Question

The $r.m.s$ speed of molecules at a temperature $27K$ and pressure $1.5$ bar is $1\times {10}^{4}cm/s$. If both temperature and pressure are raised three times, the rms speed of the gas will be :

• A. $9\times {10}^{4}cm/s$
• B. $3\times {10}^{4}cm/s$
• C. $\sqrt{3}\times {10}^{4}cm/s$
• D. $1\times {10}^{4}cm/s$

Answer 1

The correct option is A $9\times {10}^{4}cm/s$

Given,

$T_1=27K$

$V_{1}rms=1\times {10}^{4}cm/s$

The root mean square speed of the gas is given by,

$V_{rms}=\sqrt{\frac{3RT}{m}}$

$V_{rms}\propto \sqrt{T}$

$\frac{V_{1}rms}{V_{2}rms}=\sqrt{\frac{T_1}{T_2}}$

$V_{2}rms=V_{1}rms\sqrt{\frac{T_2}{T_1}}$

Now, $T_2=3T_1$

$V_{2}rms=\sqrt{3}{V}_{1}rms$

$V_{2}rms=\sqrt{3}\times {10}^{4}cm/s$

The correct option is C.