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absolute scale



0° C 10° 20" 30° 40° 50° 60° 70° 80° 90°

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0.000° 0.003 0.006 0.007 0.008 0.008 0.007 0.006 0.004 0.002 0.000

0.000° + 0.006 + 0.006 + 0.007 + 0.004

0.000 0.003 0.006 0.007 0.005 0.000


which the pressure at the freczing point of the theory of differential equations, that the water is that due to one metre of mercury, and equation so formed can always be multiplied by which is graduated according to the Centigrade some factor (the "integrating factor”) which scale so as to read 0° at the freezing point of shall cause it to become an exact differential of water and 100° at the boiling point, (1) there some function of the variables whose differmust be a constant term of 273.10° added to the entials it contains; but, so far as pure mathereading of the thermometer in order to obtain matics is concerned, it is impossible to say, in the reading of the instrument on the absolute advance, what that integrating factor will be, scale, and also (2) a small variable term, whose or what the nature of the function may prove values are given in the second and third columns to be, of which the modified expression is the of the accompanying table.

exact differential. By the aid of the second

law of thermodynamics, however, it may be Variable part of the shown that the reciprocal of the absolute temcorrection to reduce to the

perature at which the infinitesimal transformation takes place is always an integrating factor

of the differential equation in question. In Nitrogen Hydrogen thermometer | thermometer

other words, having written the differential expression for the quantity of heat absorbed by the body, we know that we only have to divide it by the absolute temperature of the body, in order to cause it to become the exact differential of some function of the variables. The function whose existence is thus indicated is called the "entropy of the body, and in the study of thermodynamics this function is a very convenient thing, because its introduction simplifies the treatment of many problems. The main difficulty that students experience in connection

with it is the difficulty of assigning to entropy The corrections here given are different any precise physical significance. It is probably from those usually quoted, but it is believed that beiter not to try to give any physical interpretathey are more accurate. In the case of the tion of this sort; for it is sufficient for many hydrogen thermometer, it will be observed that

purposes merely to recognize the existence of from 0° C. to 50° C. the corrections are positive, the function, the very fact of its existence sugwhile from 50° C. to 100° C. they are negative. gesting certain mathematical transformations However improbable this change of sign may which are exceedingly useful. The suggestion appear, it is certain that the differences between has sometimes been made, that it may prove to the readings of the hydrogen and nitrogen ther- be possible to devise an instrument which shall mometers, as deduced from the corrections

enable us to measure the value of the entropy here given, correspond very faithfully with the of a body directly, just as a thermometer measactual differences as observed at the Interna- ures the value of a temperature. If this could tional Bureau of Weights and Measures at Paris. be done, the imagination of the student of Owing to the smallness of the variable part thermodynamics would doubtless be greatly of the correction, it is usualy, in writings upon assisted; but it does not appear that the hope engineering topics and upon thermodynamics of discovering an instrument of this sort is at generally, to take note only of the large con- all well founded. stant term that is to be added, and to treat In studying the thermodynamic behavior of absolute temperature as though it were identical

a body, the state of the body is defined by givwith the temperature as read from a constant

ing as many of its measurable attributes as volume gas thermometer, save for the addition

may be necessary in order to fix the condition of the constant, 273.10° C. In other words, if t

of the body absolutely. These measurable atis the absolute temperature corresponding to a tributes are represented by letters, and are taken given reading T on the scale of a gas ther- as independent variables. Then, by treating these mometer, it is customary to assume that

iridependent variables by known mathematical AT+273.10°, if the thermometer is graduated methods, we can deduce certain conclusions on the Centigrade plan, or =T+459.58°, if the

with regard to the behavior of the body itself. graduation is according to Fahrenheit.

