Heat capacity ratio

The heat capacity ratio or adiabatic index or ratio of specific heats, 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 $$\gamma$$ (gamma) or $$\kappa$$ (kappa). The latter symbol kappa is primarily used by chemical engineers. Mechanical engineers use the more common Roman letter $$k$$.
 * $$ \gamma = \frac{C_P}{C_V}$$

where, $$C$$ is the heat capacity or the specific heat capacity of a gas, suffix $$P$$ and $$V$$ refer to constant pressure and constant volume conditions respectively.

To understand this relation, consider the following experiment:

A closed cylinder with a locked piston contains air. The pressure inside is equal to the outside air pressure. This cylinder is heated. Since the piston cannot move the volume is constant. Temperature and pressure rise. Heating is stopped and the energy added to the system, which is proportional to $$C_V$$, is noted. The piston is now freed and moves outwards, expanding without exchange of heat (adiabatic expansion). Doing this work (proportional to $$C_P$$) cools the air inside the cylinder to below its starting temperature. To return to the starting temperature (still with a free piston) the air must be heated. This extra heat amounts to about 40% of the previous amount added.

In the preceding paragraph, it may not be obvious how $$C_P$$ is involved because during the expansion and subsequent heating, the pressure does not remain constant. 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 (e.g. by moving a piston so as to compress the contents of a cylinder), or if work is done by the system which changes its volume (e.g. heating the gas in a cylinder to cause a piston to move). $$C_V$$ applies only if $$P dV$$ - that is, the work done - is zero. 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 heat capacity is constant with temperature. Accordingly we can express the enthalpy as $$H = C_P T$$ and the internal energy as $$U = C_V T$$. Thus, it can also be said that the heat capacity ratio is the ratio between the enthalpy to the internal energy:
 * $$ \gamma = \frac{H}{U}$$

Furthermore, the heat capacities can be expressed in terms of heat capacity ratio ( $$\gamma$$ ) and the gas constant ( $$R$$ ):
 * $$ C_P = \frac{\gamma R}{\gamma - 1} \qquad \mbox{and} \qquad C_V = \frac{R}{\gamma - 1}$$

It can be rather difficult to find tabulated information for $$C_V$$, since $$C_P$$ is more commonly tabulated. The following relation, can be used to determine $$C_V$$:
 * $$C_V = C_P - R$$

Relation with degrees of freedom
The heat capacity ratio ( $$\gamma$$ ) for an ideal gas can be related to the degrees of freedom ( $$f$$ ) of a molecule by:
 * $$ \gamma = \frac{f+2}{f}$$

Thus we observe that for a monatomic gas, with three degrees of freedom:
 * $$ \gamma\ = \frac{5}{3} = 1.67$$,

while for a diatomic gas, with five degrees of freedom (at room temperature):
 * $$ \gamma = \frac{7}{5} = 1.4$$.

E.g.: The terrestrial air is primarily made up of diatomic gasses (~78% nitrogen (N2) and ~21% oxygen (O2)) and, at standard conditions it can be considered to be an ideal gas. A diatomic molecule has five degrees of freedom (three translational and two rotational degrees of freedom, the vibrational degree of freedom is not involved except at high temperatures). This results in a value of
 * $$\gamma = \frac{5 + 2}{5} = \frac{7}{5} = 1.4$$.

This is consistent with the measured adiabatic index of approximately 1.403 (listed above in the table).

Real gas relations
As temperature increases, higher energy rotational and vibrational states become accessible to molecular gases, thus increasing the number of degrees of freedom and lowering $$\gamma$$. For a real gas, $$C_P$$ and $$C_V$$ usually increase with increasing temperature and $$\gamma$$ decreases. Some correlations exist to provide values of $$\gamma$$ as a function of the temperature.

Thermodynamic Expressions
Values based on approximations (particularly $$C_p - C_v = R$$) are in many cases not sufficiently accurate for practical engineering calculations such as flow rates through pipes and valves. An experimental value should be used rather than one based on this approximation, where possible. A rigorous value for the ratio $$\frac{C_p}{C_v}$$ can also be calculated by determining $$C_v$$ from the residual properties expressed as:


 * $$ C_p - C_v \ = \ -T \frac {\left(\frac{\part V}{\part P}\right)_T} \ = \ -T \frac{{ \left( {\frac{\part P}{\part T}} \right) }^2} {\frac{\part P}{\part V}} $$

Values for $$C_p$$ are readily available and recorded, but values for $$C_v$$ need to be determined via relations such as these. See here for the derviation of the thermodynamic relations between the heat capacities.

The above definition is the approach used to develop rigorous expressions from equations of state (such as Peng-Robinson), which match experimental values so closely that there is little need to develop a database of ratios or $$C_v$$ values. Values can also be determined through numerical derivatives (peturb T and P (independently!) and calculate $$\frac{\Delta V}{\Delta T}$$ and $$\frac{\Delta V}{\Delta P}$$).

Adiabatic process
This ratio also gives the important relation for an isentropic (quasistatic, adiabatic process, reversible) process of a simple compressible calorically perfect ideal gas:
 * $$ p_1{V_1}^\gamma = p_2{V_2}^\gamma = \emph{constant} $$

where, $$p$$ is the pressure and $$V$$ is the volume. The subscripts 1 and 2 refer to conditions before and after the process, or at any time during that process.