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 LV (T, V) and the

Question

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 [latex]L_V (T, V)[/latex] and the latent heat of compression [latex]L_p (T, p)[/latex].
b) Express the latent heat of compression [latex]L_p (T, p)[/latex] in terms of the specific heat at constant volume [latex]C_V (T, V)[/latex] and the specific heat at constant pressure [latex]C_p (T, p)[/latex].

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

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

Accepted Answer

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

[latex] δQ = C_V (T, V) dT + L_V (T, V) dV [/latex].

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

[latex] δQ = C_p (T, p) dT + L_p (T, p) dp [/latex].

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

[latex] 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 [/latex].

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,

[latex] L_V (T, V) = L_p (T, p) \frac{∂p (T, V)}{∂V} [/latex].

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,

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

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

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

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

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

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

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