For solids and liquids we define
the specific heat capacity as the quantity of energy that will raise the temperature of unit
mass of the body by 1 K. For gases, however, it is necessary to specify the conditions under
which the change of temperature takes place, since a change of temperature will also
produce large changes in pressure and volume.

For solids and liquids we can
neglect this pressure change and the specific heat capacity that we measure for them is
essentially one where the pressure on the body is unaltered. We call this the specific heat
capacity at constant pressure (C_{P}).

The specific heat capacity of a gas will
depend on the conditions under which it is measured and since these could vary
considerably we will restrict ourselves to the following, called the **principal specific heat capacities** of a gas:

(a) The
specific heat capacity at constant volume (c_{v}) is defined as the quantity of heat
required to raise the temperature of 1 kg of the gas by 1 K if the volume of the gas remains
constant.

(b) The specific heat capacity at constant pressure (c_{p}) is
defined as the quantity of heat required to raise the temperature of 1 kg of the gas by 1 K if
the pressure of the gas remains constant.

The specific heat capacity at constant
pressure (c_{p}) is always greater than that at constant volume (c_{v}),
since if the volume of the gas increases work must be done by the gas to push back the
surroundings.

The **molar heat capacity** at
constant volume (C_{V}) is the quantity of heat required to raise the temperature
of 1 mole of the gas by 1 K if the volume of the gas remains constant. The molar heat
capacity at constant pressure (C_{P}) is the quantity of heat required to raise the
temperature of 1 mole of the gas by 1 K if the pressure of the gas remains
constant.

The table below gives the principal specific heat capacities for some well-known gases.

Gas | Specific heat capacity at constant pressure (J kg ^{-1}K^{-1}) |
Specific heat capacity at constant volume (J kg ^{-1}K^{-1}) |

Air | 993 | 714 |

Argon | 524 | 314 |

Carbon dioxide | 834 | 640 |

Carbon monoxide | 1050 | 748 |

Helium | 5240 | 3157 |

Hydrogen | 14300 | 10142 |

Nitrogen | 1040 | 741 |

Oxygen | 913 | 652 |

Water vapour | 2020 | - |

The value of c

The gas is now heated, the volume being kept constant (Figure 1). If the rise in temperature is dT then the heat input is C

If we now return to the
initial conditions and heat the gas again but this time allow it to expand, keeping the
pressure constant, the energy input for a temperature rise of dT is
C_{V}dT.

This not only has to raise the temperature of the gas but also
must do external work in expanding it by dV. Therefore

C_{P}dT =
C_{V}dT + dW

C_{P}dT = C_{V}dT +
PdV

We assume that the gas obeys the ideal gas equation for one mole PV = RT,
and therefore PdV = RdT. Substituting for dV we have:

This formula was first derived in 1842 by Robert Mayer.

The ratio of the two principal specific heats of a gas is denoted by the Greek letter g.

Therefore:

The value of this ratio depends on the atomicity of the gas – in other words how many atoms there are in one molecule.