% this tex file was generated by the Physics Derivation Graph 
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% inference rules as newcommand for use in the body
\newcommand\declareinitialexpression[1]{Eq.~\ref{eq:#1} is an initial equation.}
\newcommand\dividebothsidesby[3]{Divide both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.}
\newcommand\substituteLHSofexprintoexpr[3]{Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.}
\newcommand\simplify[2]{Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.}
\newcommand\declarefinalexpression[1]{Eq.~\ref{eq:#1} is one of the final equations.}
\title{coefficient of isothermal compressibility using the equation of state for an ideal gas}
\date{\today}
%\author{a84c8294ad9547db4da22820fcaf8c7215485d84d522c45d981703b9995138ba}
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\begin{document}
\maketitle
\begin{abstract}
Generated by the \href{http://allofphysics.com/}{Physics Derivation Graph}.
\end{abstract}

% step ID = 0003831533
\declareinitialexpression{d0825c8cc2199e6e2b522c20456db7c2e8f247eb63df7b4c479722d68c5f4885}
\begin{equation}
\kappa_T = \frac{-1}{V} \left( \frac{ \partial V}{\partial P} \right)_T
\label{eq:d0825c8cc2199e6e2b522c20456db7c2e8f247eb63df7b4c479722d68c5f4885}
\end{equation}

% step ID = 0009872032
\declareinitialexpression{499e5eba6e32663354551010f0a44e78e79c0111e394010aefb383bbd7241a5c}
\begin{equation}
P V = n R T
\label{eq:499e5eba6e32663354551010f0a44e78e79c0111e394010aefb383bbd7241a5c}
\end{equation}

% step ID = 0003862516
\dividebothsidesby{P}{499e5eba6e32663354551010f0a44e78e79c0111e394010aefb383bbd7241a5c}{b33b2dfdc53db62deb996d8059fe1cc8fee8b694796b1bbf1ff2ba6f7fdb5514}
\begin{equation}
V = \frac{n R T}{P}
\label{eq:b33b2dfdc53db62deb996d8059fe1cc8fee8b694796b1bbf1ff2ba6f7fdb5514}
\end{equation}

% step ID = 0009225696
\substituteLHSofexprintoexpr{d0825c8cc2199e6e2b522c20456db7c2e8f247eb63df7b4c479722d68c5f4885}{b33b2dfdc53db62deb996d8059fe1cc8fee8b694796b1bbf1ff2ba6f7fdb5514}{eb7de616a981c1db6903811e0d6676ec0f8e3e71800ac33a857a16f38d2b414e}
\begin{equation}
\kappa_T = \frac{-1}{V} \left( \frac{ \partial }{\partial P}\left(\frac{nRT}{P}\right) \right)_T
\label{eq:eb7de616a981c1db6903811e0d6676ec0f8e3e71800ac33a857a16f38d2b414e}
\end{equation}

% step ID = 0006914144
\simplify{eb7de616a981c1db6903811e0d6676ec0f8e3e71800ac33a857a16f38d2b414e}{c6f45cda1a9633e9521179284b7b8c98d1771f898270c0d3f3dcb5f424f5d422}
\begin{equation}
\kappa_T = \frac{-nRT}{V} \left( \frac{ \partial }{\partial P}\left(\frac{1}{P}\right) \right)_T
\label{eq:c6f45cda1a9633e9521179284b7b8c98d1771f898270c0d3f3dcb5f424f5d422}
\end{equation}

% step ID = 0001873674
\simplify{c6f45cda1a9633e9521179284b7b8c98d1771f898270c0d3f3dcb5f424f5d422}{588c84f221f49728bb66f3126e3b13704603a22613d01af646c6c7e02688bd31}
\begin{equation}
\kappa_T = \frac{-nRT}{V} \left( \frac{-1}{P^2}\right)
\label{eq:588c84f221f49728bb66f3126e3b13704603a22613d01af646c6c7e02688bd31}
\end{equation}

% step ID = 0005391858
\substituteLHSofexprintoexpr{588c84f221f49728bb66f3126e3b13704603a22613d01af646c6c7e02688bd31}{499e5eba6e32663354551010f0a44e78e79c0111e394010aefb383bbd7241a5c}{e1dcb956cbbbcf57792819acbaea0cb9903b10447e66b8c93de30c907042810b}
\begin{equation}
\kappa_T = \frac{-PV}{V} \left( \frac{-1}{P^2}\right)
\label{eq:e1dcb956cbbbcf57792819acbaea0cb9903b10447e66b8c93de30c907042810b}
\end{equation}

% step ID = 0008648371
\simplify{e1dcb956cbbbcf57792819acbaea0cb9903b10447e66b8c93de30c907042810b}{7dd9ba86bef7d6149cdc5f93fc865842896d84982ec55316b0a253d9648f66fb}
\begin{equation}
\kappa_T = \frac{1}{P}
\label{eq:7dd9ba86bef7d6149cdc5f93fc865842896d84982ec55316b0a253d9648f66fb}
\end{equation}

% step ID = 0001059017
\declarefinalexpression{7dd9ba86bef7d6149cdc5f93fc865842896d84982ec55316b0a253d9648f66fb}
\bibliographystyle{plain}
\bibliography{pdg_derivation_citations.bib}
\end{document}
% EOF