A vessel has 6 g of oxygen at pressure p and temperature 400 K. A small hole is made in it so that oxygen…
A vessel has 6 g of oxygen at pressure p and temperature 400 K. A small hole is made in it so that oxygen leaks out.How much oxygen
has leaked out if the final pressure is $\frac{p}{2}$ and final temperature is 300 K?
5 g
4 g
2 g
3 g
Solution
Given, initial pressure, $p_i=p$
Final pressure, $p_f=\frac{p}{2}$
Initial temperature $T=400 \mathrm{~K}$
Final temperature $T^{\prime}=300 \mathrm{~K}$
Initial mass of gas, $m=6 \mathrm{~g}$
From ideal gas equation, we get
$\begin{aligned} & p V=n R T \\ & p V=\frac{m}{M} R T \\ & \left(\because \frac{m}{M}=n\right) \\ & \end{aligned}$
Initial condition, $p V=\frac{m}{M} R T$...(i)
Final condition, $p^{\prime} V=\frac{m^{\prime}}{M} R T^{\prime}$...(ii)
Dividing Eq. (ii) by Eq. (i), we get
$
\begin{aligned}
& \frac{p^{\prime}}{p}=\frac{m^{\prime} T^{\prime}}{m T} \\
& \Rightarrow \quad m^{\prime}=\frac{p^{\prime}}{p} \times \frac{T}{T^{\prime}} \times m=\frac{p}{2} \times \frac{1}{p} \times \frac{400}{300} \times 6 \\
& \Rightarrow \quad m^{\prime}=4 \mathrm{~g} \\
&
\end{aligned}
$
Mass of oxygen leaked, $\Delta m=m-m^{\prime}$
$
\Rightarrow \quad \Delta m=6-4=2 \mathrm{~g}
$