Theoretically, there is no reason why the numStudents of thermodynamics are often ber of independent variables may not be as greatly confused by the introduction of the great as we please; but in all of the more imidea of entropy); and while this subject re- portant applications of thermodynamics it is quires the higher mathematics for its adequate found to be sufficient to take two independent discussion, a few words may be given to it variables. In the case of a gas, for example, here. When a body whose state at any given it is usually, sufficient to take two such variinstant is completely defined by two independ- ables, provided the gas is in a quiescent conent variables, undergoes any infinitesimal but dition and homogeneous throughout. When the reversible change on account of corresponding possibility of internal motions is admitted or infinitesimal changes in the two defining vari- the gas differs in composition or in other reables, it will, in general, absorb or reject a spects in its different parts, it is necessary to certain infinitesimal amount of heat, and it is take more than two variables; but these cases easy to form a differential equation of the first will not be considered in the present article. order and degree, which will express the quan- (Gibbs, in his classical papers on the Equilibtity of heat that is absorbed; the expression rium of Heterogeneous Substances, published being taken negative, if there is rejection of in the Transactions of the Connecticut Acadheat instead of absorption. We know, from cmy of Sciences, just previous to 1880, dis

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cussed many of the problems that arise when stants. If a and b are both zero, this equation the composition of the substance under con- reduces to the form previously given, and the sideration departs from uniformity and homo- same is true if V is very large indeed (that is, geneity in any respect). Some latitude is per- if the gas is very rare), since in that case the missible as to the variables that are selected effects of the small constants a and b are negfor representing the state of the gas, but for ligible. If we assign to T any constant value the present we shall consider the state as being that we please, we may, m Van der Waals' thoroughly defined when we know the pressure, equation, trace all the possible relations that P, that it exerts upon each unit of area of the P and V can have, at this one temperature. wall of its containing vessel and also the vol- That is, when a, b and R are known, and we ume, V, occupied by each unit of its mass. have assigned a fixed arbitrary value to T, we Other attributes of the gas may indeed vary, as may then select any number of values of V, well as P and V; but if P and V are really suffi- and compute the value of P that corresponds cient to define the state of the gas completely, to each one of them. If we plot the values so then these other attributes that are capable of variation at the same time must all be expressible as functions of the two variables P and V. The temperature of the gas is one of the most notable physical attributes which is capable of variation; and it follows that there must be a relation connecting the temperature,

C.P. T, with the variables P and V. This equation, which is called the characteristic equation or sometimes the elastic equation, may be written, tentatively, in the general form T= F(P, V). While we know that an equation of

B this nature must exist, we do not know the

MIXED exact form of the function F for any actual substance. For gases, however, we know its approximate form, throughout certain ranges of the variables P and V. Robert Boyle showed that so long as the temperature of the gas is kept constant and the gas is not too highly compressed nor too near to its point of liquefaction, the volume varies very nearly as the

VOLUME (V) reciprocal of the pressure, and Charles discov- Fig. 1.- Isothermals of Van der Waals' Equation. ered (to express it in modern language) that so long as the pressure upon the gas remains computed, by laying off horizontal distances to constant, the volume is nearly proportional to represent the values of V, and vertical distances the absolute temperature. Taking these two to represent the corresponding values of P, we laws into account, it is evident that the form shall obtain a series of points representing the of the function F must be such that, for such

various states that the gas is capable of asvalues of the variables P and V as prevail un

suming, while I keeps its fixed value, and if der the conditions in which the laws of Boyle we make the calculated points numerous enough, and Charles are nearly true, we must have we may draw through them a curved line,


which may be taken to represent the continuous T=kPV, or P

series of states through which the gas passes, V

as the pressure is continuously varied, while

the temperature remains constant. Such lines T being the absolute temperature and k and R

are called (isothermals, on account of the conbeing constants whose values are to be deter

stancy of the temperature along them. Sevmined by experiment. For many practical pur

eral such lines, as computed for as many difposes, this relation between P, V and T is suffi

ferent values of T from Van der Waals' equaciently exact. When the gas is highly com

tion, are shown in Fig. 1. If the temperature is pressed, however, or when it is near to the high (as at T.), the isothermal line may be point of liquefaction, the foregoing equation is indistinguishable in form from the correspondfound to depart very materially from the facts

