1. The root mean square speed of gas molecules is given by

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

So, for a given gas, $v_{\text{rms}} \propto \sqrt{T}$.

2. Important point: $v_{\text{rms}}$ depends only on absolute temperature, not on pressure.

3. Convert temperatures to Kelvin:

• At $27^\circ \text{C}$: $T_1 = 27 + 273 = 300\,\text{K}$
• At $127^\circ \text{C}$: $T_2 = 127 + 273 = 400\,\text{K}$

4. Use the proportionality:

$\frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}} = \sqrt{\frac{400}{300}} = \sqrt{\frac{4}{3}} = \frac{2}{\sqrt{3}}$

Given:

$v_1 = 200\,\text{m s}^{-1}$

So,

$v_2 = 200 \cdot \frac{2}{\sqrt{3}} = \frac{400}{\sqrt{3}}\,\text{m s}^{-1}$

5. Comparing with the given form

$v_2 = \frac{x}{\sqrt{3}}\,\text{m s}^{-1}$

we get

$x = 400$

Final Answer

$\boxed{400}$

The derived answer matches the stored correct answer.