The thermodynamic state of a substance is defined by the quantities– volume, – pressure,– Temperature and– internal energy.

These four quanties are related by the equationsand one equation for the internal energy eg

This means we can choose any two of these quantities as independent andf express the other two – dependent – variables in terms of the independent variables.

The second law of thermodynamics can be written(1) whereandare independent.

Or aswhereandare independent (2)

Or aswhereandare independent (3)

Or aswhereandare independent (4)

Or aswhere U and p are independent (5)

Or aswhereandare independent. (6)

To show (1) write

Then(7)

Now let

Then(8)

Henceand

Equate this expression withandfrom (7) to give

and

Hence we can write (1) asor

he proof for (3) is similar.

To prove (4) takeandas independent, then

Then

Rearrange the second of these to give

Substitute into the first to give

Compare these two equations with equations (7) to give

and

Substitute these into (1) to obtain

(5) and (6) can be similarly proved.