# Solving for gas law partial derivative

I don't know where to begin with this question.

A scientist discovered that the state of the unknown gas can be well described by the equation of state below: $$P = \frac{RT}{\overline{V}} + \left(\frac{a + bT}{\overline{V}^2}\right)$$ Find the partial derivative $\left(\frac{\partial \overline{V}}{\partial T}\right)_P$ which is expressed in terms of $\overline{V}, R, b$, and $P$. (Hint: If $\overline{V}$ is a function of two variables $T$ and $P$ (i.e. $\bar{V} = f(T, P)$), then the total differential can be expressed as $$d\bar{V} = df(T, P) = \left(\frac{\partial \overline{V}}{\partial T}\right)_PdT + \left(\frac{\partial \overline{V}}{\partial P}\right)_TdP$$ Use a particular condition to derive the expression $\left(\frac{\partial \overline{V}}{\partial T}\right)_P$ in terms of $\left(\frac{\partial \overline{V}}{\partial P}\right)_T$ and $\left(\frac{\partial P}{\partial T}\right)_{\overline{V}}$.)

This problem is related to the physical chemistry of the gas law. What is the correct approach?

• My first thought was simply to solve the equation for T in terms of P, V, R, b, and a, then take the partial derivative. It's a fairly straightforward quotient rule. My solution still contains a though... – Jason Patterson Mar 12 '16 at 4:48
• That's not universal; what if we were not able to get the explicit form for either V or T? – Ivan Neretin Mar 12 '16 at 8:29
• This is purely a question about maths. You will probably get a better answer if you ask it on maths.SE. – bon Mar 12 '16 at 11:01
• On math.SE it would get rejected right away for not using MathJax. – Ivan Neretin Mar 12 '16 at 14:58
• Please don't remove the question. That isn't very helpful. – bon Mar 15 '16 at 13:24

Think of it this way. $$dP = \left(\partial P\over\partial T\right)_VdT+ \left(\partial P\over\partial V\right)_TdV$$ Now, we want P to be constant (that is, $dP=0$). What does that entail? A certain relation between $dV$ and $dT$ (which, BTW, are now $\partial$ rather than $d$): $$\left(\partial V\over\partial T\right)_P=-\left(\partial P\over\partial T\right)_V\left/\left(\partial P\over\partial V\right)_T\right.$$