The root mean square velocity of one mole of a monoatomic gas having molar mass $M$ is $U_{r m s}$ . The relation between average kinetic energy $(E)$ of the gas and $U_{r m s}$ is :

U_{r m s} = \sqrt{\frac{3 E}{2 M}}

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U_{r m s} = \sqrt{\frac{2 E}{3 M}}

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U_{r m s} = \sqrt{\frac{2 E}{M}}

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U_{r m s} = \sqrt{\frac{E}{3 M}}

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Solution

The correct option is C $U_{r m s} = \sqrt{\frac{2 E}{M}}$
The root mean square velocity of one mole of a monoatomic gas having molar mass M is $U_{r m s}$ . The relation between average kinetic energy (E) of the gas and $U_{r m s}$ is $U_{r m s} = \sqrt{\frac{2 E}{M}}$
The root mean square velocity of one mole of a monoatomic gas is given by the following expression.
$U_{r m s}$ = $\sqrt{\frac{3 R T}{M}}$ ......(1)
The average kinetic energy (E) of the gas is given by the following expression.
$E = \frac{3}{2} R T$
$R T = \frac{2}{3} E$ ......(2)
Substitute (2) into (1)
$U_{r m s}$ = $\sqrt{\frac{3 \times \frac{2}{3} E}{M}}$
$U_{r m s}$ = $\sqrt{\frac{2 E}{M}}$
Hence, the option (C) is the correct answer.

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The root mean square velocity of one mole of a monoatomic gas having molar mass M is Urms. The relation between average kinetic energy (E) of the gas and Urms is :

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