# Virial equation, minimum pressure point

Good evening PF! I’m having trouble figuring out how to attack this problem. I have tried two different ways but I don’t know if either of them is correct.

1. The problem statement, all variables and given/known data
Using the provided virial coefficients, determine analytically the pressure at which the graph of PV versus P for N2 at -50° C, reaches a minimum point.
Virial coefficients for N2 at -50° C:
A = 18.31
B = -2.88×10-2
C = 14.98×10-5
D = -14.47×10-8
E = 4.66×10-11

2. Relevant equations
$$PV_m=A+BP+CP^2+DP^3+EP^4$$

3. The attempt at a solution
So, in order to find the minimum point I need two differentiate the function, and I’m trying two ways of doing this. I hope at least one of them is correct.

Option A:
$$\frac {d(PV_m)}{dP} = B+2CP+3DP^2+4EP^3$$
In order to find the critical points I equate the above derivative to zero.
$$B+2CP+3DP^2+4EP^3 = 0$$
Now I have to solve this cubic equation analytically, I could solve it with the help of a CAS, but the problem is asking for an analytic solution. This is as far as I can go. I did solve the equation with a software, and got two complex solutions and a real one. I assume the only solution that is relevant to me is the real one, right?

Option B:
I cleared Vm and expressed it as a function of P first, then differentiated.
$$V_m(P) = \frac{A}{P} +B+CP+DP^2+EP^3$$
$$\frac{dV_m}{dP} = -\frac{A}{P^2} +C+2DP+3EP^2$$
Equating to zero.
$$-\frac{A}{P^2}+C+2DP+3EP^2=0$$
I have to solve this equation analytically too, but I have no idea with this one.

Well, I hope at least one of my procedures is right. Any help or insight will be greatly appreciated!

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