Lecture 10
Imperfect Gases: Virial Coefficients and
Mayer f-function
10.1
Monatomic Gases
• We neglect internal degrees of freedom which would be required for molecules. Therefore, for N = 1, U = 0 and,
Z
Z
1
Q1 (V, T ) =
· · · e−βH dp1 dr1
3
1!h
Z
βp2
V
1
= 3 e− 2m dp1
(10.1)
h
V
=
(10.2)
vQ
where,
µ
vQ =
h
2πmkT
¶3
(10.3)
is the quantum volume.
• For N > 1, we obtain the expression for the canonical partition function,
1 ZN
N
N ! vQ
µ ¶N
1 Q1
=
ZN
N! V
QN =
where,
Z
ZN =
Z
···
e−βUN dr1 dr2 . . . drN
and UN is the potential energy function.
But recall from the other day that ZN was defined in Eq. 9.12,
µ ¶N
V
QN ,
ZN = N !
Q1
(10.4)
LECTURE 10. IMPERFECT GASES: VIRIAL COEFFICIENTS AND MAYER F-FUNCTION 63
and thus we see that the ZN defined earlier was just the configurational integral.
• Thus, to calculate the virial coefficients we will need to determine these multiparticle
configurational partition functions. Starting with Z1 , we have,
Z
Z
−βU1
Z1 = e
dr1 = dr1 = V
(10.5)
Similarly, we have,
ZZ
Z2 =
e−βU2 dr1 dr2
ZZZ
Z3 =
e−βU3 dr1 dr2 dr3
(10.6)
(10.7)
..
.
etc.
• Proceed by first calculating the 2nd virial coefficient, where we need to calculate Z2
and thus U2 .
• Assume that U2 (r1 , r2 ) only depends on the particle separation (central forces), hence,
U2 = U (r12 ) , |r12 | = |r2 = r1 | = r12
(10.8)
e.g. Lennard-Jones potential or, more rigorously, we can calculate this intermolecular
potential from quantum mechanics using ab initio (from scratch) potentials.
Note: Implicit in this treatment is the assumption of ”pairwise additivity” of U (rN )
→ which means that the interaction between any two particles is not affected by the
presence of other atoms.
•
¢
1 ¡
Z2 − Z12
2V Z Z
£ −βU (r12 )
¤
1
e
− 1 dr1 dr2
= −
2V
B2 (T ) = −
(10.9)
• Note that U (r12 ) will only be appreciable over a small region where the volume elements, dr1 and dr2 are near each other. Therefore, the integral B2 → 0, except when
dr1 is near dr2 .
• We can thus change the variables of integration to r1 and r12 = r2 − r1 :
Z
Z
£ −βU (r12 )
¤
1
B2 (T ) = −
dr1
e
− 1 dr12
2V
(10.10)
Therefore, the integration over the relative separation of the two particles is independent of where the pair is in the volume V (that is, the result is independent of r1 ),
except for when the pair is near the boundary of the system.
LECTURE 10. IMPERFECT GASES: VIRIAL COEFFICIENTS AND MAYER F-FUNCTION 64
• We are interested, however, in the case where V → ∞, ∴ this surface effect is not
critical and we can carry out the integration over dr1 to get a factor of V . Using
spherical coordinates, we thus have,
Z
∞
B2 (T ) = −2π
£
¤
e−βU (r) − 1 r2 dr
(10.11)
0
• The integrand appears often in the theory of imperfect gases and is formally called the
Mayer f-function,
fij ≡ f (rij ) ≡ e−βU (rij ) − 1
(10.12)
Using the Lennard-Jones potential, this function takes on the form,
f(r)
r
−1
What about the higher order virial coefficients?
