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IDEAL AND REAL GASES 381

dharm

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It may be seen that as Joule’s law for an ideal gas states u = f(T), then

cv =

du
dT ...(8.13)
Since h = u + pv, Boyle’s law, pV = f(T) and Joule’s law u = f(T) together show, h = f(T) and
by similar argument to the above it may be seen that :

cp =

dh

dT ...(8.14)

Further as h = u + pv, then h = u + RT and by differentiation

dh

dT =

du

dT + R

Substitution from eqns. (8.13) and (8.14) gives,

cp = cv + R i.e., cp – cv = R ...(8.15)

If expressed in terms of molar quantities then eqn. (8.15) becomes

Cp – Cv = R 0 ...(8.16)

where Cp and Cv are molar specific heat capacities.
Equations for specific heat capacities of ideal gases
Since both u and h are functions of temperature, the equations to cp and cv must also be
functions of temperature. They are usually expressed in a form :
cp = a + KT + K 1 T^2 + K 2 T^3 ...(8.17)
cv = b + KT + K 1 T^2 + K 2 T^3 ...(8.18)
where a, b, K, K 1 and K 2 are constants. Values of specific enthalpy etc. are then obtained by
integration.

8.6. Real Gases

It has been observed that when experiments are performed at relatively low pressures and
temperatures most of the real gases obey Boyle’s and Charle’s laws quite closely. But the actual
behaviour of real gases at elevated pressures and at low temperatures deviates considerably.
The ideal gas equation pv = RT can be derived analytically using the kinetic theory of gases
by making the following assumptions :
(i) A finite volume of gas contains large number of molecules.
(ii) The collision of molecules with one another and with the walls of the container are
perfectly elastic.
(iii) The molecules are separated by large distances compared to their own dimensions.
(iv) The molecules do not exert forces on one another except when they collide.
As long as the above assumptions are valid the behaviour of a real gas approaches closely
that of an ideal gas.

8.7. VAN DER WAALS’ EQUATION

Van der Waals’ equation (for a real gas) may be written as :

p a

v

HFG +^2 IKJ (v – b) = RT ...[8.19 (a)]

The constants a and b are specific constants and depend upon the type of the fluid considered,

‘v’ represents the volume per unit mass and R is the gas constant.