At $$600$$ K, the root mean square (rms) speed of gas X (molar mass $$= 40$$) is equal to the most probable speed of gas Y at $$90$$ K. The molar mass of the gas Y is _____ g mol$$^{-1}$$. (Nearest integer)

Explanation

According to the kinetic theory of gases,

The root mean square speed is

$$v_{rms}=\sqrt{\frac{3RT}{M}}$$

The most probable speed is

$$v_{mp}=\sqrt{\frac{2RT}{M}}$$

The problem states that

$$(v_{rms})_X=(v_{mp})_Y$$

Substituting the respective expressions,

$$\sqrt{\frac{3RT_X}{M_X}}=\sqrt{\frac{2RT_Y}{M_Y}}$$

Squaring both sides and cancelling $$R$$,

$$\frac{3T_X}{M_X}=\frac{2T_Y}{M_Y}$$

Substituting the given values,

$$\frac{3\times600}{40}=\frac{2\times90}{M_Y}$$

$$\frac{1800}{40}=\frac{180}{M_Y}$$

$$45=\frac{180}{M_Y}$$

$$M_Y=\frac{180}{45}$$

$$M_Y=4\text{ g mol}^{-1}$$

Hence, the molar mass of gas $$Y$$ is

$$4\text{ g mol}^{-1}$$

Therefore, the correct answer is **4**.