• At the third order, B3 (T ) it is required to determine Z3 (see Eq. 10.7) which means
that we must also evaluate U3 ,
U3 (r1 , r2 , r3 ) = U (r12 ) + U (r13 ) + U (r23 ) + ∆(r12 , r13 , r23 )
(10.13)
but we will assume pairwise additivity at this level and, hence write
U3 ' U (r12 ) + U (r13 ) + U (r23 )
(10.14)
• Now, formally B3 was defined as,
B3 (T ) = 4b22 − 2b3 = −
¢
1 ¡
6V b3 − 12V b22
3V
and
6V b3 = Z3 − 3Z2 Z1 + 2Z13
(10.15)
LECTURE 10. IMPERFECT GASES: VIRIAL COEFFICIENTS AND MAYER F-FUNCTION 65
We proceed to evaluate each of these terms using Eqs. 10.6, 10.7, and 10.17
ZZZ
Z3 =
(1 + f12 ) (1 + f13 ) (1 + f23 ) dr1 dr2 dr3
ZZZ
[f12 f13 f23 + f12 f13 + f12 f23 + f13 f23 + f12 + f13 + f23 + 1] dr1 dr(10.16)
=
2 dr3
ZZ
Z1 Z2 = Z1
ZZ
(f12 + 1)dr1 dr2 = V
ZZZ
(f12 + 1)dr1 dr2 =
(f12 + 1)dr1 dr2 dr3
However, we could have equally have chosen,
ZZ
ZZZ
Z1 Z2 = V
(f13 + 1)dr1 dr3 =
(f13 + 1)dr1 dr2 dr3
or
ZZ
ZZZ
Z1 Z2 = V
(f23 + 1)dr2 dr3 =
(f23 + 1)dr1 dr2 dr3
ZZZ
∴ 3Z1 Z2 =
(f12 + f13 + f23 + 3)dr1 dr2 dr3
ZZZ
andZ3 − 3Z1 Z2 =
[f12 f13 f23 + f12 f13 + f12 f23 + f13 f23 − 2] dr1 dr2 dr3
For the 2Z13 term we have:
ZZZ
2Z13
=2
dr1 dr2 dr3
Putting these terms all together, we find that,
ZZZ
6V b3 =
[f12 f13 f23 + f12 f13 + f12 f23 + f13 f23 ] dr1 dr2 dr3
Now we must evaluate the 12V b22 term in Eq. 10.15, noting that
·Z
¸2 Z
Z
2
f12 dr12 = f12 dr12 f13 dr13
4b2 =
and thus,
Z
4V
b22
=
Z
dr1
Z
f12 dr12
ZZZ
f13 dr13 =
f12 f13 dr1 dr2 dr3
where in the very last step we have converted from relative coordinates back to regular
coordinates.
2
2
WeRneed
R R 3 of these 4V b2 terms
RRR(i.e. 12V b2 ), so it is obvious that the other two must
be,
f12 f23 dr1 dr2 dr3 and
f13 f23 dr1 dr2 dr3 .
Finally, Eq. 10.15 becomes,
1
B3 (T ) = −
3V
ZZZ
f12 f13 f23 dr1 dr2 dr3
(10.17)
• Note that we can express the type of integrals pictorally via cluster diagrams.
LECTURE 10. IMPERFECT GASES: VIRIAL COEFFICIENTS AND MAYER F-FUNCTION 66
10.2
Cluster Diagrams
• Consider the integrand in the 2nd virial coefficient, this can be represented as,
1
2
For the 3rd virial coefficient, B3 (T ), the integrand can be represented as, which means
1
2
3
that this product of Mayer f-functions will vanish unless the 3 particles are close to
each other.
• There is actually a proof for imperfect gasess that applies to cluster diagrams which
states that the virial coefficients are given by,
Bj+1 = −
j
βj
j+1
(10.18)
where,
Z
Z
1
0
βj =
· · · S1,2,...,j+1
dr1 dr2 . . . drj+1
j!V
0
represents the sum of all products of f functions that
where the symbol, S1,2,...,j+1
connect molecules 1, 2, . . . , j + 1 such that the clusters are connected in such a way
that the removal of any point, together with all the lines associated with that point,
still result in a connected graph (i.e. all remaining particles are connected).
The graphs drawn above satisfy these criteria and are referred to as stars or star
graphs (Note: the simple two-particle cluster graph is the most primitive star, but it
is a star nontheless).
• The 3 particle graphs that are not stars share the common structure,
LECTURE 10. IMPERFECT GASES: VIRIAL COEFFICIENTS AND MAYER F-FUNCTION 67
1
2
3
• Some examples of stars required for the 4th virial coefficient, that includes connections
between 4 particles are,
1
2
4
3
1
4
2
3
1
2
4
3
• The 4 particle graphs that are not stars share the common structure,
1
2
4
3
1
4
2
3
1
2
4
3
• In summary, the virial coefficients are integrals over sums of stars and, for N = 2, 3, 4,
we have,
0
S1,2
=
0
S1,2,3
= 4
0
S1,2,3,4 = 3 ¤ + 6 ¤ + £
¨
(10.19)
(10.20)
(10.21)