Question

The temperature of an ideal gas is increased from ${27}^{\circ }\text{C}$ to ${127}^{\circ }\text{C}$. Then, percentage increase in $V_{rms}$ is

• A. $37\%$
• B. $11\%$
• C. $33\%$
• D. $15.5\%$

Answer 1

The correct option is D $15.5\%$

Given,
Initial temperature of ideal gas $\left(T_1\right)=27+273=300\text{K}$
Final temperature of ideal gas $\left(T_2\right)=273+127=400\text{K}$
We know that, $v_{rms}=\sqrt{\frac{3RT}{M}}$
Then, $\%$ increase in $v_{rms}$
$=\frac{\sqrt{\frac{3R{T}_2}{M}}-\sqrt{\frac{3R{T}_1}{M}}}{\sqrt{\frac{3R{T}_1}{M}}}\times 100$
$=\frac{\sqrt{T_2}-\sqrt{T_1}}{\sqrt{T_1}}\times 100$
$=\frac{\sqrt{400}-\sqrt{300}}{\sqrt{300}}\times 100$
$=\frac{20-17.32}{17.32}\times 100$
$\therefore v_{rms}\approx 15.5\%$