Question

R.M.S velocity of an ideal gas molecules is $300\text{m/s}$ at $1\text{atm}$ pressure. If pressure is increased to four times, then R.M.S velocity becomes
$($The density of gas remains same$.)$

• A. $150\text{m/s}$
• B. $300\text{m/s}$
• C. $600\text{m/s}$
• D. $100\text{m/s}$

Answer 1

The correct option is C $600\text{m/s}$

We know that,
Pressure $\left(P\right)=\frac{1}{3}\times \frac{M}{V}\times v_{\text{rms}}^2$
Where $M=$ Molar mass
$V=$ Volume
$\therefore P\propto v_{\text{rms}}^2$ for constant $M$ and $V$
$\Rightarrow \frac{P_1}{P_2}=\frac{v_{\text{rms}\left(1\right)}^2}{v_{\text{rms}\left(2\right)}^2}$
Given $P_2=4P_1$
$\therefore \frac{P_1}{P_2}=\frac{1}{4}=\frac{v_{\text{rms}\left(1\right)}^2}{v_{\text{rms}\left(2\right)}^2}$
$\Rightarrow v_{\text{rms}\left(2\right)}^2=4v_{\text{rms}\left(1\right)}^2=4\times (300{)}^2$
$\Rightarrow v_{\text{rms}\left(2\right)}=2\times 300=600\text{m/s}$