Question

Four molecules of a gas have the velocities 3 x 10${}^4$ $cm/sec$, 4 x 10${}^4$ $cm/sec$, 2 x 10${}^4$ $cm/sec$ and 5 x 10${}^4$ $cm/sec$. The $rms$ velocity of the molecules is :

• A. $3.675\times {10}^{3}cm/sec$
• B. $36.75\times {10}^{3}cm/sec$
• C. $36.75\times {10}^{4}cm/sec$
• D. $3.675\times {10}^{2}cm/sec$

Answer 1

Answer: (B)

$36.75\times {10}^{3}cm/sec$

Let the $4$ velocities be $V_1,V_2,V_3$ and $V_4.$

$V_1=3\times {10}^4$ $cm/s,$

$V_2=4\times {10}^4$ $cm/s,$

$V_3=2\times {10}^4$ $cm/s,$

$V_4=5\times {10}^4$ $cm/s$

r.m.s velocity $V_{rms}=\sqrt{\frac{V_1^2+V_2^2+V_3^2+V_4^2}{N}}$

$N=$ No. of velocities.

$V_{rms}=\sqrt{\frac{9\times {10}^8+16\times {10}^8+4\times {10}^8+25\times {10}^8}{N}}$

$=\sqrt{\frac{{10}^8[9+16+4+25]}{4}}$

$={10}^4\sqrt{\frac{54}{4}}={10}^4\times 3.675=36.75\times {10}^3$

$\therefore V_{rms}=36.75\times {10}^3$ $cm/sec$.

Therefore, the option is B.