Browse & Filter

 Starting from the first law of thermodynamics show that
 \[C_p - C_v = \Big[P +\Big(\frac{\partial U}{\partial V}\Big)_T\Big]
\Big(\frac{\partial V}{\partial T}\Big)_P\]
In the above \(C_p\) : heat capacity at constant pressure; \(C_v\) :
heat capacity at constant volume; \(U\) : internal energy: \(V\) is
the volume. For an ideal gas show that the above reduces
to \(C_p -C_v = Nk_B\) , where \(N\) is the number of molecules and
\(k_B\) is the Boltzmann constant.

Once the canonical partition function is known all properties of a system can be derived. Find the equation of state and the heat capacity of systems for which the partition funcion is

(a) \(Z=V^N(2\pi/\beta M)^{5N/2}\);

(b) \(Z= (V-Nb)^N (2\pi/\beta M)^{3N/2} e^{\beta N^2/V} \qquad a,b \text{ are constants}\).
Identify the systems.

Atlee Jackson*