Generated by the Physics Derivation Graph.
https://www.youtube.com/watch?v=QTiqF-HtkS0 and https://www.youtube.com/watch?v=3Yls-t3B49U
Eq. \ref{eq:7368252} is an initial equation. \begin{equation} U = Q + W \label{eq:7368252} \end{equation} Eq. \ref{eq:2445123} is an initial equation. hold volume constant in first term; hold temperature constant in second term \begin{equation} dU = \left(\frac{\partial U}{\partial T}\right)_V dT + \left(\frac{\partial U}{\partial V}\right)_T dV \label{eq:2445123} \end{equation} Eq. \ref{eq:8155541} is an initial equation. \begin{equation} S = k_{\rm Boltzmann} \ln \Omega \label{eq:8155541} \end{equation} Eq. \ref{eq:4576755} is an initial equation. \begin{equation} C_V = \left(\frac{\partial U}{\partial T}\right)_V \label{eq:4576755} \end{equation} Eq. \ref{eq:7384950} is an initial equation. \begin{equation} \pi_T = \left(\frac{\partial U}{\partial V}\right)_T \label{eq:7384950} \end{equation} Substitute LHS of Eq. \ref{eq:4576755} and LHS of Eq. \ref{eq:7384950} into Eq. \ref{eq:2445123}; yields Eq. \ref{eq:5358683}. \begin{equation} dU = C_V dT + \pi_T dV \label{eq:5358683} \end{equation} Divide both sides of Eq. \ref{eq:5358683} by \(dT\); yields Eq. \ref{eq:3830663}. \begin{equation} \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} \end{equation} Eq. \ref{eq:2714175} is an initial equation. \begin{equation} \alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p \label{eq:2714175} \end{equation} Multiply both sides of Eq. \ref{eq:2714175} by \(V\); yields Eq. \ref{eq:7939101}. \begin{equation} V \alpha = \left( \frac{\partial V}{\partial T} \right)_p \label{eq:7939101} \end{equation} Substitute LHS of Eq. \ref{eq:7939101} into Eq. \ref{eq:3830663}; yields Eq. \ref{eq:1136968}. \begin{equation} \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} \end{equation} Simplify Eq. \ref{eq:1136968}; yields Eq. \ref{eq:1189259}. \begin{equation} \left(\frac{\partial U}{\partial T}\right)_p = C_V + \pi_T V \alpha \label{eq:1189259} \end{equation} Eq. \ref{eq:9670239} is an initial equation. \begin{equation} \kappa_T = \frac{-1}{V} \left( \frac{ \partial V}{\partial P} \right)_T \label{eq:9670239} \end{equation}