ing line as computed from the elastic equation as experimentally observed. Many attempts of Boyle and Charles. If, on the other hand, have been made to find a


the temperature is sufficiently low, as is indieral law connecting the temperature, pressure cated at Tı, the isothermal line will have a and volume, and such equations have been given very different shape. To trace the significance by Rankine, Van der Waals, Clausius, Sarrau of this shape, let us begin at the right-hand end and other writers. As the equation of Van of the isothermal Tı, and see what happens der Waals has been of special service and has when the pressure upon the gas is continuously figured to a great extent in the thermodynami- increased. As the pressure grows greater, the cal literature of recent times, it may be cited as

volume of the gas diminishes; but there is no an example of the attempts that have been made

notable change of any other sort until a certo firid a superior form of elastic equation. The

tain point A is reached. When this point is cquation in question has the form

attained, any attempt to further increase the RT

pressure merely results in the condensation of P=

a part of the gas; the pressure remaining con. V-b

stant (as indicated by the horizontal line ADB) where R, T, V and P have the same signifi- until

, at B, the gas is entirely condensed into cance as before, and a and b are very small con- the liquid form. Further application of pressure,



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then causes but a slight reduction of volume; often convenient, however, to assume the exa fact which is indicated by the steepness of istence of a gas of this sort, for the purpose of the isothermal line above B. We have here illustrating general principles, or of obtaining described what actually happens when the gas approximate solutions of thermodynamical is compressed along the isothermal Tı; but problems; and the ideal (but non-existent) gas it must be noted that the plot of this which fulfils the relation of Boyle and Charles isothermal from Van der Waals' equation does absolutely and under all circumstances is comnot give a straight part, ADB, but a reversed monly called a "perfect gas, though ideal gas curve between A and B, as indicated by the would appear to be a preferable name. In apdotted line. If we could actually make the gas plying the conception of a perfect gas, it is follow this dotted line, we could cause it to customary to assume the further condition that pass from the gaseous condition into the liquid when a gas of this sort changes its volume at condition, without any discontinuity in state; constant temperature, the heat that it absorbs is that is, in such a manner that it would never exactly equivalent to the external work that the be partly liquid and partly gaseous, and so gas does, in expanding against the external that we should not be able to see when the pressure that the containing vessel exerts upon conversion from one state to the other took it. In other words, it is customary to assume place. It can be shown, however, that the states that the perfect gas, in addition to obeying the of the gas which correspond to the dotted laws of Boyle and Charles perfectly, is also so part of the isothermal are essentially unstable, constituted that its internal energy depends upon so that the attempt to make the gas follow the nothing but the temperature of the gas. The dotted portion of the theoretical isothermal is characteristic equations of Van der Waals and like trying to balance a pyramid upon its point. others are decided improvements upon the The line ADB which the gas actually follows equation of Boyle and Charles, and they reprein preference to the double loop, is in such a sent, very well, the nature of the phenomena position that the areas of the two shaded loops that occur in a gas in the vicinity of the critical are equal, as was first shown by Maxwell. A point. None of them takes any account, howportion of the dotted loops in the immediate ever, of the fact that a body is capable of exvicinity of A and B can be actually realized isting in the solid state, as well as in the liquid in the laboratory, by careful experiment; but and gaseous states; and the first characteristic





Fig. 2.

FIG. 3.

