The speed of sound in an ideal gas at a given temperature T is v . The rms speed of gas molecules at that temperature is v_ rms . The ratio of the velocities v and v_ rms for helium and oxygen gases are X and X^ .

Correct answer: B. 5 21 The speed of sound at a given temperature is given by v_ sound = (r R T)/(M) v_ rms = (3 R T)/(M) For oxygen molecules; v_0 v_ rms = (r_0 R T)/(M) (3 R T)/(M) = (r_0)/(3) For helium molecules; v_ He v_ rms = r_ He (R T)/(M) (3 R T)/(M) = r_ He 3 (r_ Be =(5)/(3), r_ O =(7)/(5)) x^ x = (5)/(9) (7)/(15) = (25)/(21) = 5 21

JEE Main

Exam: JEE Main
Subject: Physics
Chapter: Kinetic Theory of Gases

JEE Main · Physics · Kinetic Theory of Gases

The speed of sound in an ideal gas at a given temperature T is v ....

The speed of sound in an ideal gas at a given temperature

T

v

. The rms speed of gas molecules at that temperature is

v_{\mathrm{rms}}

. The ratio of the velocities

v

and

v_{\text {rms }}

for helium and oxygen gases are

X

and

X^{\prime}

. respectively. Then,

\frac{X}{X^{\prime}}

is equal to

Options

A $\frac{21}{\sqrt{5}}$
B $\frac{5}{\sqrt{21}}$
C $\sqrt{\frac{5}{21}}$
D $\frac{21}{5}$

Correct answer

B. $\frac{5}{\sqrt{21}}$

Step-by-step solution

The speed of sound at a given temperature is given by

v_{\text {sound }}=\sqrt{\frac{r R T}{M}} \Rightarrow v_{\mathrm{rms}}=\sqrt{\frac{3 R T}{M}}

For oxygen molecules;

\frac{v_0}{v_{\text {rms }}}=\frac{\sqrt{\frac{r_0 R T}{M}}}{\sqrt{\frac{3 R T}{M}}}=\sqrt{\frac{r_0}{3}}

For helium molecules;

\frac{v_{\mathrm{He}}}{v_{\mathrm{rms}}}=\frac{\sqrt{r_{\mathrm{He}} \frac{R T}{M}}}{\sqrt{\frac{3 R T}{M}}}=\sqrt{\frac{r_{\mathrm{He}}}{3}}\left(r_{\mathrm{Be}}=\frac{5}{3}, r_{\mathrm{O}}=\frac{7}{5}\right)\frac{x^{\prime}}{x}=\frac{\sqrt{\frac{5}{9}}}{\sqrt{\frac{7}{15}}}=\sqrt{\frac{25}{21}}=\frac{5}{\sqrt{21}}

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