An approximation of the ideal gas law for real gases

[pentazoid]

Homework Statement

Even at low density, real gases don't quite obey the ideal gas law. A systematic way to account for deviations from ideal behavior is the virial expansion,

PV=nRT(1+B(T)/(V/n) + C(T)/(V/n)^2+...)

where functions B(T), C(T) and so on are called the virial coefficients.

Any proposed relation between P,V, and T , like the ideal gas law or the virial equationm is called an equation of state. Another famous equation of state , which is qualitative accurate even for dense fluids, is the van der waals equations

(P+an^2/V^2)(V-nb)=nRT

where a and b are constants that depend on the type of gas. Calculate the second and third virial coefficients(B and C) for a gas obeying the van der waals equation , and terms of a and b. (hint: The binomial expansion says that (1+x)^p=1+px+1/2p(p-1)x^2, provided that abs(px)<<1. Apply this approximation to the quantity [1-nb/V]^-1.)

Homework Equations

The Attempt at a Solution

(P+an^2/V^2)(V-nb)=nRT==> PV-Pnb-an^3b/V^2+an^2/V=nRT==> PV=nRT+Pnb+an^3/V^2-an^2/V ==> PV=nRT(1+Pnb/nRT-an/RTV+an^3b/nRTV^2) ==> PV=nRT(1+Pnb/PV-an/RTV+an^3b/nRTV^2)==> PV=nRT(1+nb/(V/n)-(a/RT)/(V/n)+(ab/RT)V^2/n^2). Therefore the coefficients of B and C are:

B(T)=(b-a/RT) and C(T)=ab/RT right?

 

[Alewhey]

Homework Statement

Even at low density, real gases don't quite obey the ideal gas law. A systematic way to account for deviations from ideal behavior is the virial expansion,

PV=nRT(1+B(T)/(V/n) + C(T)/(V/n)^2+...)

where functions B(T), C(T) and so on are called the virial coefficients.

Any proposed relation between P,V, and T , like the ideal gas law or the virial equationm is called an equation of state. Another famous equation of state , which is qualitative accurate even for dense fluids, is the van der waals equations

(P+an^2/V^2)(V-nb)=nRT

where a and b are constants that depend on the type of gas. Calculate the second and third virial coefficients(B and C) for a gas obeying the van der waals equation , and terms of a and b. (hint: The binomial expansion says that (1+x)^p=1+px+1/2p(p-1)x^2, provided that abs(px)<<1. Apply this approximation to the quantity [1-nb/V]^-1.)

Homework Equations

The Attempt at a Solution

(P+an^2/V^2)(V-nb)=nRT==> PV-Pnb-an^3b/V^2+an^2/V=nRT==> PV=nRT+Pnb+an^3/V^2-an^2/V ==> PV=nRT(1+Pnb/nRT-an/RTV+an^3b/nRTV^2) ==> PV=nRT(1+Pnb/PV-an/RTV+an^3b/nRTV^2)==> PV=nRT(1+nb/(V/n)-(a/RT)/(V/n)+(ab/RT)V^2/n^2). Therefore the coefficients of B and C are:

B(T)=(b-a/RT) and C(T)=ab/RT right?

Instead ofmultiplying out like you I wrote (as suggested)
$$(V-nb) = V(1-\frac{nb}{V} )\\$$
and took the bracket to the other side, then:
$$nRT(1-\frac{nb}{V} )^-^1 = nRT(1 + \frac{nb}{V} +\frac{n^2b^2}{V^2} + ...) = PV +\frac {an^2}{V} = PV + \frac{n}{V}. \frac {a}{RT}.nRT$$
Giving
$$B = b - \frac{a}{RT} , C = b^2$$