Alternative Formulation of the Second Law of Thermodynamics
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
Equate this expression withandfrom (7) to give
Hence we can write (1) asor
he proof for (3) is similar.
To prove (4) takeandas independent, then
Rearrange the second of these to give
Substitute into the first to give
Compare these two equations with equations (7) to give
Substitute these into (1) to obtain
(5) and (6) can be similarly proved.