Statistical Thermodynamics
Z. Suo
Chemical Potential
A system that can exchange particles with the rest of the world. A glass of wine can
warm up, expand, and evaporate. The interaction between the system and its environment may
be registered with three variables: the energy U, the volume V, and the number of particles N of
the glass of wine
There may be many ways for the environment to do work to the system, so
that we need to register variables other than the volume of the system. There may be many
species of particles, so that we need register the number of particles of every species. To
illustrate the method, however, we will consider only the three variables, U, V, and N, and block
all other modes of interaction between the glass of wine and its environment. When the three
variables are held constant, the system is an isolated system. As usual, let Ω be the number of
states of the isolated system. Consequently, the function Ω(U ,V , N ) characterizes a family of
isolated systems.
As before, we call S = log Ω the entropy of an isolated system. For a family of isolated
systems registered with the three variables, (U ,V , N ) , the entropy is a function, S (U ,V , N ) . As
before, this function may be inverted to U (S ,V , N ) .
The two functions S (U ,V , N ) and
U (S ,V , N ) contain the same information and characterize the same family of isolated system.
Chemical potential. Associated with small changes dS , dV and dN , the energy of the
system changes by
dU =
∂U (S ,V , N )
∂U (S ,V , N )
∂U (S ,V , N )
dS +
dV +
dN .
∂S
∂V
∂N
We have interpreted the first differential coefficients as the temperature,
T=
∂U (S ,V , N )
,
∂S
and the second differential as the pressure,
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Statistical Thermodynamics
Z. Suo
−p=
∂U (S ,V , N )
.
∂V
We will call the third differential coefficient the chemical potential:
µ=
∂U (S ,V , N )
.
∂N
In words, the chemical potential of a species is the increase in the energy of the system when the
system gains one particle of the species, while all other variables are held constant. Chemical
potential has the unit of energy.
Using the above definitions, we write
dU = TdS − pdV + µdN .
We can also write the above relation in terms of the function S (U ,V , N ) :
dS =
µ
1
p
dU + dV − dN .
T
T
T
This leads to an alternative but equivalent definition of the chemical potential:
µ = −T
System A'
Ω' (U ' ,V ' , N ')
∂S (U ,V , N )
.
∂N
U
V
N
System A' '
Ω ' ' (U ' ' , V ' ' , N ' ')
We can use the fundamental postulate to show that the chemical potential behaves in a
way analogous to the temperature and the pressure.
The chemical potential directs the
reallocation of particles, the temperature directs the reallocation of energy, and the pressure
directs the reallocation of volume. When two systems can exchange particles, energy, and
volume, equilibrium is reached when the two systems have the identical chemical potential,
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Statistical Thermodynamics
Z. Suo
identical temperature and identical pressure. When two systems of the same temperature and the
same pressure were allowed to exchange a species of molecules, the molecules will go from a
system with a high chemical potential to a system with a low chemical potential. For example,
water molecules may escape from a glass of wine, and then diffuse into a piece of cheese, or the
other way around, depend whether the wine of the cheese has the high chemical potential.
Ideal gas. An ideal gas, of N molecules and in a container of volume V, is subject to
pressure p and temperature T. We have derived the ideal gas law:
pV = NT .
Another fundamental relation for an ideal gas is
U = γNT ,
where γ is a dimensionless constant specific to the molecule, a constant known as the heat
capacitance per molecule.
We now change the energy and the volume of the gas, but keep the number of molecules
constant. Thus
dU = TdS − pdV .
Inserting the above relations specific for the ideal gas, we obtain that
γ
dU dS dV
=
−
.
U
N
V
Integrating from one configuration (U 0 ,V0 ) to another configuration (U ,V ) , we obtain that
S (U ,V , N ) = S (U 0 ,V0 , N ) + γN log(U / U 0 ) + N log(V / V0 ) .
Recall the definition of the chemical potential, we obtain that
µ (U ,V , N ) = µ (U 0 ,V0 , N ) − γT log(U / U 0 ) − T log(V / V0 )
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Statistical Thermodynamics
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Gas is a homogeneous medium. At a constant temperature and pressure, the volume, energy and
entropy all scale linearly with the number of molecules. Consequently, the chemical potential is
a function of the pressure and temperature only. Rewrite the above relation, and we obtain that
µ ( p, T ) = µ ( p0 , T0 ) − (γ + 1)T log(T / T0 ) + T log( p / p0 ) .
This expression determines the chemical potential of an ideal gas up to an additive constant.
Experimental determination of chemical potential. While temperature is measured by
allowing two systems to exchange energy, chemical potential of a species is measured by
allowing two systems to exchange the species. For example, to measure the chemical potential
of water molecules in a wine, we can use a membrane that allows water molecules to go through
while blocks all other species of molecules. A water vapor will appear above the membrane.
We then allow the wine to equilibrate with water vapor, and then measure the pressure of the
water vapor. The measured pressure gives the chemical potential of water molecules up to an
additive constant.
Lagendre transformation. Define the Gibbs free energy by
G = U − TS + pV .
Its differential is
dG = − SdT + Vdp + µdN .
Consequently, the Gibbs free energy is a function of three variables, G (T , p, N ) . The chemical
potential can also be defined as
µ=
∂G (T , p, N )
.