FIG. 4. the instability speedily becomes too marked to equation complete enough to take the solid state permit of the experiments being carried far, into account also, has yet to be proposed. At temperatures intermediate between Ti and When a body passes from one condition of T. the isothermals have a character intermedi- pressure and density to another, it either ab. ate between those shown for those tem- sorbs or emits heat, unless certain special condiperatures. As we proceed upward from T1, tions are fulfilled. To avoid circumlocution, the loops on the isothermals grow less and less we may speak of it as always "absorbing heat; pronounced, as is indicated by the partial, the emission of heat being considered to be dotted isothermal, and we presently arrive at merely a case of negative absorption. Suppose, one particular isothermal, T2, where the loops for example, that a body is in the state correjust cease to exist. At any temperature higher sponding to A, in Fig. 2; and for definiteness let than T, it is, therefore, impossible to liquefy us suppose that the body under consideration is the gas by the application of any pressure what- a gas, although the reasoning will apply equally ever, no matter how great. Hence the tempera- well to a liquid or to a homogeneous, isotropic ture T, is the critical temperature) of the gas. solid. The height of A above the horizontal (See Critical Point). There is one point on reference line then represents, on some conthis critical isothermal (marked "C. P.”), at venient scale, the pressure to which each unit which the isothermal iş precisely horizontal and of the bounding surface of the gas is exposed; where it also has a point of infection; and this and the distance of A from the vertical referpoint corresponds to the critical point of the ence line at the left corresponds, upon some gas; its temperature being the critical tempera- other convenient scale, to the volume occupied ture, its volume the critical volume and its

by a unit mass of the gas. Now if the gas be pressure the critical pressure.

caused to pass from the condition represented Although gases obey the characteristic equa- by A to that which is represented by B, by passtion of Boyle and Charles very closely when ing through all the intermediate conditions that they are not too highly compressed and not too are represented by the points that are intermedinear to condensation, there is no gas which ate to A and B on the line ACB, the gas is said obeys it rigorously under all conditions. It is to pass from the state A to the state B along

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the "path" ACB. In genera, a change of this and if we cause it to pass from Ti to T, along sort will be accompanied by an absorption of the vertical line AB, we are heating it while its heat; the heat which is absorbed being partly volume remains constant. If the difference in expended in increasing the internal energy of temperature between Ti and Te is one degree, the gas, and partly in the performance of ex- and the mass of the gas is (as we have already ternal work. It is a consequence of the first law assumed) unity, then the quantity of heat abof thermodynamics that the change in the in- sorbed along AB is the specific heat ternal energy of the gas is entirely independent stant volume, and the quantity absorbed along of the shape of the path ACB, and depends only AC is the "specific heat at constant pressure." upon the positions of the points A and B. That That these two specific heats are really different portion of the absorbed heat which goes to will be evident from the fact (1) that the inincrease the internal energy of the gas, there- ternal energy of the gas in the states B and C fore, depends upon nothing but the positions of are not necessarily the same, unless the gas is a A and B. The case is different, however, with "perfect gas”; and also from the fact (2) that that portion of the absorbed heat which is con- the heat that is absorbed along the path AC has sumed in the performance of external work. to be partially expended in doing the external Consider, for example, the state of the gas at work represented by the shaded area; while the point C. The pressure upon the gas, per along the path AB there is no external work unit of area of the containing vessel, is repre

done. sented by the vertical line CE; and when the The fact that in an ideal reversible heat envolume of the gas increases by the slight gine the efficiency does not depend at all upon amount EF, the external work that the gas does the nature of the substance whose expansion is represented by the product of the pressure does the work, is sometimes hard for the beand the increase in volume; that is, it is repre- ginner in thermodynamical reasoning to undersented by the area of the little rectangle CDFE. stand, for the reason that objections occur to We may regard the area ACBNM as made up him which appear to controvert the principle, of an infinite number of infinitesimal rectangles, and to be themselves unanswerable. There is each of which is typified by the little rectangle an answer, however, to every objection that can that is shown; and hence it follows that the be urged. One of the commonest of the diffitotal quantity of external work done by the gas culties is this: In a steam engine, water is as it passes from the condition A to the condi- pumped into the boiler, and is then evaporated tion B, along the path ACB, is represented by by the expenditure of a large amount of heat. the area included between the curve ACB and The steam is next passed to the cylinder of the the straight lines AM, MN and NB. Obviously engine and expanded, after which it is turned this area depends upon the form of the path into the condenser and re-converted into water. ACB; and hence the external work that is done The quantity of heat which is expended upon by the gas also depends upon the form of that the water in merely converting it into steam path, and so also does that part of the heat (and which is known as the latent heat of absorbed along ACB, which is consumed in per- vaporization”) appears to be wasted in large forming this work.

