Question

A rigid diatomic ideal gas undergoes an adiabatic process at room temperature. The relation between temperature and volume of this process is TV\[^x\] = constant, then what is the value of x?
A. $\dfrac{5}{3}$
B. $\dfrac{2}{5}$
C. $\dfrac{2}{3}$
D. $\dfrac{3}{5}$

Answer 1

Hint:The adiabatic process in gases depends upon the ratio of the specific heats of the gas also known as the heat capacity ratio, represented by $\gamma $. The value of x can be found with the help of $\gamma $.

Complete step by step answer:
An adiabatic process is a thermodynamic process in which there is no transfer of energy and mass between the system and the surrounding. For ideal gas, the process is governed by the equation,
$P{V^\gamma }$ = constant .…equation (1)
Since the gas in the system is ideal, we can apply the ideal gas equation on this system. The ideal gas equation is given as follows,
$PV = nRT$ …equation (2)
Where, P is the pressure, V is the volume, n is the number of moles of the ideal gas, R is the universal gas constant and T is the temperature of the system of gas.
Since the question asks us about the relation for adiabatic process in terms temperature and volume, but equation (1) is in terms of pressure and volume. Therefore, we will use equation (2) to obtain an expression of pressure in terms of temperature and volume. We obtain,
$P = \dfrac{{nRT}}{V}$ …equation (3)
We substitute the expression of pressure obtained in equation (3) into equation (1), we get,
$\dfrac{{nRT}}{V}{V^\gamma } = const. \Rightarrow nRT{V^{\gamma - 1}} = const.$
However, the number of moles of the gas and the universal gas constant do not change; therefore, they can be transferred to the right side of the equation, making it,
$T{V^{\gamma - 1}}$ = constant
Therefore, the value of x in TV\[^x\] = constant is equal to $\gamma $ - 1.
The heat capacity ratio, $\gamma $, can also be expressed in terms of the degrees of freedom of the gas. It is given as,
$\gamma = \dfrac{{f + 2}}{f} = 1 + \dfrac{2}{f}$
A diatomic gas molecule has 5 degrees of freedom at normal temperatures, which can be decomposed into three translational and two rotational degrees of freedom. Hence, the value of specific heat ratio for diatomic gas is,
$\gamma = 1 + \dfrac{2}{5} = \dfrac{7}{5}$
$ \therefore \gamma - 1 = \dfrac{7}{5} - 1 = \dfrac{2}{5}$
Therefore, the numerical value of x is $\dfrac{2}{5}$.

Hence, the correct answer is option B.

Additional Information:
The degree of freedom for monoatomic gases is three, which is equal to three translational degrees of freedom. Therefore, the value of $\gamma $ for monoatomic gas is $1 + \dfrac{2}{3} = \dfrac{5}{3}$. If the diatomic gas in the system is replaced by monoatomic gas, then the relation between temperature and volume of the process will be given by,
$T{V^{\gamma - 1}}$ = constant, where $\gamma - 1 = \dfrac{5}{3} - 1 = \dfrac{2}{3}$
At higher temperatures, the degrees of freedom of diatomic gas increase by one because of the vibration in molecules.

Note: While solving questions where we need to find the unknown power of a variable, we need to remember the equations that define the process mentioned in the question. In case the question does not mention the type of gas, we assume the gas to be ideal and we can then apply the ideal gas equation. But if the question exclusively mentions that the gas is real, the ideal gas equation cannot be applied.