QJEEAdv23Ph2e5a: JEE Advanced 2023 Physics - Class 11 JEEAdv23Ph2e5a Solution: An ideal gas is in thermodynamic equilibrium. The number of…

1. Formula for Sound Velocity:

- The velocity of sound, denoted as $$v$$, is determined by the equation $$v=\sqrt{\frac{\gamma RT}{M}}$$, where:

- $$\gamma$$ is a factor given by $$\gamma =1+\frac{2}{n}$$.

- $$R$$ is the gas constant.

- $$T$$ is the temperature.

- $$M$$ is the molar mass of the gas.

2. Relation of $$\gamma$$ with $$n$$:

- The value of $$\gamma$$ is inversely related to $$n$$. This means as $$n$$ increases, $$\gamma$$ decreases.

3. Internal Energy Relation:

- The internal energy, denoted as $$U$$, for a mole of gas is given by $$U=\frac{nRT}{2}$$.

4. Effect of Increasing $$n$$ on Sound Velocity and Internal Energy:

When the value of $$n$$ increases:

- The factor $$\gamma$$ decreases, according to its formula.

- As a result, the velocity of sound $$v$$ also decreases, as it is dependent on $$\sqrt{\gamma }$$.

- Conversely, the internal energy $$U_n$$ increases with the increase in $$n$$, as evident from the internal energy formula.

Therefore, in summary, a higher value of $$n$$ leads to a lower velocity of sound and higher internal energy in the context of these formulas.