What is Mayer’s relation?

Short Answer

Mayer’s relation is a thermodynamic equation that shows the connection between the specific heat capacities of an ideal gas. It states that the difference between the specific heat at constant pressure (Cp) and the specific heat at constant volume (Cv) is equal to the universal gas constant (R).

This relation is written as Cp – Cv = R. Mayer’s relation helps in understanding how gases absorb heat under different conditions and is very important in the study of heat capacity and thermodynamics of ideal gases.

Detailed Explanation :

Mayer’s Relation

Mayer’s relation is a fundamental equation in thermodynamics that connects two important properties of an ideal gas: specific heat at constant pressure (Cp) and specific heat at constant volume (Cv). It states that:

Cp – Cv = R

where R is the universal gas constant. This relation was established by the German physicist Julius Robert Mayer and plays a key role in understanding the energy changes in gases when heat is supplied under different conditions.

Specific heat capacity tells us how much heat is required to raise the temperature of a unit mass of a substance by one degree. Since gases behave differently under constant pressure and constant volume, they have two different specific heats—Cp and Cv. Mayer’s relation explains the exact difference between these two values.

Meaning of Cp and Cv

To understand Mayer’s relation more clearly, it is essential to understand what Cp and Cv represent.

Specific Heat at Constant Volume (Cv)

Cv is the amount of heat required to raise the temperature of 1 unit mass of a gas by 1°C (or 1 K) when the gas is kept at constant volume.
Because volume does not change, the gas does not do any external work.
So, all the heat supplied goes into increasing the internal energy of the gas.

Specific Heat at Constant Pressure (Cp)

Cp is the amount of heat required to raise the temperature of 1 unit mass of a gas by 1°C (or 1 K) when the gas is kept at constant pressure.
In this case, the gas expands, and some heat is used to do external work against the surroundings.

This is why Cp is always greater than Cv for an ideal gas.

Derivation of Mayer’s Relation (Conceptual Explanation)

When heat is supplied to a gas at constant volume, the gas does not expand, so no work is done.
The first law of thermodynamics gives:

Qv = mCvΔT = ΔU

Where:

  • Qv is heat supplied at constant volume
  • ΔU is the change in internal energy

When heat is supplied at constant pressure, the gas expands and does work.
So,

Qp = mCpΔT = ΔU + W

Where W is work done by the gas.

Work done during expansion at constant pressure is:

W = PΔV

Using the ideal gas law, PΔV = RΔT (for 1 mole)

Substituting this into the equation:

CpΔT = CvΔT + RΔT

Divide both sides by ΔT:

Cp = Cv + R

This gives Mayer’s relation:

Cp – Cv = R

This shows that the difference between the two specific heats depends only on the universal gas constant and is the same for all ideal gases.

Importance of Mayer’s Relation

Mayer’s relation is widely used in thermodynamics and gas laws because:

  1. Helps to Calculate Heat Capacities

If either Cp or Cv is known, the other can be calculated using Mayer’s relation.

  1. Important for Thermodynamic Processes

In isothermal, adiabatic, and other gas processes, Cp and Cv play a major role.

  1. Used in Engines and Compressors

Engine cycles, such as Otto and Diesel cycles, require values of Cp and Cv for performance calculations.

  1. Helps Understand Energy Distribution in Gases

It explains why gases require more heat to raise temperature at constant pressure.

  1. Helps Derive Other Thermodynamic Relations

Many important equations, like the adiabatic equation PVᵞ = constant, use Cp and Cv.

Why Cp Is Greater Than Cv

Cp is always greater than Cv because:

  • At constant volume, the gas only increases internal energy.
  • At constant pressure, the gas must increase internal energy and do work by expanding.

So, more heat is needed at constant pressure.

This extra heat requirement is exactly equal to R, the universal gas constant.

Mayer’s Relation and Ideal Gas Behaviour

Mayer’s relation holds perfectly only for ideal gases, because ideal gases follow the assumptions:

  • No intermolecular forces
  • Internal energy depends only on temperature
  • PV = RT relation holds for all conditions

Real gases follow Mayer’s relation only approximately, especially under low pressure and high temperature, where they behave nearly ideally.

Conclusion

Mayer’s relation, expressed as Cp – Cv = R, shows the simple and clear relationship between the specific heats of an ideal gas. It explains that more heat is required at constant pressure than at constant volume because the gas must do external work. This relation is essential for understanding thermodynamic processes, calculating heat capacities, and analyzing gas behaviour in engines and scientific systems. Mayer’s relation forms the foundation for many important results in thermodynamics.