Answer
See explanation
Work Step by Step
The relations for entropy differential are $$
\begin{aligned}
& d s=c_v \frac{d T}{T}+\left(\frac{\partial P}{\partial T}\right)_v d v \\
& d s=c_p \frac{d T}{T}-\left(\frac{\partial v}{\partial T}\right)_P d P
\end{aligned}
$$ For fixed $s$, these basic equations reduce to $$
\begin{aligned}
& c_v \frac{d T}{T}=-\left(\frac{\partial P}{\partial T}\right)_v d v \\
& c_P \frac{d T}{T}=\left(\frac{\partial v}{\partial T}\right)_P d P
\end{aligned}
$$ Also, when $s$ is fixed, $$
\frac{\partial v}{\partial P}=\left(\frac{\partial v}{\partial P}\right)_x
$$ Forming the specific heat ratio from these expressions gives $$
k=-\frac{\left(\frac{\partial v}{\partial T}\right)_P\left(\frac{\partial T}{\partial P}\right)_v}{\left(\frac{\partial v}{\partial P}\right)_s}
$$ The cyclic relation is $$
\left(\frac{\partial v}{\partial T}\right)_P\left(\frac{\partial P}{\partial v}\right)_T\left(\frac{\partial T}{\partial P}\right)_v=-1
$$ Solving this for the numerator of the specific heat ratio expression and substituting the result into this numerator produces $$
k=\frac{\left(\frac{\partial v}{\partial P}\right)_T}{\left(\frac{\partial v}{\partial P}\right)_s}=-\frac{v \alpha}{\left(\frac{\partial v}{\partial P}\right)_s}
$$
