Question

‘N’ moles of the diatomic gas in the cylinder are at a temperature ‘T’. Heat is supplied to the cylinder such that the temperature remains constant but n moles of the diatomic gas get converted into monatomic gas. What is the total change in kinetic energy of the gas?
A. \[\dfrac{5}{2}nRT\]
B. \[\dfrac{1}{2}nRT\]
C. \[0\]
D. \[\dfrac{3}{2}nRT\]

Answer 1

Hint: Mono-atomic gas are the atoms which have one atomicity. The diatomic has two atomicities. The atomicity is the number of atoms combined to form the molecule to remain in stable state.

Complete solution
Kinetic energy of the gas is the amount of energy required to move. It shows the randomness of the molecule in the tank.
Heat is supplied to the container but the temperature is constant. This is due to the heat being used up to break the bond between the molecules to form the atom. Diatomic molecules are converted to the monatomic gas. Here, the gaseous molecules will be following all the assumptions of the kinetic theory of gas.
The difference in the value of the internal kinetic energy and the final kinetic energy is the value of the total kinetic energy.
As per the law of equipartition of energy “the system is in equilibrium at the absolute temperature T, the total energy distribution is done equally in different energy modes of absorption, the energy in each mode being equal to \[\dfrac{1}{2}{k_B}T\]. Each vibrational frequency has two modes of energy (kinetic energy and potential) with corresponding energy equal to \[{k_B}T\].
Specific heat of the gas at constant volume:\[{C_v} = \dfrac{{fR}}{2}\]
f is degree of freedom and R is gas constant.
Number of diatomic gases is N; hence the number of monoatomic gases is 2N.
As T is constant and volume is constant work done is zero.
Change in internal energy is equal to heat supplied
Change in total kinetic energy \[ = \;{n_{f\;}}{C_{vf\;}}T\; - \;{n_{i\,\;}}{C_{vi\;}}T\]
\[
{n_i}\; = \;N,\;{C_{vi}} = \;\dfrac{{5R}}{2}\\
{n_i}\; = \;2N,\;{C_{vf}} = \;\dfrac{{3R}}{2}\\
\therefore \Delta K = \;3NRT - \dfrac{5}{2}NRT = \;\dfrac{1}{2}NRT
\]

Note:
The value of $C_v$ is different for both monoatomic and diatomic gases.Interchanging these will affect the solution.