg. C.,

we may perform a similar series of operations with air in the

cylinder, instead of water and steam. Suppose the cylinder to contain

1 gramme of air at 100 deg. C. and 760 mm. pressure instead of water.

Compress it without loss of heat (adiabatically), so as to raise its

temperature to 101 deg. C. Place it in contact with the hot body at

101 deg. C., and allow it to expand at this temperature, absorbing

heat from the hot body, until its volume is increased by 1/374th part

(the expansion per degree at constant pressure). The quantity of heat

absorbed in this expansion, as explained in S 14, will be the

difference of the specific heats or the latent heat of expansion R' =

.069 calorie. Remove the hot body, and allow the gas to expand further

without gain of heat till its temperature falls to 100 deg. C.

Compress it at 100 deg. C. to its original volume, abstracting the

heat of compression by contact with the cold body at 100 deg. C. The

air is now in its original state, and the process has been carried out

in strict accordance with Carnot's rule. The quantity of external work

done in the cycle is easily obtained by the aid of the indicator

diagram ABCD (fig. 5), which is approximately a parallelogram in this

instance. The area of the diagram is equal to that of the rectangle

BEHG, being the product of the vertical height BE, namely, the

increase of pressure per 1 deg. at constant volume, by the increase of

volume BG, which is 1/273rd of the volume at 0 deg. C. and 760 mm., or

2.83 c.c. The increase of pressure BE is 760/373, or 2.03 mm., which

is equivalent to 2.76 gm. per sq. cm. The work done in the cycle is

2.76 X 2.83 = 7.82 gm. cm., or .0782 gram-metre. The heat absorbed at

101 deg. C. was .069 gram-calorie, so that the work obtained is

.0782/.069 or 1.13 gram-metre per gram-calorie, or 1.13 kilogrammetre

per kilogram-calorie. This result is precisely the same as that

obtained by using steam with the same range of temperature, but a very

different kind of cycle. Carnot in making the same calculation did not

obtain quite so good an agreement, because the experimental data at

that time available were not so accurate. He used the value 1/267 for

the coefficient of expansion, and .267 for the specific heat of air.

Moreover, he did not feel justified in assuming, as above, that the

difference of the specific heats was the same at 100 deg. C. as at the

ordinary temperature of 15 deg. to 20 deg. C., at which it had been

experimentally determined. He made similar calculations for the vapour

of alcohol, which differed slightly from the vapour of water. But the

agreement he found was close enough to satisfy him that his

theoretical deductions were correct, and that the resulting ratio of

work to heat should be the same for all substances at the same

temperature.

[Illustration: FIG. 5.--Elementary Carnot Cycle for Gas.]

17. _Carnot's Function. Variation of Efficiency with Temperature._--By

means of calculations, similar to those given above, Carnot

endeavoured to find the amount of motive power obtainable from one

unit of heat per degree fall at various temperatures with various

substances. The value found above, namely 1.13 kilogrammetre per

kilo-calorie per 1 deg. fall, is the value of the efficiency per 1

deg. fall at 100 deg. C. He was able to show that the efficiency per

degree fall probably diminished with rise of temperature, but the

experimental data at that time were too inconsistent to suggest the

true relation. He took as the analytical expression of his principle

that the efficiency W/H of a perfect engine taking in heat H at a

temperature t deg. C., and rejecting heat at the temperature 0 deg.

C., must be some function Ft of the temperature t, which would be the

same for all substances. The efficiency per degree fall at a

temperature t he represented by F't, the derived function of Ft. The

function F't would be the same for all substances at the same

temperature, but would have different values at different

temperatures. In terms of this function, which is generally known as

Carnot's function, the results obtained in the previous section might

be expressed as follows:--

"The increase of volume of a mixture of liquid and vapour per

unit-mass vaporized at any temperature, multiplied by the increase of

vapour-pressure per degree, is equal to the product of the function

F't by the latent heat of vaporization.

"The difference of the specific heats, or the latent heat of expansion

for any substance multiplied by the function F't, is equal to the

product of the expansion per degree at constant pressure by the

increase of pressure per degree at constant volume."

Since the last two coefficients are the same for all gases if equal

volumes are taken, Carnot concluded that: "The difference of the

specific heats at constant pressure and volume is the same for equal

volumes of all gases at the same temperature and pressure."

Taking the expression W = RT log _e r for the whole work done by a gas

obeying the gaseous laws pv = RT in expanding at a temperature T from

a volume 1 (unity) to a volume r, or for a ratio of expansion r, and

putting W' = R log _e r for the work done in a cycle of range 1 deg.,

Carnot obtained the expression for the heat absorbed by a gas in

isothermal expansion