measure, because the greater part of it is not When a gas (or other body) describes a converted into mechanical energy by the engine, closed path, such as is shown in Fig. 3, and but is merely rejected into the condenser. Enreturns finally to its original state, then the gines have been designed and built, in which internal energy of the gas Iso returns to its the water that is commonly used is replaced by original value; and the total quantity of heat some other liquid (such as ether or carbon that is absorbed by the gas during its passage

disulphide) which has a much smaller latent around the closed path is therefore represented heat of vaporization, in the belief that this apentirely by the external work that the gas does. parently large source of loss could be avoided; That is, it is represented by the area of the but such engines have invariably proved disapclosed path, as shown shaded in Fig. 3. A pointing, any trifling superiority that they may closed path of this sort is called a "cycle, and have shown from time to time being attributable the consideration of cycles of various kinds is to other causes than the smaller latent heat of very important in many branches of thermo- vaporization of the working fluid. The reason dynamical reasoning. If AB and DC, in Fig. 3, for this is, that there is an intimate relation represent isothermal lines, and AD and BC between the pressure of a saturated vapor at a represent adiabatic lines (that is, lines along given temperature, and the latent heat of vaporwhich there is no absorption or rejection of heat ization of the liquid. This relation is sometimes by the gas), then the cycle ABCDA is called known as the second thermodynamic relation,” a "Carnot cycle, because it is the kind of a and sometimes as “Clapeyron's equation. The cycle that Carnot imagined his ideal, reversible elucidation of this matter requires a knowledge engine to describe. (See Carnot's principle, of the infinitesimal calculus, and reference must enunciated in the earlier part of this article). be made for it to the standard works on

When a gas is heated from a temperature thermodynamics. Much of the practical experiTi to another temperature T2, the quantity of mental work that has been done upon the heat absorbed in the process will depend upon hot-air engine has probably been inspired by ignothe precise way in which the passage from one rance of the existence, or at least of the sigof these temperatures to the other is effected. nificance, of this "second thermodynamic relaThus let A, in Fig. 4, represent the initial state tion.” See Heat; SPECIFIC HEAT; THERMOMEof the gas, and let the curved lines, Ti and T2, TER; THERMOMETRY; GASES, KINETIC THEORY represent the isothermals corresponding to the OF; and other similar articles in this encyclotemperatures Ti and T2. If we cause the gas to pedia. pass from the isothermal Ti to the isothermal Bibliography.-- Bryan, Thermodynamics'; T: along the horizontal line AC, we are heating Clausius, Mechanical Theory of Heat' the gas while its pressure remains constant; (Browne's translation); Findlay, The Phase

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Rule); Mach, Principien der Wärmelehre); force will be proportional to the difference in
Magie, The Second Principle of Thermody- temperature between the two junctions. But it
namics'; Maxwell, Theory of Heat); Poincaré is also evident that when the average tem-
Thermodynamique): Preston, (Theory of perature of the two junctions is such that the
Heat'; Tyndall, Heat as a Mode of Motion.' relation

b+c(T, +T)=0 Director Technical Research, The Travelers

is fulfilled (or, in other words, when the averInsurance Company, Hartford, Conn.

age temperature of the two junctions is numerTHERMO-ELECTRICITY. If an elec- ically equal to -b/2c), there will be no thertrical circuit is constructed partly of one metal mo-electromotive force in the circuit (and, and partly of another, and one of the points of

