7.2 The Virial Equation of State
371
or
ρ =
ζ
k B T
∂P
∂ζ
V ,T
.
(7.2.15)
This result provides the connection between Eq. (7.2.9), our series expansion for
the pressure, and the number density, so that ρ is given by the series expansion
ρ =
∞
j =1
jb j ζ
j .
(7.2.16)
We may now employ a method known as regression of series (see also
Appendix B.1) to obtain P as a power series in ρ: we begin by assuming that
ζ itself can be written as a power series in ρ as
ζ = a 1 ρ + a 2 ρ
2
+ a 3 ρ
3
+ · · · ,
(7.2.17)
and then substituting this power series into Eq. (7.2.16), followed by a matching of
the coefficients of the powers of ρ on the left- and right-hand sides of the resultant
equation. This process gives
a 1 = 1 , a 2 = −2b 2 , a 3 = −3b 3 + 8b
2
2 .
(7.2.18)
Now we find that ζ has the power series representation
ζ = ρ − 2b 2 ρ
2
+ (8b
2
2 − 3b 3 )ρ
3
+ · · · ,
(7.2.19)
which we may now substitute into Eq. (7.2.12) for P to obtain
P
k B T
= ρ − b 2 ρ
2
+
4b
2
2 − 2b 3
ρ
3
+ · · ·
= ρ −
1
2V
Z 2C − Z
2
1C
ρ
2
−
1
3!V
V
Z 3C − 3Z 2C Z 1C + 2Z
3
1C
− 3
Z 2C − Z
2
1C
ρ
3
+ · · · .
(7.2.20)
If we now compare this expression with the well-known virial expansion of
thermodynamics, viz.,
P V
Nk B T
= 1 + B 2 (T )ρ + B 3 (T )ρ
2
+ · · · ,
(7.2.21)