A cyclinder with a movable piston contains `3 mol s` of hydrogen at standard temperature and pressure. The walls of the cyclinder are made of a heat insulator, and the piston is insulated by having a pile of sand on it. By what factor does the pressure of the gas increases, if the gas is compressed to half its original volume? Given `gamma= 1.4`.

Explanation

To solve the problem, we will use the principles of thermodynamics, specifically the equations governing adiabatic processes. Here’s a step-by-step solution:

### Step 1: Understand the Adiabatic Process
Since the walls of the cylinder are insulated, the process is adiabatic, meaning there is no heat exchange with the surroundings. For an adiabatic process, the relationship between pressure and volume is given by the equation:
\[ P_1 V_1^\gamma = P_2 V_2^\gamma \]
where \( P_1 \) and \( V_1 \) are the initial pressure and volume, \( P_2 \) and \( V_2 \) are the final pressure and volume, and \( \gamma \) is the heat capacity ratio (given as 1.4 for hydrogen).

### Step 2: Define Initial and Final Volumes
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