Step 1: Understanding the Concept:
The gas behaves ideally, so it obeys the ideal gas equation.
During the leakage, the volume of the vessel remains constant, and the temperature is explicitly stated to remain constant.
The drop in pressure is directly proportional to the number of moles that have left the vessel.
Step 2: Key Formula or Approach:
The Ideal Gas Law: $PV = nRT$.
We calculate the initial number of moles ($n_1$) and the final number of moles ($n_2$) inside the vessel.
The number of moles leaked is $\Delta n = n_1 - n_2$.
Step 3: Detailed Explanation:
Initial State:
Pressure $= P$, Volume $= V$, Temperature $= T$.
Using the ideal gas equation, the initial number of moles $n_1$ is:
\[ n_1 = \frac{PV}{RT} \]
Final State:
Pressure $= P'$, Volume $= V$ (vessel size doesn't change), Temperature $= T$ (assumed constant).
Using the ideal gas equation, the final number of moles $n_2$ is:
\[ n_2 = \frac{P'V}{RT} \]
Leaked Gas:
The number of moles of gas that leaked out is the difference between initial and final moles:
\[ \Delta n = n_1 - n_2 \]
Substitute the expressions for $n_1$ and $n_2$:
\[ \Delta n = \frac{PV}{RT} - \frac{P'V}{RT} \]
Factor out the common terms $\frac{V}{RT}$:
\[ \Delta n = \frac{V}{RT}(P - P') \]
Step 4: Final Answer:
The number of moles leaked is $\frac{\text{V}}{\text{RT}}(\text{P} - \text{P}')$.