Eq. \ref{eq:7368252} is an initial equation. $$U = Q + W \label{eq:7368252}$$ Eq. \ref{eq:2445123} is an initial equation. hold volume constant in first term; hold temperature constant in second term $$dU = \left(\frac{\partial U}{\partial T}\right)_V dT + \left(\frac{\partial U}{\partial V}\right)_T dV \label{eq:2445123}$$ Eq. \ref{eq:8155541} is an initial equation. $$S = k_{\rm Boltzmann} \ln \Omega \label{eq:8155541}$$ Eq. \ref{eq:4576755} is an initial equation. $$C_V = \left(\frac{\partial U}{\partial T}\right)_V \label{eq:4576755}$$ Eq. \ref{eq:7384950} is an initial equation. $$\pi_T = \left(\frac{\partial U}{\partial V}\right)_T \label{eq:7384950}$$ Substitute LHS of Eq. \ref{eq:4576755} and LHS of Eq. \ref{eq:7384950} into Eq. \ref{eq:2445123}; yields Eq. \ref{eq:5358683}. $$dU = C_V dT + \pi_T dV \label{eq:5358683}$$ Divide both sides of Eq. \ref{eq:5358683} by $$dT$$; yields Eq. \ref{eq:3830663}. $$\left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T \left( \frac{\partial V}{\partial T} \right)_p \label{eq:3830663}$$ Eq. \ref{eq:2714175} is an initial equation. $$\alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p \label{eq:2714175}$$ Multiply both sides of Eq. \ref{eq:2714175} by $$V$$; yields Eq. \ref{eq:7939101}. $$V \alpha = \left( \frac{\partial V}{\partial T} \right)_p \label{eq:7939101}$$ Substitute LHS of Eq. \ref{eq:7939101} into Eq. \ref{eq:3830663}; yields Eq. \ref{eq:1136968}. $$\left(\frac{\partial U}{\partial T}\right)_p = C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T V \alpha \label{eq:1136968}$$ Simplify Eq. \ref{eq:1136968}; yields Eq. \ref{eq:1189259}. $$\left(\frac{\partial U}{\partial T}\right)_p = C_V + \pi_T V \alpha \label{eq:1189259}$$ Eq. \ref{eq:9670239} is an initial equation. $$\kappa_T = \frac{-1}{V} \left( \frac{ \partial V}{\partial P} \right)_T \label{eq:9670239}$$