174
4 Mean Values and Thermodynamics
in which we can now identify X r as a generalized force. If we now make the
association of ‘macroscopic’ with ‘average behaviour’, then we can take the
following steps:
δW = −−X dx;
; X ≡
−
∂E r
∂x
,
(4.1.9)
in which the macroscopic generalized force X is related to the partition function
via
X ≡
r
p r X r =
r
e −βE r
Z(β)
−
∂E r
∂x
.
We may also write this expression in the form
X =
1
β
∂ ln Z
∂x
,
(4.1.10)
so that ultimately the incremental thermodynamic work can be expressed in terms
of the partition function and its derivatives as
δW = −
1
βZ
∂Z
∂x
dx = −
1
β
∂ ln Z
∂x
dx .
(4.1.11)
For the moment, we shall consider only the case in which the external parameter
is the ‘volume’. In this case, we obtain explicitly for the incremental work the
expression
δW = −
1
β
∂ ln Z
∂V
dV .
(4.1.12a)
If there are only volume changes in a system such as a gas, then work is related to
the average pressure P and the volume change that occurs, namely
δW = −P dV .
(4.1.12b)
From the above two expressions for the incremental work for pressure–volume
changes, we may deduce that the equation relating the pressure to the partition
function is
P =
1
β
∂ ln Z(β, V )
∂V
= k B T
∂ ln Z(β, V )
∂V
.
(4.1.13)
In thermodynamics when we have an expression for the pressure P in terms of
the temperature and volume, we call that expression the equation of state for the
system.