# Virial equation of state

## Homework Statement

It's just that in my textbook, for section titled "Second virial coefficients can be used to determine intermolecular potentials," I have an equation that I do NOT understand how it was derived---I tried to do it over and over, but couldn't quite figure how. If anyone could explain, thank you!!!

B2v(T) = RT * B2p(T), where B2v(T) and B2p(T) are virial coefficient

## Homework Equations

B2v(T) = RT * B2p(T), where B2v(T) and B2p(T) are virial coefficient

## The Attempt at a Solution

I assume the equation comes from Z, the compressibility factor:

Z = PV/RT, where it can be expanded to:

Z = 1 + B2v(T)/V + B3v(T)/V2 + ...
Z = 1 + B2p(T)*P + B3p(T)*P2 + ...

So, I equivocated 1 + B2v(T)/V + ... = 1 + B2p(T)P + ...
B2v(T)/V + ... = B2p(T)P + ...
...and tried multiplying both side by V or divide by P to cancel out other virial coefficients with numbers larger than 2 (meaning B3v and B3p), but I don't know how to progress anymore.......pls give me hints! I want to understand how the original equation B2v(T) = RT * B2p(T) was obtained!

BvU
Homework Helper
Perhaps the idea isn't to let all higher order terms cancel, just to equate the first order coefficients ?

Quantum Defect
Homework Helper
Gold Member

## Homework Statement

It's just that in my textbook, for section titled "Second virial coefficients can be used to determine intermolecular potentials," I have an equation that I do NOT understand how it was derived---I tried to do it over and over, but couldn't quite figure how. If anyone could explain, thank you!!!

B2v(T) = RT * B2p(T), where B2v(T) and B2p(T) are virial coefficient

## Homework Equations

B2v(T) = RT * B2p(T), where B2v(T) and B2p(T) are virial coefficient

## The Attempt at a Solution

I assume the equation comes from Z, the compressibility factor:

Z = PV/RT, where it can be expanded to:

Z = 1 + B2v(T)/V + B3v(T)/V2 + ...
Z = 1 + B2p(T)*P + B3p(T)*P2 + ...

So, I equivocated 1 + B2v(T)/V + ... = 1 + B2p(T)P + ...
B2v(T)/V + ... = B2p(T)P + ...
...and tried multiplying both side by V or divide by P to cancel out other virial coefficients with numbers larger than 2 (meaning B3v and B3p), but I don't know how to progress anymore.......pls give me hints! I want to understand how the original equation B2v(T) = RT * B2p(T) was obtained!

What happens if you use the compressibility (written in the volume version of the virial expansion) to solve for P. Take this series expansion for P and plug into the pressure version of the virial expansion. Compare this equation with the virial expansion in terms of volume. Coefficients in front of the same powers of 1/V must be the same for the two expressions to be the same:

A x + B x^2 + ... = alpha x + beta x^2 + ... is true iff A = alpha, B = beta, ....