Part 31

r + δr = N(t + dt + 267) log v.

Subtracting equation (2), we have

δr = N log vdt. (3)

Let e be the quantity of heat employed to maintain the temperature of the gas constant during its dilatation. According to the reasoning of page 69, δr will be the power developed by the fall of the quantity e of heat from the degree t + td to the degree t. If we call u the motive power developed by the fall of unity of heat from the degree t to the degree zero, as, according to the general principle established page 68, this quantity u ought to depend solely on t, it could be represented by the function Ft, whence u = Ft.

When t is increased it becomes t + td, u becomes u + du; whence

u + du = F(t + dt).

Subtracting the preceding equation, we have

du = F(t + dt) − Ft = F′tdt.

This is evidently the quantity of motive power produced by the fall of unity of heat from the temperature t + dt to the temperature t.

If the quantity of heat instead of being a unit had been e, its motive power produced would have had for its value

edu = eF′tdt. (4)

But edu is the same thing as δr; both are the power developed by the fall of the quantity e of heat from the temperature t + dt to the temperature t; consequently,

edu = δr,

and from equations (3), (4),

eF′tdt = N log vdt;

or, dividing by F′tdt,

e = (N)/(F′t) log v = T log v.

Calling T the fraction (N)/(F′t) which is a function of t only, the equation

e = T log v

is the analytical expression of the law stated pp. 80, 81. It is common to all gases, since the laws of which we have made use are common to all.

If we call s the quantity of heat necessary to change the air that we have employed from the volume 1 and from the temperature zero to the volume v and to the temperature t, the difference between s and e will be the quantity of heat required to bring the air at the volume 1 from zero to t. This quantity depends on t alone; we will call it U. It will be any function whatever of t. We shall have

s = e + U = T log v + U.

If we differentiate this equation with relation to t alone, and if we represent it by T′ and U′, the differential coefficients of T and U, we shall get

(ds)/(dt) = T′ log v + U′; (5)

ds/dt is simply the specific heat of the gas under constant volume, and our equation (1) is the analytical expression of the law stated on page 86.

If we suppose the specific heat constant at all temperatures (hypothesis discussed above, page 92), the quantity ds/dt will be independent of t; and in order to satisfy equation (5) for two particular values of v, it will be necessary that T′ and U′ be independent of t; we shall then have T′ = C, a constant quantity. Multiplying T′ and C by dt, and taking the integral of both, we find

T = Ct + C{1}_;

but as T = N/F′t, we have