Eq. \ref{eq:4239912} is an initial equation. $$\kappa_T = \frac{-1}{V} \left( \frac{ \partial V}{\partial P} \right)_T \label{eq:4239912}$$ Eq. \ref{eq:4454896} is an initial equation. $$P V = n R T \label{eq:4454896}$$ Divide both sides of Eq. \ref{eq:4454896} by $$P$$; yields Eq. \ref{eq:5840241}. $$V = \frac{n R T}{P} \label{eq:5840241}$$ Substitute LHS of Eq. \ref{eq:5840241} into Eq. \ref{eq:4239912}; yields Eq. \ref{eq:5196207}. $$\kappa_T = \frac{-1}{V} \left( \frac{ \partial }{\partial P}\left(\frac{nRT}{P}\right) \right)_T \label{eq:5196207}$$ Simplify Eq. \ref{eq:5196207}; yields Eq. \ref{eq:3915956}. $$\kappa_T = \frac{-nRT}{V} \left( \frac{ \partial }{\partial P}\left(\frac{1}{P}\right) \right)_T \label{eq:3915956}$$ Simplify Eq. \ref{eq:3915956}; yields Eq. \ref{eq:6275836}. $$\kappa_T = \frac{-nRT}{V} \left( \frac{-1}{P^2}\right) \label{eq:6275836}$$ Substitute LHS of Eq. \ref{eq:4454896} into Eq. \ref{eq:6275836}; yields Eq. \ref{eq:1003658}. $$\kappa_T = \frac{-PV}{V} \left( \frac{-1}{P^2}\right) \label{eq:1003658}$$ Simplify Eq. \ref{eq:1003658}; yields Eq. \ref{eq:2206759}. $$\kappa_T = \frac{1}{P} \label{eq:2206759}$$ Eq. \ref{eq:2206759} is one of the final equations.