a) The Mayer relation yields,
C_p = C_V + T \frac{\partial p}{\partial T} \frac{\partial V}{\partial T} .
According to the cyclic rule,
\frac{\partial p}{\partial T} \frac{\partial T}{\partial V} \frac{\partial V}{\partial p} = -1 thus \frac{\partial p}{\partial T} = – \frac{\partial V}{\partial T} \frac{\partial p}{\partial V} = \frac{α}{κ_T} .
Moreover,
\frac{\partial V}{\partial T} = α V .
Thus, the Mayer relation can be recast as,
C_p = C_V + \frac{α^2}{κ_T} V T .
b) Using the mathematical definition (4.77),
\frac{\partial f}{\partial y}\mid _z ≡ \frac{\partial f \Bigl(x(y,z),y\Bigr) }{\partial x (y,z)} \frac{\partial x (y,z)}{\partial y } + \frac{\partial f \Bigl(x(y,z),y\Bigr) }{\partial y } .
\frac{\partial f}{\partial z}\mid _y ≡ \frac{\partial f \Bigl(x(y,z),y\Bigr) }{\partial x (y,z)} \frac{\partial x (y,z)}{\partial z } . (4.77)
\frac{∂^2F (T, V)}{∂V^2} = \frac{d}{dV} \Biggl(\frac{dF\Bigl(T(S,V),V\Bigr) }{dV}\Biggr) = – \frac{dp\Bigl(T(S,V),V\Bigr) }{dV}
= – \frac{∂p}{∂T} \frac{∂T}{∂V}-\frac{∂p}{∂V} = – \frac{∂p}{∂T} \frac{∂}{∂V} \Bigl(\frac{∂U}{∂S}\Bigr) + \frac{∂}{∂V} \Bigl(\frac{∂U}{∂V}\Bigr) .
Thus,
\frac{∂^2F (T, V)}{∂V^2} = \frac{∂^2U}{∂V^2} – \frac{∂p}{∂T} \frac{∂^2U}{∂S ∂V } .
According to the cyclic rule,
\frac{\partial p}{\partial T} \frac{\partial T}{\partial S} \frac{\partial S}{\partial p} = -1 thus \frac{\partial p}{\partial T} = – \frac{\partial p}{\partial S} \frac{\partial S}{\partial T} .
which is then recast as,
\frac{\partial p}{\partial T} = – \frac{\frac{∂p}{∂S }}{\frac{∂T}{∂S } } = \frac{\frac{\partial }{\partial S}\Bigl(\frac{\partial U}{\partial V} \Bigr) }{\frac{\partial }{\partial S} \Bigl(\frac{\partial U}{\partial S} \Bigr)} = \frac{\frac{\partial^2 U}{\partial S \partial V} }{\frac{\partial^2 U}{\partial S^2} } .
Hence,
\frac{\partial^2 F(T,V) }{\partial V^2 }= \frac{\frac{\partial^2 U }{\partial S^2 }\frac{\partial^2 U}{\partial V^2} – \Bigl(\frac{\partial^2 U}{\partial S \partial V } \Bigr)^2}{\frac{\partial^2 U}{\partial S^2} } .
c) According to relation (6.32), the compressibility coefficient κ_T is expressed as,
\frac{\partial^2 F}{\partial T^2} = -\frac{\partial S}{\partial T} = – \frac{C_V}{T} \leq 0 and \frac{\partial^2 F}{\partial V^2} = -\frac{\partial p}{\partial V} = \frac{1}{κ_T V} ≥ 0 . (6.32)
κ_T = V \Bigl(\frac{\partial^2 F}{\partial V^2} \Bigr)^{-1} .
According to relation (6.22), the numerator in the expression for ∂^2F (T, V)/∂V^2 is positive and according to relation (6.14), the denominator is also positive. Thus, since the volume V is positive, the compressibility coefficient κ_T is positive, i.e. κ_T ≥ 0. Furthermore, since the temperature is positive, the Mayer relation requires that C_p ≥ C_V. Since C_V is positive according to relation (6.29), this in turn implies that C_p is positive, i.e. C_p ≥ 0.
\frac{\partial^2 U}{\partial S^2} \frac{\partial^2 U}{\partial V^2} – \Bigl(\frac{\partial^2 U}{\partial S ∂V} \Bigr)^2 ≥0 . (6.22),
\frac{\partial^2 U}{\partial S^2} ≥0 . (6.14)
\frac{\partial^2 U}{\partial S^2} = \frac{\partial T}{\partial S} = \frac{ T}{C_V}≥0 and \frac{\partial^2 U}{\partial V^2} = -\frac{\partial p}{\partial V} = \frac{ 1}{k_S V}≥0 . (6.29)