# Virial equation, minimum pressure point

**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 N_{2} at -50° C, reaches a minimum point.

Virial coefficients for N_{2} 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**

[tex]PV_m=A+BP+CP^2+DP^3+EP^4[/tex]

**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:

[tex]\frac {d(PV_m)}{dP} = B+2CP+3DP^2+4EP^3[/tex]

In order to find the critical points I equate the above derivative to zero.

[tex]B+2CP+3DP^2+4EP^3 = 0[/tex]

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 V_{m} and expressed it as a function of P first, then differentiated.

[tex]V_m(P) = \frac{A}{P} +B+CP+DP^2+EP^3[/tex]

[tex]\frac{dV_m}{dP} = -\frac{A}{P^2} +C+2DP+3EP^2[/tex]

Equating to zero.

[tex]-\frac{A}{P^2}+C+2DP+3EP^2=0[/tex]

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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