Mayer's relation (Mayer's law)

Mayer’s relation (Mayer’s law) is the relation between molar heat capacities at constant pressure Cp and at constant volume CV for an ideal gas.

The figure below shows two reversible processes (or transformations) of an ideal gas. The process AB is isochoric (at constant volume) and the process AC is isobaric (at constant pressure). The temperature of the gas in state A is T1. The states B and C lie on the same isotherm which corresponds to the temperature T2; therefore, the gas reaches the same final temperature after the two processes.

Since the temperature increases for both processes (as T1 < T2) a certain amount of heat will be absorbed by the gas.

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The heat exchanged in an isochoric process is given by:

Where CV is the molar heat capacity at constant volume.

The heat exchanged in an isobaric process is given by:

Where Cp is the molar heat capacity at constant pressure.

Using statistical mechanics it can be proved that these heat capacities are:

Ideal gas CV Cp Cp – CV γ = Cp/CV
Monoatomic (3/2) R (5/2) R R 5/3
Diatomic (5/2) R (7/2) R R 7/5

Where R is the ideal gas constant which in SI units is: R = 8.31 J/mol K.

You can see that the difference between heat capacities is R (the penultimate column in the previous table). This relationship is known as Mayer’s relation (or Mayer’s law). In order to prove it, we will apply the differential form of the first law of Thermodynamics to the two processes represented in the PV diagram.

The work in the isochoric process AB is zero because there is no volume change.

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On the other hand, as the gas is ideal, its internal energy depends only on the temperature. Its variation is the same for the two processes because they both have the same initial and final temperatures. After equating and substituting heat and work we get:

Using the equation of state of an ideal gas to deduce the value of pdV, we get:

And by substituting in the previous expression, we get:

which proves the Mayer’s relation.

The Cp/CV ratio that appears in the last column of the table is called the heat capacity ratio or adiabatic index γ. It appears in the equation that describes a reversible adiabatic process of an ideal gas.

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