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In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure (C_P) to heat capacity at constant volume (C_V). It is sometimes also known as the isentropic expansion factor and is denoted by γ (gamma) for an ideal gas^{[note 1]} or κ (kappa), the isentropic exponent for a real gas. The symbol γ is used by aerospace and chemical engineers.

${\displaystyle \gamma ={\frac {C_{P}}{C_{V}}}={\frac {{\bar {C}}_{P}}{{\bar {C}}_{V}}}={\frac {c_{P}}{c_{V}}},}$

where C is the heat capacity, ${\displaystyle {\bar {C}}}$ the molar heat capacity (heat capacity per mole), and c the specific heat capacity (heat capacity per unit mass) of a gas. The suffixes P and V refer to constant-pressure and constant-volume conditions respectively.

The heat capacity ratio is important for its applications in thermodynamical reversible processes, especially involving ideal gases; the speed of sound depends on this factor.

Thought experiment

To understand this relation, consider the following thought experiment. A closed pneumatic cylinder contains air. The piston is locked. The pressure inside is equal to atmospheric pressure. This cylinder is heated to a certain target temperature. Since the piston cannot move, the volume is constant. The temperature and pressure will rise. When the target temperature is reached, the heating is stopped. The amount of energy added equals C_V ΔT, with ΔT representing the change in temperature.

The piston is now freed and moves outwards, stopping as the pressure inside the chamber reaches atmospheric pressure. We assume the expansion occurs without exchange of heat (adiabatic expansion). Doing this work, air inside the cylinder will cool to below the target temperature.

To return to the target temperature (still with a free piston), the air must be heated, but is no longer under constant volume, since the piston is free to move as the gas is reheated. This extra heat amounts to about 40% more than the previous amount added. In this example, the amount of heat added with a locked piston is proportional to C_V, whereas the total amount of heat added is proportional to C_P. Therefore, the heat capacity ratio in this example is 1.4.

Another way of understanding the difference between C_P and C_V is that C_P applies if work is done to the system, which causes a change in volume (such as by moving a piston so as to compress the contents of a cylinder), or if work is done by the system, which changes its temperature (such as heating the gas in a cylinder to cause a piston to move). C_V applies only if ${\displaystyle P\,\mathrm {d} V=0}$, that is, no work is done. Consider the difference between adding heat to the gas with a locked piston and adding heat with a piston free to move, so that pressure remains constant.

In the second case, the gas will both heat and expand, causing the piston to do mechanical work on the atmosphere. The heat that is added to the gas goes only partly into heating the gas, while the rest is transformed into the mechanical work performed by the piston.

In the first, constant-volume case (locked piston), there is no external motion, and thus no mechanical work is done on the atmosphere; C_V is used. In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case.

Ideal-gas relations

For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., ${\displaystyle U=U(n,T)}$, where n is the amount of substance in moles. In thermodynamic terms, this is a consequence of the fact that the internal pressure of an ideal gas vanishes.

Mayer's relation allows us to deduce the value of C_V from the more easily measured (and more commonly tabulated) value of C_P:

${\displaystyle C_{V}=C_{P}-nR.}$

This relation may be used to show the heat capacities may be expressed in terms of the heat capacity ratio (γ) and the gas constant (R):

${\displaystyle C_{P}={\frac {\gamma nR}{\gamma -1}}\quad {\text{and}}\quad C_{V}={\frac {nR}{\gamma -1}},}$

Real-gas relations

As noted above, as temperature increases, higher-energy vibrational states become accessible to molecular gases, thus increasing the number of degrees of freedom and lowering γ. Conversely, as the temperature is lowered, rotational degrees of freedom may become unequally partitioned as well. As a result, both C_P and C_V increase with increasing temperature.

Despite this, if the density is fairly low and intermolecular forces are negligible, the two heat capacities may still continue to differ from each other by a fixed constant (as above, C_P = C_V + nR), which reflects the relatively constant PV difference in work done during expansion for constant pressure vs. constant volume conditions. Thus, the ratio of the two values, γ, decreases with increasing temperature.

However, when the gas density is sufficiently high and intermolecular forces are important, thermodynamic expressions may sometimes be used to accurately describe the relationship between the two heat capacities, as explained below. Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate.

Adiabatic process

This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas:

${\displaystyle PV^{\gamma }}$ is constant

Using the ideal gas law, ${\displaystyle PV=nRT}$:

${\displaystyle P^{1-\gamma }T^{\gamma }}$ is constant

${\displaystyle TV^{\gamma -1}}$ is constant

where P is the pressure of the gas, V is the volume, and T is the thermodynamic temperature.

In gas dynamics we are interested in the local relations between pressure, density and temperature, rather than considering a fixed quantity of gas. By considering the density ${\displaystyle \rho =M/V}$ as the inverse of the volume for a unit mass, we can take ${\displaystyle \rho =1/V}$ in these relations. Since for constant entropy, ${\displaystyle S}$, we have ${\displaystyle P\propto \rho ^{\gamma }}$, or ${\displaystyle \ln P=\gamma \ln \rho +\mathrm {constant} }$, it follows that

${\displaystyle \gamma =\left.{\frac {\partial \ln P}{\partial \ln \rho }}\right|_{S}.}$

For an imperfect or non-ideal gas, Chandrasekhar^[3] defined three different adiabatic indices so that the adiabatic relations can be written in the same form as above; these are used in the theory of stellar structure:

${\displaystyle {\begin{aligned}\Gamma _{1}&=\left.{\frac {\partial \ln P}{\partial \ln \rho }}\right|_{S},\\[2pt]{\frac {\Gamma _{2}-1}{\Gamma _{2}}}&=\left.{\frac {\partial \ln T}{\partial \ln P}}\right|_{S},\\[2pt]\Gamma _{3}-1&=\left.{\frac {\partial \ln T}{\partial \ln \rho }}\right|_{S}.\end{aligned}}}$

All of these are equal to ${\displaystyle \gamma }$ in the case of an ideal gas.

See also

• Relations between heat capacities
• Heat capacity
• Specific heat capacity
• Speed of sound
• Thermodynamic equations
• Thermodynamics
• Volumetric heat capacity

References

Notes