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Question 5.8: The thermal response of a homogeneous system subjected to an...

The thermal response of a homogeneous system subjected to an infinitesimal heat transfer δQ is characterised by coefficients defined in equations (5.4) and (5.17) when either the state variables (T, V) or (T, p) are used.
a) Find a relation between the latent heat of expansion L_V (T, V) and the latent heat of compression L_p (T, p).
b) Express the latent heat of compression L_p (T, p) in terms of the specific heat at constant volume C_V (T, V) and the specific heat at constant pressure C_p (T, p).

δQ = T dS(T, V) ≡ C_V (T, V) dT + L_V (T, V) dV .                (5.4)

δQ = T dS(T, p) ≡ C_p (T, p) dT + L_p (T, p) dp .                  (5.17)

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a) The thermal response is written in terms of the temperature T and the volume V as,

δQ = C_V (T, V) dT + L_V (T, V) dV .

The thermal response is written in terms of the temperature T and the pressure p as,

δQ = C_p (T, p) dT + L_p (T, p) dp .

which can be recast in terms of the temperature T and the volume V as,

δQ = C_p (T, p) dT + L_ p (T, p) \biggl(\frac{\partial p (T,V) }{\partial T }dT + \frac{\partial p (T,V) }{\partial V }dV\biggr) = \biggl(C_p (T,p) + L_p (T,p) \frac{\partial p (T,V)}{\partial T} \biggr)dT + L_p (T,p) \frac{\partial p (T,V)}{\partial V} dV .

The identification of the terms multiplying the volume differential dV in the two expressions for the thermal response δQ written in terms of the temperature T and the volume V yields the relation,

L_V (T, V) = L_p (T, p) \frac{∂p (T, V)}{∂V} .

b) The identification of the terms multiplying the temperature differential dT in the two expressions for the thermal response δQ written in terms of the temperature T and the volume V yields the relation,

 C_V (T, V) = C_p (T, p) + L_p (T, p) \frac{∂p (T, V)}{∂T} .

Using relation (4.80) for the inverse of a partial derivative, this result can be recast as,

\frac{\partial x (y,z)}{\partial y} = \biggl(\frac{\partial y (z,x) }{\partial x  }\biggr)^{-1}

\frac{\partial x (y,z)}{\partial z} = \biggl(\frac{\partial z (x,y) }{\partial x  }\biggr)^{-1} .                (4.80)

L_p (T, p) = \biggl(C_V (T, V) − C_p (T, p)\biggr) \frac{\partial T (p,V) }{\partial p  } .

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