The classical virial expansion expresses the pressure of a many-particle system in equilibrium as a power series in the number density. The virial expansion, introduced in 1901 by Heike Kamerlingh Onnes, is a generalization of the ideal gas law. He wrote that for a gas containing
N
atoms or molecules,
p
k
B
T
=
ρ
+
B
2
(
T
)
ρ
2
+
B
3
(
T
)
ρ
3
+
⋯
,
where
p
is the pressure,
k
B
is the Boltzmann constant,
T
is the absolute temperature, and
ρ
≡
N
/
V
is the number density of the gas. Note that for a gas containing a fraction
n
of
N
A
(Avogadro's number) molecules, truncation of the virial expansion after the first term leads to
p
V
=
n
N
A
k
B
T
=
n
R
T
, which is the ideal gas law.
Writing
β
=
(
k
B
T
)
−
1
, the virial expansion can be written as
β
p
ρ
=
1
+
∑
i
=
1
∞
B
i
+
1
(
T
)
ρ
i
.
The virial coefficients
B
i
(
T
)
are characteristic of the interactions between the particles in the system and in general depend on the temperature
T
. Virial expansion can also be applied to aqueous ionic solutions, as shown by Harold Friedman.
The Van der Waals equation can be used to derive the approximation
B
2
(
T
)
≈
b
N
A
−
a
N
A
2
k
B
T
with the Van der Waals constants a and b.
And when
T
B
=
a
b
⋅
k
B
N
A
=
a
b
R
then
B
2
(
T
B
)
≈
0
, see Boyle temperature.
According to Van der Waals constants (data page) the constants for hydrogen gas are for example a = 0.2476 L2bar/mol2 and b = 0.02661 L/mol and therefore the estimation of the Boyle temperature for hydrogen is
T
B
=
0.2476
⋅
10
−
1
m
6
P
a
m
o
l
2
0.02661
⋅
10
−
3
m
3
m
o
l
⋅
8.3145
J
m
o
l
⋅
K
=
112
K
. (The real value for hydrogen is 110 K. In nitrogen the difference is bigger.)
