Question

The specific heat of air at constant pressure is \[1.005\text{ kJ/kg K}\] and the specific heat of air at constant volume is \[0.718\text{ kJ/kg K}\]. Find the specific gas constant.
(A). \[0.287\text{ kJ/kg K}\]
(B). \[0.21\text{ kJ/kg K}\]
(C). \[0.34\text{ kJ/kg K}\]
(D). \[0.19\text{ kJ/kg K}\]

Answer 1

- Hint: Use this relation to find the specific gas constant, \[{{c}_{p}}-{{c}_{v}}=R\], where \[{{c}_{p}}\] is the specific heat at constant pressure and \[{{c}_{v}}\] is the specific heat at constant volume.

Complete step-by-step solution -

Specific gas constant is the amount of mechanical work obtained by heating gas of unit mass through a unit temperature rise at constant pressure. The famous relation of specific heat with specific gas constant is given below.
\[\text{Specific gas constant = Specific heat at constant pressure - Specific heat at constant volume}\]
i.e.
\[{{c}_{p}}-{{c}_{v}}=R\]
We can assign the given values in this equation
\[1.005-0.718=R\]
\[1.005-0.718=0.287\text{ kJ/kg K}\]
Hence the option (A) is correct.

Additional information:
For all ideal gases, the specific heat at constant volume and specific heat at constant pressure is a function of temperature only. Their units are depending upon the mass considered. An ideal gas with constant values of specific heats and independent of temperature is referred to as perfect gas. Monoatomic gases and diatomic gases are considered as perfect gases at ordinary temperatures.
Monoatomic gases such as He, Ne, Ar, etc. possess constant specific heat at constant pressure over a wide range of temperature. It’s about \[\dfrac{5}{2}R\]. Specific heat at constant volume will be \[\dfrac{3}{2}R\] over a wide range of temperature.
Diatomic gases such as $H_2$, $O_2$, $N_2$, etc. possess constant specific heat at constant pressure at ordinary temperatures. It’s about \[\dfrac{7}{2}R\]. Specific heat at constant volume will be \[\dfrac{5}{2}R\] at ordinary temperature.
For polyatomic gases, specific heats will vary with the temperature. It will vary differently for each gas.
For an ideal gas, Julius Von Mayer formulated the relation between the specific heat at constant pressure and the specific heat at constant volume. Thus, it is known as Mayer’s relation.

Note: The Mayer’s relation is used for ideal gases. If we are dealing with homogeneous substances, the relation will become
\[{{C}_{p}}-{{C}_{v}}=VT\dfrac{\alpha _{V}^{2}}{{{\beta }_{T}}}\], where V is the volume, T is the temperature, \[{{\alpha }_{V}}\] is the thermal expansion coefficient and \[{{\beta }_{T}}\] is the isothermal compressibility.