Question

A cylindrical container is divided in two equal parts by a diathermic piston. Different ideal gases are filled in the two parts. Find the ratio of the mass of the molecules of the gas in the lower part to that of the upper part, if the root mean square velocity of molecules in the lower part is equal to the mean velocity of molecules in the upper part.

• A. $1.224$
• B. $1.178$
• C. $1.288$
• D. $1.128$

Answer 1

The correct option is B $1.178$

As piston is diathermic the two gases must be in thermal equilibrium.

Let $M_1=$ Molecular mass of the gas filled in the lower part.
$M_2=$ Molecular mass of the gas filled in the upper part.


We know that
Average velocity ($V_{avg}$) = $\sqrt{\frac{8RT}{\pi M}}$
Root mean square velocity ($V_{rms}$) = $\sqrt{\frac{3RT}{M}}$

From the question,
$V_{rms}$ of lower part $=V_{avg}$ of upper part
where,
$\Rightarrow \sqrt{\frac{3R{T}_1}{M_1}}=\sqrt{\frac{8R{T}_1}{\pi M_2}}$
$[T_1=T_2\text{because piston is diathermic}]$
$\Rightarrow \sqrt{\frac{3}{M_1}}=\sqrt{\frac{8}{\pi M_2}}$
$\Rightarrow \frac{M_1}{M_2}=\frac{3\pi }{8}=1.178$
The ratio of the mass of the molecules and that of molecular masses is the same.
Thus, the required ratio is $1.178$.