Basic thermodynamics 77andE). If the number of particles is increased fromN 1 toN 2 > N 1 , then chemical
work
W 12 (chem)=∑N^2
N=N 1μ(N) (3.2.5)is done on the system. Clearly, the number of particles in a system can only change by
integral amounts, ∆N. However, the changes we wish to consider are so small compared
to the total particle number (N∼ 1023 ) that they can be regarded approximately as
changes dN in a continuous variable. Therefore, the chemical work corresponding
to such a small change dNcan be expressed as dWrev(chem) =μ(N)dN. Again, we
suppress the explicit dependence ofμonN(as well as onVandE) and write simply
dWrev(chem)=μdN. Therefore, the total reversible work done on the system is givenby
dWrev= dWrev(mech)+ dWrev(chem)=−PdV+μdN, (3.2.6)so that the total change in energy is
dE=TdS−PdV+μdN. (3.2.7)By writing eqn. (3.2.7) in the form
dS=1
T
dE+P
T
dV−μ
TdN, (3.2.8)it is clear that the state function we are seeking is just the entropyof the system,
S=S(N,V,E), since the change inSis related directly to the change in the three
control variables of the ensemble. However, sinceSis a function ofN,V, andE, the
change inSresulting from small changes inN,V, andEcan also be written using
the chain rule as
dS=(
∂S
∂E
)
N,VdE+(
∂S
∂V
)
N,EdV+(
∂S
∂N
)
V,EdN. (3.2.9)Comparing eqn. (3.2.9) with eqn. (3.2.8) shows that the thermodynamic quantitiesT,
P, andμcan be obtained by taking partial derivatives of the entropy with respect to
each of the three control variables:
1
T=
(
∂S
∂E
)
N,V,
P
T
=
(
∂S
∂V
)
N,E,
μ
T=−
(
∂S
∂N
)
V,E. (3.2.10)
We now recall that the entropy is a quantity that can be related to the number of
microscopic states of the system. This relation was first proposedby Ludwig Boltz-
mann in 1877, although it was Max Planck who actually formalized the connection. Let
Ω be the number of microscopic states available to a system. The relation connecting
Sand Ω states that
S(N,V,E) =kln Ω(N,V,E). (3.2.11)
SinceSis a function ofN,V, andE, Ω must be as well. The constant,k, appearing in
eqn. (3.2.11) is known as Boltzmann’s constant; its value is 1.3806505(24)× 10 −^23 J·K−^1.