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Mayer’s Relation (thermodynamics)

1842
  • Julius Robert von Mayer
19th century laboratory with thermodynamic instruments and Mayer's relation equations.

(generated image for illustration only)

Mayer’s relation connects the specific heats of a perfect gas to the specific gas constant (\(R_s\)). The relation is \(c_p – c_v = R_s\). For molar specific heats (\(C_p\) and \(C_v\)), the relation is \(C_p – C_v = R\), where \(R\) is the universal gas constant. This shows that \(c_p\) is always greater than \(c_v\).

Mayer’s relation is a direct consequence of the first law of thermodynamics applied to a perfect gas. It quantifies the difference between the specific heat at constant pressure (\(c_p\)) and the specific heat at constant volume (\(c_v\)). When a gas is heated at constant volume, all the added heat goes into increasing its internal energy. However, when heated at constant pressure, the gas must expand to keep the pressure constant. This expansion requires work to be done on the surroundings. Therefore, additional heat energy must be supplied to perform this expansion work, in addition to the heat required to raise the internal energy.

The difference, \(c_p – c_v\), is precisely the amount of work done by one unit mass of the gas when its temperature is raised by one degree at constant pressure. For a perfect gas, this work is equal to the specific gas constant, \(R_s\). The relation is derived from the definitions of enthalpy (\(h = u + Pv\)) and the perfect gas law (\(Pv = R_s T\)). Differentiating with respect to temperature gives \(dh/dT = du/dT + R_s\), which directly translates to \(c_p = c_v + R_s\). This simple yet elegant relationship is fundamental in thermodynamics.

UNESCO Nomenclature: 2212
– Thermodynamics

Type

Physical Law

Disruption

Substantial

Usage

Widespread Use

Precursors

  • first law of thermodynamics
  • concept of specific heat (joseph black)
  • ideal gas law (clapeyron)
  • definition of enthalpy
  • work of sadi carnot on heat engines

Applications

  • calculating unknown specific heats from known values
  • determining the heat capacity ratio (gamma) for gas dynamic calculations
  • thermodynamic property tables generation
  • educational tool for demonstrating the first law of thermodynamics
  • fundamental equation in the analysis of gas power cycles

Patents:

NA

Potential Innovations Ideas

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Related to: Mayer’s relation, specific heat, heat capacity, gas constant, thermodynamics, first law of thermodynamics, enthalpy, internal energy, perfect gas, Cp-Cv.

Historical Context

Mayer’s Relation (thermodynamics)

1834
1836
1839-01-01
1842
1847
1850
1850
1834
1835
1838
1841
1845
1850
1850
1850

(if date is unknown or not relevant, e.g. "fluid mechanics", a rounded estimation of its notable emergence is provided)

Related Invention, Innovation & Technical Principles

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