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https://notendur.hi.is/hj/EE2/HD1lausn.pdf
Eq. \ref{eq:4239912} is an initial equation. \begin{equation} \kappa_T = \frac{-1}{V} \left( \frac{ \partial V}{\partial P} \right)_T \label{eq:4239912} \end{equation} Eq. \ref{eq:4454896} is an initial equation. \begin{equation} P V = n R T \label{eq:4454896} \end{equation} Divide both sides of Eq. \ref{eq:4454896} by \(P\); yields Eq. \ref{eq:5840241}. \begin{equation} V = \frac{n R T}{P} \label{eq:5840241} \end{equation} Substitute LHS of Eq. \ref{eq:5840241} into Eq. \ref{eq:4239912}; yields Eq. \ref{eq:5196207}. \begin{equation} \kappa_T = \frac{-1}{V} \left( \frac{ \partial }{\partial P}\left(\frac{nRT}{P}\right) \right)_T \label{eq:5196207} \end{equation} Simplify Eq. \ref{eq:5196207}; yields Eq. \ref{eq:3915956}. \begin{equation} \kappa_T = \frac{-nRT}{V} \left( \frac{ \partial }{\partial P}\left(\frac{1}{P}\right) \right)_T \label{eq:3915956} \end{equation} Simplify Eq. \ref{eq:3915956}; yields Eq. \ref{eq:6275836}. \begin{equation} \kappa_T = \frac{-nRT}{V} \left( \frac{-1}{P^2}\right) \label{eq:6275836} \end{equation} Substitute LHS of Eq. \ref{eq:4454896} into Eq. \ref{eq:6275836}; yields Eq. \ref{eq:1003658}. \begin{equation} \kappa_T = \frac{-PV}{V} \left( \frac{-1}{P^2}\right) \label{eq:1003658} \end{equation} Simplify Eq. \ref{eq:1003658}; yields Eq. \ref{eq:2206759}. \begin{equation} \kappa_T = \frac{1}{P} \label{eq:2206759} \end{equation} Eq. \ref{eq:2206759} is one of the final equations.