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.
Du = cu 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 = cu dt / dt + R =cu + R
cp = cu + R =or cp – cu = R
To note dh = cP dT also holds good for ideal gas even when pressure changes. but it is true for other substance for constant pressure process.
The ratio cp / cu is written as γ
Cp / cu = γ
valid for ideal gas and is used in many computation.
Equations (9.16) and (9.17) give rise to
Cp = γR / γ -1 , and cu = R / γ -1
The S.I. unit of c1, and C1 is KJ/ kg K.
Molar or molal specific heats:
cp= M cp and C v =M cu.
Where cp, and cu” are molar or molal specific heats at constant pressure and constant
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.