Question

For one mole of gas the average kinetic energy is given as E. The $U_{rms}$ of gas is:

• A. $\sqrt{\frac{2E}{M}}$
• B. $\sqrt{\frac{3E}{M}}$
• C. $\sqrt{\frac{2E}{3M}}$
• D. $\sqrt{\frac{3E}{2M}}$

Answer 1

The correct option is A $\sqrt{\frac{2E}{M}}$

Solution:- (A) $\sqrt{\frac{2E}{M}}$

Given that:-

$U_{av.}=E.....\left(1\right)$

As we know that,

$U_{av.}=\frac{3}{2}RT.....\left(2\right)$

$U_{rms}=\sqrt{\frac{3RT}{M}}.....\left(3\right)$

From ${eq}^n\left(1\right)\&\left(2\right)$, we have

$E=\frac{3}{2}RT$

$\Rightarrow 3RT=2E.....\left(4\right)$

Now, from ${eq}^n\left(3\right)\&\left(4\right)$, we have

$U_{rms}=\sqrt{\frac{2E}{M}}$

Hence, $U_{rms}=\sqrt{\frac{2E}{M}}$.