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First Law of Thermodynamics

The first law can be represented as

U is the internal energy of the system. This is defined exactly by the extrinsic variables of the system. For our purposes we can assume the internal is only a function of temperature where . This is true for an ideal gas. W is the work done on the system and Q is the heat transferred to the system.

The equation of state for an ideal gas is

This is very useful in defining how changes can happen in a hydrostatic system.

The differential form of the first law is

If U is a function of the thermodynamic coordinates of the system T, P and V then we have

and

But as then

and

For a hydrostatic system, the work done on the system is given by

as the change in volume implies compression is a negative change when the fluid is compressed.

Therefore we can rewrite the first law as

So for an isochoric change, V is constant, we have

or

Giving

where is the specific heat capacity of the fluid at constant volume.

So for an isobaric change, P is constant, we have

so

differentiating with respect to T

where

so

For molar specific heat capacities

This is a definite observable result. Most gases behave like ideal gases at low pressure and high temperature (room temperature). So looking at observable measurements of and for known gases we find

for oxygen and so and