∂N
Ideal gas once more. We now vary the pressure by dp, but hold the temperature and the
number of molecules constant. Associated with this change, the Gibbs free energy changes by
dG = Vdp .
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Statistical Thermodynamics
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Inserting the ideal gas law pV = NT , and integrating from one pressure p0 to another pressure
p , we obtain that
G (T , p, N ) = G (T , p0 , N ) + NT log( p / p0 )
Recall the definition of the chemical potential,
µ=
∂G (T , p, N )
.
∂N
For an ideal gas, G is linear in N. We obtain that
µ (T , p ) = µ (T , p0 ) + T log( p / p0 ) .
Thus, by changing the (partial) pressure of a species of molecules, we can vary the chemical
potential.
Ideal solution. Consider the isolated system composed of ink particles and wine once
more. The particles explore the space within the wine by Brownian motion. The chemical
potential of the ink particles takes the same form as that for the ideal gas, with the pressure
interpreted as the osmotic pressure. Recall that pV = NT , so that we can replace the above
relation in terms of the concentration of the ink particles, c = N / V , namely,
µ (T , c ) = µ (T , c0 ) + T log(c / c0 ) .
Hydrogel (or poroelasticity or elastic solution). A hydrogel is a cross-linked polymer
capable of uptaking water. It is analogous to sponge, or soil. Cheese behaves like a hydrogel, so
do many tissues. Consider a piece of hydrogel, subject to a force f and a moist environment in
which the chemical potential of water is µ .
At a fixed temperature, when the hydrogel
equilibrate with the force and the moist environment, the hydrogel is characterized by a
Helmholtz free energy F (l , N ) , where l is the displacement caused by the force, and N the
number of water molecules in the hydrogel. By definition, we have
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Statistical Thermodynamics
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dF = fdl + µdN ,
so that
f =
∂F (l , N )
,
∂l
µ=
∂F (l , N )
.
∂N
A change in the chemical potential of the moist environment will cause a change in the
displacement, and a change in the force can cause water to diffuse into or out of the hydrogel.
Just like thermal expansion of a material may be used to measure temperature, hydrogel can be
used to measure chemical potential of water molecules.
So far as the above description is concerned, there is nothing special about the gel or
water. The description fits any elastic solid capable of uptaking any species. For example, the
crystalline iron uptakes carbon atoms.
A system in contact with a reservoir of energy, volume and particles. We can also
make the chemical potential as an independent variable. For example, define a function
Λ = U − TS + pV − µN .
Its differential is
dΛ = − SdT + Vdp − Ndµ .
The coefficients in the differential can be defined by the partial derivatives of the function
Λ(T , p, µ ) .
If the above change of variable makes you feel disoriented, you can always go back to the
fundamental postulate. Holding the system at a constant temperature, pressure and chemical
potential means that the system is in contact with a reservoir of energy, volume and particles.
When the reservoir has fixed values of energy U R , volume VR , and number of particles N R , the
reservoir is an isolated system, with it the number of states being Ω R (U R ,VR , N R ) .
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Statistical Thermodynamics
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We view the composite of the small system and the reservoir as an isolated system, with
fixed total energy U tot , total volume Vtot and total number of particles N tot . When the small
system is held at U, V and N, the reservoir has U tot − U , Vtot − V and N tot − N . Following the
same procedure as before, we find that the number of states of the reservoir is
⎛ U + pRV − µ R N ⎞
⎟⎟ .
Ω R (U tot − U ,Vtot − V , N tot − N ) = Ω R (U tot ,Vtot , N tot )exp⎜⎜ −
TR
⎠
⎝
Upon losing energy U , volume V and N particles to the small system, the reservoir reduces its
number of states by a factor
⎛ U + pRV − µ R N ⎞
⎟⎟ ,
exp⎜⎜ −
TR
⎠
⎝
known as the Gibbs factor.
Let
Ω(U ,V , N )
be
the
number
of
state
of
the
small
system,
and
S (U ,V , N ) = log Ω(U ,V , N ) be the entropy of the small system. The number of states of the
composite is
Ωcom = Ω R (U tot − U ,Vtot − V , N tot − N )Ω(U ,V , N ) .
Thus,
⎛ U − TR S + pRV − µ R N ⎞
⎟⎟ .
Ω com = Ω R (U tot ,Vtot , N tot )exp⎜⎜ −
TR
⎠
⎝
We regard the composite as isolated system, and (U ,V , N ) as internal variables. Note that the
number of states of the composite Ω com is a function of (U ,V , N ) . According to the fundamental
postulate, the most probable values of (U ,V , N ) maximizes Ωcom , or equivalently maximizes the
function
U − TR S (U ,V , N ) + pRV − µ R N .
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Statistical Thermodynamics
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This will lead to the familiar equilibrium conditions
1 ∂S (U ,V , N )
=
,
TR
∂U
pR ∂S (U ,V , N )
=
,
TR
∂V
−
µR
TR
=
∂S (U ,V , N )
.
∂N
Consider the following scenario. The system has been in contact with the reservoir long
enough to equalize temperature, pressure and chemical potential, so that we can drop the
subscript R. However, there is still a constraint internal to the system. Upon lifting this
constraint, an internal variable Y can changes, the number of states of the composite can further
increase. after a long time, the more probably value of the internal variable has a smaller value
of the following function.
Λ(T , p, µ , Y ) = U − TS (U ,V , N , Y ) + pV − µN .
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