How is the virial equation of state derived?

[terp.asessed]

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!

B_2v(T) = RT * B_2p(T), where B_2v(T) and B_2p(T) are virial coefficient

Homework Equations

B_2v(T) = RT * B_2p(T), where B_2v(T) and B_2p(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 + B_2v(T)/V + B_3v(T)/V² + ...
Z = 1 + B_2p(T)*P + B_3p(T)*P² + ...

So, I equivocated 1 + B_2v(T)/V + ... = 1 + B_2p(T)P + ...
B_2v(T)/V + ... = B_2p(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 B_3v and B_3p), but I don't know how to progress anymore...pls give me hints! I want to understand how the original equation B_2v(T) = RT * B_2p(T) was obtained!

 

[BvU]

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

 

[Quantum Defect]

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!

B_2v(T) = RT * B_2p(T), where B_2v(T) and B_2p(T) are virial coefficient

Homework Equations

B_2v(T) = RT * B_2p(T), where B_2v(T) and B_2p(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 + B_2v(T)/V + B_3v(T)/V² + ...
Z = 1 + B_2p(T)*P + B_3p(T)*P² + ...

So, I equivocated 1 + B_2v(T)/V + ... = 1 + B_2p(T)P + ...
B_2v(T)/V + ... = B_2p(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 B_3v and B_3p), but I don't know how to progress anymore...pls give me hints! I want to understand how the original equation B_2v(T) = RT * B_2p(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, ...