Question

The average speed of gas molecules is v at pressure p if by keeping temperature constant the pressure of gas is doubled then average speed will become

Answer 1

Average speed of gas molecules is given by $v=\sqrt{\frac{kT}{m\pi }}$

$k=\frac{R}{N_A}$

Thus $v=\sqrt{\frac{RT}{m{N}_A\pi }}$

For 1 mole of gas molecule $PV=RT$ (from ideal gas equation) and here n=1

$v=\sqrt{\frac{RT}{m{N}_A\pi }}$

As temperature is constant

$PV=P_2{V}_2$

$V_2=\frac{PV}{P_2}=\frac{PV}{2P}=\frac{V}{2}$ as per question

Now when pressure is increased to double, volume V is reduced to half

Hence the new average velocity will be

$v_n=\sqrt{\frac{2P\frac{V}{2}}{m{N}_A\pi }}=\sqrt{\frac{PV}{m{N}_A\pi }}$

Thus $v_n=v$