Ideal Gases

For situations where the gas being considered can be modelled as ideal, a specific expression is required.

By substituting two key relations for an ideal gas (du = CvdT, P = RT/V) into the Tds relation (ds = du/T + Pdv/T), the following equation is obtained:

This can be adjusted to give the entropy change for a whole process by integrating both sides:

If it is instead more desirable to find the entropy change using Cp values, this can be done again by substituting two relations for an ideal gas (dh = CpdT,     v = RT/p) into the other Tds relation (ds = dh/T - vdP/T):

HOWEVER, the specific heats (Cp and Cv) depend on temperature (with the exception of monatomic gases). This means that the above equations cannot be used unless we know the relationship between specific heat and temperature.

Even when we know what Cv(T) and Cp(T) are, it is not practical to constantly perform these difficult integrals to find the change in entropy.

This leaves two ways to find the entropy change.

Option 1: Assume Constant Specific Heat:

By assuming the specific heat of the gas is constant, the analysis becomes much more simple but also less accurate. But the error due to this is small in cases where specific heat increases almost linearly with temperature (the specific heat at the average temperature can be taken). Also if the temperature range is less than a few hundred degrees, calculations will still be sufficiently accurate.

and:

Entropy change can also be represented per unit mole instead of per unit mass. This is done by multiplying through by the molar mass of the gas:

and:

Option 2: Variable Specific Heats:

This is a much more exact analysis of the change in entropy of a system.

Typically this would be used for processes with large temperature changes, or where specific heat increases non-linearly.

For these situations the CV/p(T) terms must be included:

Or if it is preferable to calculate in terms of moles:

Where:

It is important to remember that entropy does not solely depend on temperature, but also specific volume and pressure (particularly important when obtaining values from tabulated data).

Isentropic Processes of Idea Gases

For isentropic process of ideal gases, the entropy-change relations are greatly simplified as they are equal to zero.

There are three isentropic relations that can be derived from the equations for constant specific heat:

These can be expressed much more simply by the following:

When it cannot be assumed that specific heat is constant for an isentropic process, the equations which account for the variation in specific heat simplify to the following:

This equation works well so long as the pressure ratio is known, but if instead a volume ratio is given, it becomes very difficult. To deal with this two new dimensionless quantities are introduced.

The first is the relative pressure, Pr:

The other is the relative specific volume:

These two equations are ONLY valid for isentropic processes involving an ideal gas. They account for changes in the specific heats and are therefore much more accurate than those which assume specific heat is constant.