Eq. \ref{eq:5888046} is an initial equation. $$\alpha = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p \label{eq:5888046}$$ Eq. \ref{eq:5927974} is an initial equation. $$V = \frac{n R T}{P} \label{eq:5927974}$$ Substitute LHS of Eq. \ref{eq:5927974} into Eq. \ref{eq:5888046}; yields Eq. \ref{eq:7236464}. $$\alpha = \frac{1}{V} \frac{nR}{P} \left( \frac{\partial T}{\partial T} \right)_P \label{eq:7236464}$$ Simplify Eq. \ref{eq:7236464}; yields Eq. \ref{eq:4600503}. $$\alpha = \frac{nR}{VP} \label{eq:4600503}$$ Eq. \ref{eq:5130250} is an initial equation. $$P V = n R T \label{eq:5130250}$$ Divide both sides of Eq. \ref{eq:5130250} by $$T$$; yields Eq. \ref{eq:8283443}. $$\frac{PV}{T} = nR \label{eq:8283443}$$ Substitute RHS of Eq. \ref{eq:8283443} into Eq. \ref{eq:4600503}; yields Eq. \ref{eq:7845152}. $$\alpha = \frac{PV}{T} \frac{1}{VP} \label{eq:7845152}$$ Simplify Eq. \ref{eq:7845152}; yields Eq. \ref{eq:2491768}. $$\alpha = \frac{1}{T} \label{eq:2491768}$$ Eq. \ref{eq:2491768} is one of the final equations.