therefore, no current), no matter what the junction between the dissimilar metals is heated difference in temperature between the two while the other is kept cool, a current of elec- junctions may be. This average temperature, tricity will be caused to flow in the circuit. for which there is no thermo-electric effect in This fundamental fact was discovered by See- a circuit, has a definite value for every pair of beck in 1821. The electricity thus generated metals, and is known as the “neutral temis not in any wise different from that which perature for that pair. The values of the is generated by an ordinary galvanic battery; constants b and c, in the foregoing formulæ, but on account of its mode of production it is could be determined experimentally, and called "thermo-electricity.” The electromotive recorded in tabular form for various pairs of force that is set up in a circuit under the cir- metals. It is usual, however, to record the cumstances here described is always quite small, experimental data in a somewhat different manand its intensity depends (1) upon the nature ner, as we proceed to explain. If the average of the metals of which the circuit is composed, of the two temperatures T, and T, be denoted (2) upon the difference in temperature between by To, then the formula for the effective electhe two junctions where the dissimilar metals tromotive force, F, may be written come together and (3) upon the average tem

F=(T2-T,) (b +2cT.). perature of these junctions. For the sake of definiteness, let the two metals of which the

The constants b and c refer, it will be uncircuit is composed be designated by the letters

derstood, to a particular pair of metals; but it X and Y. The phenomena of thermo-electricity

is found that their values can be satisfactorily may then be described in the following mathe

represented as the differences between constants matical language: It is known from experiment

which can be stated for the two metals septhat when the two metals X and Y are brought arately. Thus b can be expressed in the form together so that their point of contact has the

b=B' B" and 2c be expressed in temperature T, an electromotive force exists be

the form 2c=('C"; B' and C being tween the two, which tends to send a current

constants whose values depend solely upon (say) from X into Y; and it is also known

the metal X, and B" and C" being constants that the magnitude of this electromotive force

whose values depend, in a similar manner, can be expressed as a parabolic function of the solely upon the metal Y. The expression for temperature, T. Thus if E is the electromotive

the effective electromotive force F can, thereforce in question, the facts of experiment can

fore, be written thus: be adequately expressed by a relation of the F= (T.-T.) [(B'- B") + (C-C") T.). form Ė=a+bf + ct'; where a, b and c

The values of the constants B and C for the are constants whose values depend upon the natures of the metals X and Y. In the actual

different metals vary somewhat with the physcircuit there are necessarily two junctions across

ical conditions of the metals; but the data which electromotive forces of this character

given in the accompanying table will show the

general nature of these constants, and will also exist. Let the temperatures of these junctions be respectively T, and T.. Then the foregoing proximation, the actual magnitude of the ther

suffice to represent, with some degree of apformula shows that across the junction whose temperature is T. there is an electromotive

mo-electric effects that may be expected from

circuits composed of the metals there repreforce of intensity, E. =a + b1; + cT’ı, tending

sented. In applying this table, temperatures to send a current from X into Y; and across the junction whose temperature is T, there is

are supposed to be expressed on the ordinary a similar electromotive force of intensity

Centigrade scale, which define the freezing

point of water to be 0°, and the boiling point E,= a +bTz + cT?

to be 100°; and the results are expressed in also tending to send a current from X into Y. hundred-millionths of a volt, so that to reThese electromotive forces being opposed to duce them to volts it is necessary to divide each other, so far as the production of a current them by 100,000,000. around the circuit is concerned, the effective To illustrate the use of this table, let us electromotive force around the circuit is the compute the electromotive force of a circuit difference between E, and Ez; and if we denote composed of iron and copper, when one of the this effective electromotive force by the letter junctions is kept at 0° C., and the other at F, we have

100°C. For iron we have B=+ 1734 and F=B-Eb (1,-1) + c(T; -T),

C=-4.87; and for copper we have B=+ 136

and C=+0.95. Hence we see that for this F=(T.-T.) (b + c(T. + T.)]. pair of metals b= + 1734 — 136= + 1598, and

20 =

4.87 -0.95 From this last equation it is evident that so

- 5.82. The thermo-eleclong as the average temperature of the two

tromotive force in the circuit is therefore, junctions is constant (or, in other words, so

F=(T,T.) (1593 — 5.827.). long as T.+ Tis constant), the electromotive But we have assumed that T = 100° and T. =


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