Partial Differential from Implicit Expression

Question

I have an expression which is explicit in $p$ given by $$p=\frac{\rho^{2}RT}{M^{2}}\left(1-\frac{{\rho}c}{MT^{3}}\right)\left(\frac{M}{\rho}+B_{o}-\frac{{\rho}bB_{o}}{M}\right)-\frac{\rho^{2}A_{o}}{M^{2}}\left(1-\frac{{\rho}a}{M}\right)$$ where $R$, $c$, $b$, $B_{o}$, $A_{o}$ and $a$ are constants. So we have $p=p(\rho,T)$, $\rho=\rho(p,T)$ and $T=T(p,\rho)$ that is each can be expressed a function of the other variables. Now the above expression is explicit in $p$ but implicit in $\rho$. How would I determine $\left(\frac{\partial{\rho}}{\partial{T}}\right)_{p}$?

Answer 1

Taking into account the fact that you want to obtain $$\left(\frac{\partial{\rho}}{\partial{T}}\right)_{p}$$ as Chinny84 commented, you need to consider that $p$ is constant. So, you have $$F(\rho,T)=-\frac{A \rho ^2 \left(1-\frac{a \rho }{M}\right)}{M^2}+\frac{\rho ^2 R T \left(-\frac{b B \rho }{M}+B+\frac{M}{\rho }\right) \left(1-\frac{c \rho }{M T^3}\right)}{M^2}-p=0$$ So, compute $$\left(\frac{\partial{F(\rho,T)}}{\partial{T}}\right)_{\rho}$$ $$\left(\frac{\partial{F(\rho,T)}}{\partial{\rho}}\right)_{T}$$ and, as usual, compute $$\left(\frac{\partial{\rho}}{\partial{T}}\right)_{p}=- \frac {\left(\frac{\partial{F(\rho,T)}}{\partial{T}}\right)_{\rho}}{\left(\frac{\partial{F(\rho,T)}}{\partial{\rho}}\right)_{T}}$$