The second term of this expression is represented graphically, as in fig. 94, by the limiting ratio of the area of the band A, B, B, An to the difference between the absolute temperatures corresponding to the upper and lower edges of that band.

Applying the thermodynamic function to the determination, in foot-pounds, of the whole quantity of heat d H, which must be communicated to one pound of the fluid in order to produce simultaneously the indefinitely small variation of temperature d«, and the indefinitely small variation of volume dv, we find,

which is the general equation of the expansive action of heat in a fluid.

If this expression be analyzed, it is found to consist of the following parts:

I. The variation of the actual heat of unity of weight of the fluid κατ.

II. The heat which disappears in producing work by mutual molecular actions depending on change of temperature and not on change of volume,

The lower limit of this integral is made to correspond to the state of indefinite rarefaction; that is, of perfect gas, in which

those actions are null. Let D

1

be the density, or weight of

v

unity of volume of the fluid; then we have, as a more convenient form of the integral,

d2 p

d r2 · d D...................... (3.) 0 D2

III. The latent heat of expansion,-that is, heat which disappears in performing work, partly by the forcible enlargement of the vessel containing the fluid, partly by mutual molecular actions

The heat, expressed in units of work, which must be communicated to unity of weight of a fluid to produce any given finite

changes of temperature and volume, is found by integrating the expression 2. Now that expression is not the exact differential of any function of the temperature and volume; consequently its integral does not depend solely on the initial and final condition of the fluid as to temperature and volume, but also upon the mode of intermediate variation of those quantities. The graphic representation of that integral is the indefinitely prolonged area MACBN in fig. 93.

247. Intrinsic Energy of a Fluid.—Another mode of analyzing the expression 2 of Article 246 is as follows:

I. The variation of actual heat, as before, kd

T.

II. The external work performed, p d v, represented by an elementary vertical band of the area VA ACB VB, fig. 93.

III. The internal work performed in overcoming molecular forces, viz. :

v p

.

• f and v dr + (-dp-p) de

T

pdv.

Now this last quantity is the exact differential of a function of the temperature and volume, viz. :

A given value of S expresses the work required to overcome molecular forces, in expanding unity of weight of a fluid from a given state, to that of perfect gas; and the excess of the actual heat of the fluid above this quantity, or

is the intrinsic energy of the fluid, or the energy which it is capable of exerting against a piston, in changing from a given state as to temperature and volume, to a state of total privation of heat and indefinite expansion. In fig. 93, the values of the intrinsic energy of the fluid in the conditions A and B are represented respectively by the indefinitely prolonged areas X VA A M, X VB BN. The quantity above denoted by S is the same with that denoted by the same symbol in Article 239. Let the suffixes a, b, denote the states of the fluid at the beginning and end of any given series of changes of temperature and volume, and Ha, b, the supply of heat from an external source necessary to produce those changes, expressed in foot-pounds; then

that is to say, the excess of the heat absorbed above the external work

performed is equal to the increase of the intrinsic energy; so that this excess depends on the initial and final states only, as already shown in Article 239.

248. Expression of the Thermodynamic Function in Terms of the Temperature and Pressure.—The volume of unity of weight of a fluid v, its expansive pressure p, and its absolute temperature, form a system of three quantities, of which, when any two are given, the third is determined. In the preceding Articles, the volume and temperature are taken as independent variables, and the pressure is expressed as a function of them. In some investigations it is convenient to take the pressure and temperature as independent variables, the volume being expressed as their function. The following expression of the thermodynamic function in terms of this pair of independent variables is taken from an unpublished paper, which has been in the hands of the Royal Society of Edinburgh since 1855 (see their Proceedings for 1855, p. 287). Let before, be the absolute temperature of melting ice; Po vo the product of the pressure and volume of unity of weight of the fluid, in the perfectly gaseous state, at that temperature (of which quantity examples are given in Table II., at the end of the volume); then

as

By the aid of the above equation, and of the following well known theorem:

Pa

v Pb

+

all the equations of the preceding sections are easily transformed. The graphic representation of the quantity denoted by the second

point A, on the diagram, whose co-ordinates are O V1

1

=

= v, and

OP=V1 A1 p; the absolute temperature being T. Let A1 T1 be the isothermal curve of 7. Then the indefinitely extended area XOPA, T, is what is represented by

fvdp.

Let A, T2 be the isothermal curve corresponding to the absolute temperature - AT, and cutting A, POX in A, Then the symbol

pd v

d p

represents the limit towards which the quotient

area To A, A, T,

A T

approximates, when A is indefinitely diminished.

By using the form of the thermodynamic function explained in this Article, the general equation of the expansive action of heat in a fluid is made to take the following form:

:

a form which is convenient in cases where the pressure and its mode of variation are amongst the primary data of the problem. It will be shown in a subsequent Article, that the constant part

k +
Po vo
Το

of the co-efficient of dr, is the dynamical specific heat of the fluid, in the state of perfect gas, under a constant pressure.

249. Principal Applications of the Laws of the Expansive Action of Heat. The relation between the temperature, pressure, and volume of one pound of any particular substance being known by experiment, the principles of the preceding Articles serve to compute the quantity of heat which will be absorbed or rejected by one pound of that substance under given circumstances; and conversely, in some cases when the quantities of heat absorbed or rejected under given circumstances are known by experiment, the same principles serve to determine relations between the temperature, pressure, and density of the substance. The chief subjects to which the principles of the expansive action of heat are applicable, are the following:-Real and apparent specific heat; the heating and cooling of gases and vapours by compression and expansion; the

velocity of sound in gases; the free expansion of gases; the flow of gases through orifices and pipes; the latent and total heat of evaporation of fluids, the latent heat of fusion; the efficiency of thermodynamic engines. The last of those subjects is that to which this treatise specially relates; but in order to make it intelligible, it is necessary in the first place to give a summary of the principles of the subjects enumerated before it.

250. Real and Apparent Specific Heat.-These terms have been explained in a previous Article. The symbolical expression for the apparent specific heat of a given substance, stated in units of work per degree of temperature in unity of weight, is as follows:

In which the term k is the real specific heat, or that which actually makes the substance hotter, being a constant quantity; while the other term represents the heat which disappears in performing work, internal and external, for each degree of rise of temperature.

The co-efficients

d: Ф dr

and

d.

d U

d, represent respectively the ατ

complete rates of variation with temperature of the thermodynamic function and heat-potential, under the circumstances of the particular case. With respect to liquids and solids, it is impossible to regulate artificially the mode of variation of the thermodynamic function to an extent appreciable in practice. For substances in these states, the apparent specific heat increases with rise of temperature at a rate which is slow, but which appears, as theory would lead us to expect, to be connected with the rate of expansion. For gases, the mode of variation of the thermodynamic function with temperature may be regulated artificially in an arbitrary manner, so as to vary the apparent specific heat in an indefinite number of ways. It is customary, however, to restrict the term "Specific heat" in speaking of gases, to two particular cases; that in which the volume is maintained constant during the variation of temperature, and that in which the pressure is maintained constant, as formerly explained in Article 210. The specific heat at constant volume, is expressed as follows, in units of work per degree, being deduced from the expression for the thermodynamic function in Article 246, equation 1 :—

•v d2 p 72

K, = k + ↑

dr2

dv... (2.)