# Equation for temperature of gas at Joule-Thompson Inversion State

Assume that a gas obeys the following equation of state:

$\nu = {\frac{RT}{P}}-{\frac{a}{T}} + b$

Where $a , b$ are constant.

Derive an equation for the temperature at the Joule-Thompson inversion state.

My professor gave us the following hint:

$T = {\frac{\mu_{JT}c_p + \nu}{\frac{\partial \nu}{\partial T}}}$

Where ${\frac{\partial \nu}{\partial T}}$ is held constant at P

My attempt at the solution starts here:

When I differentiate the equation of state wrt T, I get

${\frac{\partial \nu}{\partial T}} = {\frac{RT}{P}} + {\frac{a}{T^2}}$

Plugging this back into the equation for T leads to:

$T = {\frac{\mu_{JT}c_p + \nu}{\frac{RT}{P}+{\frac{a}{T^2}}}}$

Is this the correct approach to take to solve for the temperature at the Joule - Thompson inversion state? It seems I have need to take a cubic root for the temperature so I was not sure if this is what the intended answer is.

• How do we know what the professor has decided? – Solar Mike Jul 25 '18 at 5:36
• Hi @SolarMike, do you mind explaining your question a bit more? – ganondorf29 Jul 27 '18 at 14:44
• Why don't you explain the "intended answer"... – Solar Mike Jul 27 '18 at 14:46
• When you attempt to solve the equation for T, you get an analytical solution that I couldn't analytically solve without help of a computer (at least the answer per WolframAlpha). In addition, we did not discuss using numerical methods to solve for non-linear equations so I did not expect this answer. – ganondorf29 Jul 28 '18 at 20:25