Question

The root mean square velocity of an ideal gas in a closed container of fixed volume is increased from $5\times {10}^{4}cm{s}^{-1}$ to $10\times {10}^{4}cm{s}^{-1}$. Which of the following statements correctly explains how the change is accomplished?

• A. By heating the gas, the temperature is doubled.
• B. By heating the gas, the pressure is quadrupled.
• C. By heating the gas, the temperature is quadrupled.
• D. By heating the gas, the pressure is doubled.

Answer 1

Answer: (B), (C)

By heating the gas, the pressure is quadrupled.
By heating the gas, the temperature is quadrupled.

${\mu }_{rms}=\sqrt{\frac{3RT}{M}}$ or $\sqrt{\frac{3PV}{M}}$

$({\mu }_{rms}{)}_1=5\times {10}^{4}cm{s}^{-1}$

$({\mu }_{rms}{)}_2=10\times {10}^{4}cm{s}^{-1}$

$\therefore P_1=\frac{({\mu }_{rms}{)}_1^2\times M}{3V}=(5\times {10}^4{)}^2\times \frac{M}{3V}$

$\therefore P_2=\frac{({\mu }_{rms}{)}_2^2\times M}{3V}=(10\times {10}^4{)}^2\times \frac{M}{3V}$

or $\frac{P_2}{P_1}=\frac{(10\times {10}^4{)}^2}{(5\times {10}^4{)}^2}=\frac{100}{25}=4$

$\therefore P_2=4\times P_1$

Similarily, $T_2=4\times T_1$

Hence, the correct options are $\text{B}$ and $\text{C}$