**Problem : **

Suppose that we have a fuel cell battery, in which electrons flow from a
terminal in one half cell to a terminal in another. Explain this
phenomenon in terms of the chemical potential.

We can look at the battery as two systems in diffusive contact through
the connecting wire. The electrons simply then flow from the cell with
the higher chemical potential to that with the lower until an
equilibrium is reached, if ever.

**Problem : **

Show that the units of pressure as we have defined it agree with those
of the conventional understanding of pressure.

The conventional units are . We have defined
pressure so that we have an energy in the numerator and a volume in the
denominator. But remember that energy has the same units as work,
namely *FORCE*×*LENGTH*, and therefore we have
= .

**Problem : **

Forcing a system into a small volume makes the energy of the system
grow, whereas expanding the system, colloquially speaking, gives the
particles more room to relax, and the energy of the system decreases
(all for a process at constant entropy). Using the definition of
pressure we've investigated, show what happens to the pressure at large
volumes and very small volumes of the system. Does this agree with your
intuition?

For a system of small volume for the number of particles, the energy is
high. Increasing the volume some small amount, *δV*, will cause a
great decrease in the energy *U*. Therefore, the pressure is:

*p* = - (

)

_{σ} = -

= big positive

For a system of great volume for the number of particles, the energy is
already low. Increasing the volume some small amount, *δV*, will
only cause a small decrease in the energy *U*. Therefore, the pressure
is:

*p* = - (

)

_{σ} = -

= small positive

This makes sense to us. We expect a cramped system to have a high
pressure and sprawling system to have low pressure.

**Problem : **

Is the energy of the system *U* an intensive or extensive variable?

Doubling the system should double the energy, so *U* is an extensive
variable.

**Problem : **

Explain why the entropy is an extensive variable.

Remember that entropy was defined as *σ* = log *g* where *g* was the
multiplicity function. We defined the
entropy in this manner so that the entropies of two systems in contact
would add together, since their individual *g* functions multiply
together. So, doubling the system means that
*σ*_{new} = *σ*_{original} + *σ*_{duplicate} = 2*σ*_{original}.
Therefore entropy is an extensive variable.