Statistical Mechanics - Physics at Oregon State University

3.2. EXAMPLES OF THE USE OF THE CHEMICAL POTENTIAL. 47
and at the positive electrode
P bO2 + 2H + + H2SO4 + 2e − → P bSO4 + 2H2O + 3.2eV (3.16)
We ignore free electrons in the electrolyte, and hence the only place where we
can take electrons or dump electrons is in the P b of the electrodes. Therefore,
if the terminals of the battery are not connected by an external wire, these
reactions have as a net result a transport of electrons from the positive electrode
to the negative electrode. The amount of energy gained by transferring one
electron from the positive electrode to the negative electrode through these
reactions is 2 eV. Therefore, the internal chemical potential is higher at the
positive electrode, and ˆµ+ − ˆµ− = 2eV . Of course, after a number of electrons
are transported from the positive electrode to the negative electrode there will
be a difference in electrical potential energy between the electrodes. Suppose
the potentials at these electrodes are V+ and V−. The flow of electrons in
this unconnected battery will have to stop when the total chemical potential is
constant, or
ˆµ+ + (−e)V+ = ˆµ− + (−e)V− ⇒ V+ − V− = 2V (3.17)
Do we know how much charge has been transferred? No, unless we know the
capacitance of the battery, which depends on the shape and geometry of the
plates and the space between them.
If we connect the terminals of the battery through an outside resistor, electrons
will flow in the outside circuit. This will reduce the number of electrons
in the negative electrode, and as a result the chemical reactions will continue to
make up for the electrons leaving through the outside circuit. The actual difference
in potential between the electrodes will be less than 2V and will depend
on the current through the outside circuit. Of course, in real life the potential
difference is always less than 2V because of internal currents through the
electrolyte.
Gravity and atmospheric pressure.
A second example is a very simplified study of the pressure in the atmosphere.
Consider a column of gas in a gravitational field. There is only
one type of molecule in this gas. The gravitational potential at a height h in
the column is mgh, where m is the mass of a molecule. We assume that the
temperature T is the same everywhere, in other words this is a model of a
isothermal atmosphere. Probably not very realistic, but not a bad start. The
density n = N
V of the molecules is a function of h. Next, we focus on a slice
of gas at a height h with a thickness dh. The value of dh is small enough that
the density n(h) does not vary appreciably within this slice. But it is also large
enough that the slice of the gas contains many molecules. This is possible only
because the gravitational potential doe not very much on the scale of the interatomic
distance! We assume that the gas inside this slice through the column