A metal jar has a gas of volume $10^{- 3} m 3$ at a pressure of $2 \times 10^{5}$ Pa and temperature 400K. The jar has small hole and hence the gas leaks into atmosphere. The pressure and temperature of atmosphere is $10^{5}$ Pa and 300 K respectively. If R is the gas constant , the number of moles of the gas that has leaked into atmosphere is

\frac{1}{5 R}

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\frac{1}{6 R}

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\frac{1}{7 R}

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\frac{1}{8 R}

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Solution

The correct option is C $\frac{1}{6 R}$
Here, $P_{1} = 2 \times 10^{5} P a, V_{1} = 10^{- 3} m^{3}, T_{1} = 400 K$

$P_{2} = 10^{5} P a, T_{2} = 300 K, V_{2} = 10^{- 3} m^{3}$

Using ideal gas equation , $P V = n R T$

$n_{1} = \frac{P_{1} V_{1}}{R T_{1}} a n d n_{2} = \frac{P_{2} V_{2}}{R T_{2}}$

So number of moles leaked, $n_{1} - n_{2} = \frac{P_{1} V_{1}}{R T_{1}} - \frac{P_{2} V_{2}}{R T_{2}}$

$n_{1} - n_{2} = \frac{2 \times 10^{5} \times 10^{- 3}}{400 R} - \frac{10^{5} \times 10^{- 3}}{300 R}$

On solving, we will get $Δ n = \frac{1}{6 R}$

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