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 need to understand the relationship between pressure, volume, and the adiabatic process. 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. In an adiabatic process, the relationship between pressure (P) and volume (V) is given by the equation:
\[ P V^\gamma = \text{constant} \]
where \( \gamma \) (gamma) is the heat capacity ratio.

### Step 2: Set Up the Initial and Final Conditions
...