A closed vessel contains 10 g of an ideal gas X at 300 K, which exerts 2 atm pressure. At the same temperature, 80 g of another ideal gas Y is added to it and the pressure becomes 6 atm. The ratio of root mean square velocities of X and Y at 300 K is

The correct option is 2 ∶ 1.

Let us find the ratio of the root mean square (rms) velocities of gas X and gas Y step-by-step.

First, the formula for the root mean square velocity ($u_{\text{rms}}$) of an ideal gas is given by:
$u_{\text{rms}}=\sqrt{\frac{3RT}{M}}$
where $R$ is the universal gas constant, $T$ is the temperature in Kelvin, and $M$ is the molar mass of the gas.

Since both gases X and Y are at the same temperature ($T=300\text{K}$), the ratio of their root mean square velocities is:
$\frac{u_X}{u_Y}=\sqrt{\frac{M_Y}{M_X}}$
where $M_X$ and $M_Y$ are the molar masses of gas X and gas Y, respectively.

Now, let us find the relation between the molar masses using the ideal gas equation, $PV=nRT$.

For the initial state containing only gas X:
The pressure $P_X=2\text{atm}$.
The mass of gas X, $w_X=10\text{g}$.
The number of moles of gas X is:
$n_X=\frac{10}{M_X}$
According to the ideal gas equation, we have:
$P_X=\frac{n_{X}RT}{V}\Rightarrow 2=\frac{10}{M_X}·\frac{RT}{V}$ --- (Equation 1)

After adding 80 g of gas Y, the total pressure becomes $6\text{atm}$.
According to Dalton's law of partial pressures, the total pressure is the sum of the partial pressures:
$P_{\text{total}}=P_X+P_Y$
$6=2+P_Y\Rightarrow P_Y=4\text{atm}$

The mass of gas Y added is $w_Y=80\text{g}$.
The number of moles of gas Y is:
$n_Y=\frac{80}{M_Y}$
Applying the ideal gas equation for gas Y:
$P_Y=\frac{n_{Y}RT}{V}\Rightarrow 4=\frac{80}{M_Y}·\frac{RT}{V}$ --- (Equation 2)

Dividing Equation 1 by Equation 2:
$\frac{2}{4}=\frac{(10/M_X)}{(80/M_Y)}$
$\frac{1}{2}=\frac{10}{80}·\frac{M_Y}{M_X}$
$\frac{1}{2}=\frac{1}{8}·\frac{M_Y}{M_X}$
Multiplying both sides by 8:
$\frac{M_Y}{M_X}=4$

Now, substitute this ratio back into the root mean square velocity ratio formula:
$\frac{u_X}{u_Y}=\sqrt{\frac{M_Y}{M_X}}=\sqrt{4}=2$

Thus, the ratio of root mean square velocities of X and Y is 2 ∶ 1.