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
But as then
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
where is the specific heat capacity of the fluid at constant volume.
So for an isobaric change, P is constant, we have
differentiating with respect to T
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