Chemical Potential

Let us now turn to look at the situation where two systems can exchange particles.What is the condition of equilibrium in this case?Let two systems S₁ and S₂ be in thermal contact and in diffusive contact.By diffusive contact we mean that particles can flow from S_l to S₂ and vice versa.We maintain a constant t by placing both systems in thermal contact with a large reservoir, R.We saw earlier that for a system in thermal equilibrium with a reservoir, the Helmholtz free energy F will be a minimum.Thus for the two systems in contact with R, we must have that F₁ + F₂ must be a minimum.The change in F = F₁ + F₂ as the number of particles varies is given by

formula

But if we treat S_l + S₂ as a closed system, we have that N₁ + N₂ = N is a constant.

Thus dN₁ = -dN₂ and the requirement that dF = 0 becomes

(10.1)

We define the chemical potential of a system to be

(10.2)

Then the condition of equilibrium reduces to

m₁ = m₂

(10.3)

Strictly speaking, the true definition of m is in terms of the difference

m = F(t,V,N) - F(t,V,N-1)

since particles are a discrete set.However we are usually dealing with such large quantities of particles that we are able to treat m as a continuous function.

Example:

The chemical potential of an ideal gas is found from

F = -t ln(Z_n)

with Z_n = Z₁^N / N! and Z₁ = n_q V.So

F = -t [N ln(n_qV) - N ln(N) + N]

and thus

(10.4)

Connection to Potential Energy

To better understand what a chemical potential is, recall that the Helmhotz free energy is defined to be

F = U - ts

where U is the total energy of the system.If the system is subjected to an external force, this will give rise to either a potential energy or dissipative work.In either case, the Helmholtz free energy can be written as

F = U_int + NU_ext - ts

where the N comes from the fact that the external energy acts on each particle in the system.Now consider two systemsS₁ and S₂ which are at the same temperature and are capable of exchanging particles, but which are not yet in diffusive equilibrium.We can assume that initially m₂ > m₁, and we can denote the initial, non-equilibrium difference in chemical potential as Dm_i = m_2i - m_1i.Now let a difference in potential energy be established between the systems, such that the potential energy of each particle in S₁ is raised exactly by Dm_i. Such a difference could come from gravitational potential energy, or electrodynamical potential energy, for example.

By establishing a potential energy step, we are adding a potential energy NDm_i to the energy of each state in S_l.Thus the final chemical potential energy of S₁ is

(10.5)

since we didn't change the energy in S₂.But now m_lf = m_2f, so the systems are in

diffusive equilibrium.Thus, we see that the chemical potential is equivalent to a true potential energy in that the difference in chemical potential between two systems is equal to the potential barrier that will bring the two systems into diffusive equilibrium.Since the chemical potential has the same properties as potential energy, we see that we cannot define an absolute chemical potential.Only differences in chemical potential have any physical meanings.From the argument above, we see that if an external potential step exists, the total chemical potential of a system is the sum

m_tot = m_int + m_ext

wherem_ext is the potential energy per particle in the external potential, and m_int is the chemical potential that would be present if the external potential were zero.Finally, since we can only physically discuss differences in chemical potential, the physical condition of diffusive equilibrium becomes

Dm_ext = Dm_int

(10.6)

Example:

What is the condition for diffusive equilibrium of a column of ideal gas at temperature t?

drawing

Let n(0) be the number of particles in the bottom chamber, and n(h) be the number of particles in the top chamber.We want m(0) = m(h), where we must have m = m_ext + m_int.Then from (10.4), we have that

(10.7)

Once we have the number of particles at a height h, we can determine the pressure by recalling that the pressure of an ideal gas is proportional to the concentration of particles,

p = nt

Thus the pressure of an ideal gas at height h is

(10.8)

Example:

What is the chemical potential of an ideal gas in a magnetic field B?

drawing

In a magnetic field we can view the gas as being made up of two types, or species, of particles; those with spin up, and those with spin down.If the spin is up, the potential energy is -mB, while the potential energy of the spin down particles is mB.For the container out of the magnetic field the chemical potentials are

formula

while those in the magnetic field have

formula

we also have that

formula

and

Using the condition of diffusive equilibrium, we get

formula

and

formula

n(B) = n(0) cosh(mB/t)

Example:

For a lead acid battery, the relationship between chemical potentials and potential steps is even more apparent.In a normal lead acid battery, the negative electrode consists of metallic lead, Pb, and the positive electrode is a layer of lead oxide, Pb0₂, on a Pb substrate.The electrodes are immersed in diluted sulfuric acid, H₂SO₄, which is partially ionized into H⁺ ions and S0₄^-- ions.

In the discharge process, both the metallic Pb of the negative electrode and the Pb0₂ of the positive electrode are converted to lead sulfate, PbS0₄, via the two reactions

Negative electrode:

Positive electrode:

Thus, the negative electrode acts as a sink for the S0₄^-- ions, and the positive electrode acts as a sink for the H⁺ ions.If the battery terminals are not connected, electrons are depleted from the positive electrode and accumulate in the negative electrode, thereby charging them both.As a result, electrochemical potential steps develop at the electrode?electrolyte interfaces, steps of exactly the correct magnitude to equalize the chemical potential steps and stop the diffusion of ions, which stops the chemical reactions from proceeding further.If an external current is permitted to flow, the reactions resume.

If we denote by DV- and DV₊ the differences in electrostatic potential of the negative and positive electrodes relative to the common electrolyte, diffusion of the sulfate ions will stop when (remembering that the sulfate carries two negative charges)

-2eDV_- = Dm(SO₄^--)

Similarly, diffusion of the H⁺ ions will stop when

eDV₊= Dm(H⁺)

drawing drawing

The electrostatic potential as a function of position is seen to be

drawing

The two potentials, DV_? and DV₊, are called half-cell potentials, and have magnitudes of DV- = -0.4 V and DV₊ = 1.6 V.Thus, the total electrostatic potential that develops across the cell in order to stop the diffusion reaction is

DV = DV₊ - DV- = 2 V

This is the open-circuit voltage, or EMF, of the battery.

Chemical Potential and Entropy

Recall that we defined the chemical potential as

Substituting in the definition of F, we get

We can replace the right hand side of this by considering the following argument.

Let the entropy be a function of the independent variables U, V and N, s = s(U,V,N).

Then the differential of s is given by

Let dV = 0 for the process under consideration, and require the changes in ds, dU and dN be inter?related in such a way that dt = 0. Thus we get that

or, after dividing by (dN)_t and using the definition of 1/t,

(10.9)

Returning to m, we can solve (10.9) for to yield

(10.10)

What is the difference between this expression and the one involving F?Recall that F is a function of the variables t, V and N, so m = m(t,V,N) in this form.In (10.10), m = m(U,V,N), i.e. it has a dependence on a different set of variables.A third formulation that we will not prove in class is

(10.11)

We can now derive a more general form of the thermodynamic identity we encountered earlier.Once again consider an infinitesimal change in entropy, where now the entropy depends on U, V and N

but, as before, we recall that , and .So the change in entropy becomes

(10.12)