Question

An ideal gas $(\gamma =1.5)$ is expanded adiabatically. How many times has the gas to be expanded to reduce the root mean square velocity of the molecules $2.0$ times?

• A. $4$ times
• B. $16$ times
• C. $8$ times
• D. $2$ times

Answer 1

Answer: (B)

$16$ times

Root mean square velocity, $v=\sqrt{\frac{2RT}{m}}$ $⟹T\propto v^2$

$\therefore$ $\frac{T_2}{T_1}=\frac{v_2^2}{v_1^2}=\frac{(v_1/2{)}^2}{v_1}=\frac{1}{4}$ ..........(1)

From $T{V}^{\gamma -1}=$ constant

$⟹$ $\frac{V_2}{V_1}=(\frac{T_1}{T_2}{)}^{\frac{1}{\gamma -1}}$

$\therefore$ $\frac{V_2}{V_1}=(4{)}^{\frac{1}{1.5-1}}=4^2=16$

Hence the gas has to be expanded by 16 times.