The molecules of a given mass of a gas have a rms velocity of 200 m/sec at 27°C and 1.0 x 10⁵ N /m² pressure. When the temperature is 127°C and pressure is 0.5x10⁵N /m² , the rms velocity in m/sec will be

The correct option is 400/√3.

Step-by-step Explanation:

The root-mean-square (rms) velocity of the molecules of a gas is given by the formula:

$v_{\text{rms}}=\sqrt{\frac{3RT}{M}}$
where:
• $R$ is the universal gas constant,
• $T$ is the absolute temperature of the gas in Kelvin, and
• $M$ is the molar mass of the gas.

From this formula, we can see that for a given gas (constant molar mass $M$), the rms velocity depends only on the absolute temperature $T$ and is completely independent of the pressure of the gas. Therefore, the change in pressure has no effect on the rms velocity.

Thus, we can write the proportionality relation:

$v_{\text{rms}}\propto \sqrt{T}$
Using this relation, the ratio of the final rms velocity ($v_2$) to the initial rms velocity ($v_1$) is:

$\frac{v_2}{v_1}=\sqrt{\frac{T_2}{T_1}}$

Given Data:
• Initial velocity, $v_1=200\text{m/sec}$
• Initial temperature, $T_1=27°\text{C}=27+273=300\text{K}$
• Final temperature, $T_2=127°\text{C}=127+273=400\text{K}$

Substituting these values into the ratio formula:

$\frac{v_2}{200}=\sqrt{\frac{400}{300}}$

$\frac{v_2}{200}=\sqrt{\frac{4}{3}}=\frac{2}{\sqrt{3}}$

$v_2=200\times \frac{2}{\sqrt{3}}=\frac{400}{\sqrt{3}}\text{m/sec}$