Specific heat ratio

The specific heat ratio of a gas is the ratio of the specific heat at constant pressure, $$C_p$$, to the specific heat at constant volume, $$C_v$$. It is sometimes referred to as the adiabatic index or the heat capacity ratio or the isentropic expansion factor or the adiabatic exponent or the isentropic exponent.

Either $$\kappa$$ (kappa), $$k$$ (Roman letter k) or $$\gamma$$ (gamma) may be used to denote the specific heat ratio:


 * $$\kappa = k = \gamma = \frac{C_p}{C_v}$$

where:


 * $$C$$ = the specific heat of a gas
 * $$p$$ = refers to constant pressure conditions
 * $$v$$ = refers to constant volume conditions



Ideal gas relations
In thermodynamic terminology, $$C_p$$ and $$C_V$$ may be expressed as:


 * $$C_p=\left(\frac{\partial H}{\partial T}\right)_p $$    and     $$C_V=\left(\frac{\partial U}{\partial T}\right)_V$$

where $$T$$ stands for temperature, $$H$$ for the enthalpy and $$U$$ for the internal energy. 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 specific heat ratio of an ideal gas is the ratio of the enthalpy to the internal energy:


 * $$\kappa = \frac{H}{U}$$

The specific heats at constant pressure, $$C_p$$, of various gases are relatively easy to find in the technical literature. However, it can be difficult to find values of the specific heats at constant volume, $$C_v$$. When needed, given $$C_p$$, the following equation can be used to determine $$C_v$$ :


 * $$C_v = C_p - R$$

where $$R$$ is the molar gas constant (also known as the Universal gas constant). This equation can be re-arranged to obtain:


 * $$ C_p = \frac{\kappa R}{\kappa - 1} \qquad \mbox{and} \qquad C_v = \frac{R}{\kappa - 1}$$

Relation with degrees of freedom
The specific heat ratio ( $$k$$ ) for an ideal gas can be related to the degrees of freedom ( $$f$$ ) of a molecule by:


 * $$ \kappa = \frac{f+2}{f}$$

Thus for a monatomic gas, with three degrees of freedom:


 * $$ \kappa = \frac{5}{3} = 1.67$$

and for a diatomic gas, with five degrees of freedom (three translational and two rotational degrees of freedom, the vibrational degree of freedom is not involved except at high temperatures):


 * $$ \kappa = \frac{7}{5} = 1.4$$.

Earth's atmospheric air is primarily made up of diatomic gases with a composition of ~78% nitrogen (N2) and ~21% oxygen (O2). At 20 °C and an absolute pressure of 101.325 kPa, the atmospheric air can be considered to be an ideal gas. This results in a value of:
 * $$ \kappa = \frac{5 + 2}{5} = \frac{7}{5} = 1.4$$

The specific heats of real gases (as differentiated from ideal gases) are not constant with temperature. As temperature increases, higher energy rotational and vibrational states become accessible to molecular gases, thus increasing the number of degrees of freedom and lowering $$\kappa$$. For a real gas, $$C_p$$ and $$C_v$$ usually increase with increasing temperature and $$\kappa$$ decreases. Some correlations exist to provide values of $$\kappa$$ as a function of the temperature.

Isentropic compression or expansion of ideal gases
The specific heat ratio plays an important part in the isentropic process of an ideal gas (i.e., a process that occurs at constant entropy):


 * (1)   $$ p_1{V_1}^\kappa = p_2{V_2}^\kappa $$

where, $$p$$ is the absolute 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.

Using the ideal gas law, $$pV = nRT$$, equation (1) can be re-arranged to:


 * (2)   $$\frac{p_1}{p_2} = \left(\frac{V_2}{V_1}\right)^\kappa = \left(\frac{T_2}{T_1}\right)^\kappa  \left(\frac{p_1}{p_2}\right)^\kappa$$

where $$T$$ is the absolute temperature. Re-arranging further:


 * (3)   $$\left(\frac{T_2}{T_1}\right)^\kappa = \left(\frac{p_1}{p_2}\right)\left(\frac{p_2}{p_1}\right)^\kappa = \left(\frac{p_2}{p_1}\right)^{\kappa-1}$$

we obtain the equation for the temperature change that occurs when an ideal gas is isentropically compressed or expanded:


 * (4)    $$\frac{T_2}{T_1} = \left(\frac{p_2}{p_1}\right)^{(\kappa-1)/\kappa}$$

Equation (4) is widely used to model ideal gas compression or expansion processes in internal combustion engines, gas compressors and gas turbines.

Determination of $$C_v$$ values
Values for $$C_p$$ are readily available, but values for $$C_v$$ are not as available and often need to be determined. Values based on the ideal gas relation of $$C_p - C_v = R$$ are in many cases not sufficiently accurate for practical engineering calculations. If at all possible, an experimental value should be used.

A rigorous value can be calculated by determining $$C_v$$ from the residual property functions (also referred to as departure functions)  using this relation:


 * $$C_v = C_p + T \frac{{\; \left( {\frac{\part p}{\part T}} \right) }^2_V} {\left( \frac{\part p}{\part V} \right)_T} $$

Equations of state (EOS) (such as the Peng-Robinson equation of state) can be used to solve this relation and to provide values of $$C_v$$ that match experimental values very closely.