1Fundamentals of statistical mechanics
II Statistical Physics
1.6 Extensive and intensive properties
So far, we have defined a lot of different quantities —
p, V, µ, N, T, S
etc. In
general, we can separate these into two different types. Quantities such as
V, N
scale with the size of the volume, while µ and p do not scale with the size.
Definition
(Extensive quantity)
.
An extensive quantity is one that scales pro-
portionally to the size of the system.
Definition
(Intensive quantity)
.
An intensive quantity is one that is independent
of the size of the system.
Example. N, V, E, S are all extensive quantities.
Now note that the entropy is a function of
E, V, N
. So if we scale a system
by λ, we find that
S(λE, λV, λN) = λS(E, V, N).
Example. Recall that we defined
1
T
=
∂S
∂E
V,N
.
So if we scale the system by
λ
, then both
S
and
E
scale by
λ
, and so
T
does
not change. Similarly,
p = T
∂S
∂V
T,N
, µ = −T
∂S
∂N
T,V
Example. The free energy is defined by
F = E − T S.
Since
E
and
S
are both extensive, and
T
is intensive, we find that
F
is extensive.
So
F (T, λV, λN) = λF (T, V, N).
Similarly, the grand canonical potential is
Φ = F − µN.
Since F and N are extensive and µ are intensive, we know Φ is extensive:
Φ(T, λV, µ) = λΦ(T, V, µ).
This tells us something useful. We see that Φ must be proportional to
V
.
Indeed, taking the above equation with respect to λ, we find
V
∂Φ
∂V
T,µ
(T, λV, µ) = Φ(T, V, µ).
Now setting λ = 1, we find
Φ(T, V, µ) = V
∂Φ
∂V
T,µ
= −pV.
Here p is an intensive quantity, it cannot depend on V . So we have
Φ(T, V, µ) = −p(T, µ)V.
This is quite a strong result.