Advanced Thermodynamics Lectures

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In this course we had problem sets like derivation of joule thomson coefficient from van der waal equation of state and etc. My solutions for this were like below :

\color{#0000CD}\ {\mu }_t\coloneqq \ {\left(\frac{\partial T}{\partial P}\right)}_h=\frac{1}{C_P}(T\left({\left(\frac{\partial v}{\partial T}\right)}_P-V\right)=\frac{1}{C_P}\left(\frac{2a}{RT}-b\right)

First Solution :
van der waals equation : \left(P+\frac{a}{V^2}\right)\left(V-b\right)=RT

P=\frac{RT}{V-b}-\frac{a}{V^2} \xrightarrow[]{\text{P=cte}} 0=\frac R{V-b}\frac{\mathit{dT}}{\mathit{dV}}-\frac{\mathit{RT}}{\left(V-b\right)^2}+\frac{2a}{V^3}

\frac R{V-b}\frac{\mathit{dT}}{\mathit{dV}}=\frac{\mathit{RT}}{\left(V-b\right)^2}-\frac{2a}{V^3}

\frac{\mathit{dT}}{\mathit{dV}}=\frac{(\frac{\mathit{RT}}{\left(V-b\right)^2}-\frac{2a}{V^3})}{(\frac R{V-b})}

\frac{dV}{dT}=(\frac R{V-b})(\frac{\mathit{RT}}{\left(V-b\right)^2}-\frac{2a}{V^3})^{-1}

T\frac{\mathit{dV}}{\mathit{dT}}-V=\left(\frac{RT}{V-b}\right)\left(\frac{\mathit{RT}V^3-2a\left(V-b\right)^2}{\left(V-b\right)^2V^3}\right)^{-1}-V

~~~~~ ~~~~~ ~~~~~ ~ =\left(\mathit{RT}\right)\left(\frac 1{\left(\frac{\mathit{RT}}{V-b}\right)-\frac{2a\left(V-b\right)}{V^3}}\right)-V

~~~~~ ~~~~~ ~~~~~ ~ =\left(\frac{\mathit{RT}}{\left(\frac{\mathit{RT}}{V-b}\right)\left(1-\frac{2a\left(V-b\right)^2}{\mathit{RT}V^3}\right)}\right)-V

~~~~~ ~~~~~ ~~~~~ ~ =\left(V-b\right)\left(1-\frac{2a\left(V-b\right)^2}{\mathit{RT}V^3}\right)^{-1}-V

~~~~~ ~~~~~ ~~~~~ ~ {\cong}\left(V-b\right)\left(1+\frac{2a\left(V-b\right)^2}{\mathit{RT}V^3}\right)-V

~~~~~ ~~~~~ ~~~~~ ~ =V-b+\frac{2a\left(V-b\right)^3}{\mathit{RT}V^3}-V\xrightarrow[]{\text{V}\gg\text{b}}{\cong}-b+\frac{2a}{\mathit{RT}}

~~~~~ ~~~~~ ~~~~~ \Longrightarrow \bf\color{#0000CD} \mu _t=\frac 1{C_P}\left(\frac{2a}{\mathit{RT}}-b\right)

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