Question 11.9: Using the Berthelot equation of state given in Example 11.8,...

From Eq. (11.28) we have what we seek,

(\frac{∂c_v}{∂v})_T = T(\frac{∂^2p}{∂T^2 })_v

and from Example 11.8, the Berthelot equation of state can be written as

p = \frac{RT}{v −b} – \frac{a}{Tv^2}

so that

(\frac{∂p}{∂T})_v = \frac{R}{v −b} + \frac{a}{T^2v^2}

and

(\frac{∂^2p}{∂T^2 })_v = – \frac{2a}{T^3v^2 }

then,

(\frac{∂c_v}{∂v})_T = −\frac{2a}{T^2v^2}

and, to find an explicit c_v = c_v (T, v ) equation, the preceding equation can be integrated from a reference state specific volume v_o to give

c_v = − \frac{2a}{T^2} \int_{v_0}^{v} \frac{dv}{v^2} = 2a(v_0 − v)/(T^2v_0v) + f (T)

where f (T) is a function of integration. Note that c_v is independent of v only in the case where a = 0 in the Berthelot equation of state.