The ideal gas is having its specific heats constant. It follows the equation Pv = RT. But real gas is having different specific heats at different temperatures and pressures. The change is less in case of pressure variation.

We have from 1st law

dq = du + dw

or Tds = du + pdu

or ds = du / T + pdu / T

The specific internal energy u is assumed to be a function of T and v, i.e.,

u = f (T , u)

Differentiating equation (9.8) w.r.t. T when v is const.

Thus, u does not change when v changes at constant temperature.

We can also show if u = f (T, P);

Thus u does not change with P at aP T

constant temperature. It means u does not change unless T changes.

So u = f(T)

The equation (9.14) is valid for ideal gas only. Equation (9.14) is known as Joule’s law.

If u=f(T,v)

Du = c_{u} dt

Equation (9.15) is valid for ideal gas and for any process of ideal gas. For other substance

equation (9.15) is valid for constant volume process only.

Now h = u + Pv = u + RT

dh = du + RdT

dh / dt = c_{u} dt / dt + R =c_{u} + R

c_{p} = c_{u} + R =or c_{p} – c_{u} = R

To note dh = c_{P }dT also holds good for ideal gas even when pressure changes. but it is true for other substance for constant pressure process.

The ratio c_{p} / c_{u} is written as γ

C_{p} / c_{u }= γ

valid for ideal gas and is used in many computation.

Equations (9.16) and (9.17) give rise to

C_{p} = γR / γ -1 , and c_{u} = R / γ -1

The S.I. unit of c1, and C1 is KJ/ kg K.

Molar or molal specific heats:

c_{p}= M c_{p} and C _{v} =M c_{u}.

Where c_{p}, and c_{u}” are molar or molal specific heats at constant pressure and constant

volume respectively.

Value of γ : γ = 5 /3 for monoatomic gas

= 7 / for diatomic gas

= 4 /3 for polyatomic (more than two atoms) gas.

Thus 1 < γ < 4 /3

So γ depends on the molecular structure of the gas.

cp and c1, for ideal gas depend only on y and R. C _{v} and _{cp} are independent of temperature

and pressure of the gas.