Question

The molecules of a given mass of a gas have root mean square speed of 120 m/s at 88°C and 1 atmospheric pressure. The root mean square speed of the molecules at 127°C and 2 atmospheric pressure is:

• A. 105.2 m/s
• B. 1.443 m/s
• C. 126.3 m/s
• D. 88/127 m/s

Hint:

The root mean square speed is directly proportional to the square root of the temperature in Kelvin. Changes in pressure do not affect the rms speed for an ideal gas when only temperature is changing.

Answer 1

The Correct Option is C

The root mean square (rms) speed of gas molecules is related to temperature by the following equation:
\[ v_{\text{rms}} \propto \sqrt{T} \] where \(v_{\text{rms}}\) is the root mean square speed and \(T\) is the temperature in Kelvin.
Given:
- Initial speed \(v_{\text{rms1}} = 120 \, \text{m/s}\) at \(T_1 = 88^\circ \text{C} = 88 + 273 = 361 \, \text{K}\). - Final temperature \(T_2 = 127^\circ \text{C} = 127 + 273 = 400 \, \text{K}\).
- The pressure change does not affect the rms speed directly in this case since we are comparing the same gas under ideal conditions.
Now, using the relation \( \frac{v_2}{v_1} = \sqrt{\frac{T_2}{T_1}} \), we can solve for \(v_2\): \[ \frac{v_2}{120} = \sqrt{\frac{400}{361}} \] \[ v_2 = 120 \times \sqrt{\frac{400}{361}} = 120 \times \sqrt{1.107} \approx 120 \times 1.053 \approx 126.3 \, \text{m/s} \] Thus, the root mean square speed at 127°C and 2 atmospheric pressure is approximately 126.3 m/s.