Question

Pressure of the gas remaining same, the temperature at which r.m.s. speed of the gas molecules is double its value at \( 27^\circ\text{C} \) is

• A. $1200^\circ$C
• B. $927^\circ$C
• C. $627^\circ$C
• D. $300^\circ$C

Hint:

- $v_{\text{rms}} \propto \sqrt{T}$ - Doubling speed $\Rightarrow$ temperature becomes four times

Answer 1

The Correct Option is B

Concept:
Root mean square speed: \[ v_{\text{rms}} = \sqrt{\frac{3kT}{m}} \;\Rightarrow\; v_{\text{rms}} \propto \sqrt{T} \]

Step 1: Initial temperature in Kelvin.
\[ T_1 = 27 + 273 = 300\ \text{K} \]

Step 2: Given condition.
\[ v_2 = 2v_1 \;\Rightarrow\; \frac{v_2}{v_1} = 2 = \sqrt{\frac{T_2}{T_1}} \]

Step 3: Square both sides.
\[ 4 = \frac{T_2}{300} \;\Rightarrow\; T_2 = 1200\ \text{K} \]

Step 4: Convert to Celsius.
\[ T_2 = 1200 - 273 = 927^\circ\text{C} \]