Consider a cubical vessel of edge a having a small hole in one of its walls. The total thermal resistance of the walls is r. At time `t=0` , it contains air at atmospheric pressure `p_(a)` and temperature `T_(0)` . The temperature of the surrounding air is `T_(a)``(gtT_(0))` . Find the amount of the gas (in moles) in the vessel at time t. Take `C_(v)` of air to be `5 R//2`.

Explanation

To solve the problem of finding the amount of gas (in moles) in a cubical vessel with a small hole at time \( t \), we can follow these steps:

### Step 1: Understand the System
We have a cubical vessel with edge length \( a \), containing air at atmospheric pressure \( P_a \) and initial temperature \( T_0 \). The surrounding air temperature is \( T_a \) (where \( T_a > T_0 \)). The total thermal resistance of the walls is \( r \).

### Step 2: Volume of the Vessel
The volume \( V \) of the cubical vessel can be calculated as:
\[
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