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.

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 = .

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:

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:

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

Doubling the system should double the energy, so U 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 = 2σ _original . Therefore entropy is an extensive variable.