The volume of a given mass of a gas in vessel is reduced to half at constant temperature. What is the charge in pressure?

Explanation

To solve the problem of how the pressure changes when the volume of a gas is reduced to half at constant temperature, we can use Boyle's Law. Here's a step-by-step solution:

### Step 1: Understand Boyle's Law
Boyle's Law states that for a given mass of gas at constant temperature, the product of pressure (P) and volume (V) is constant. Mathematically, it can be expressed as:
\[ P_1 V_1 = P_2 V_2 \]
where:
- \( P_1 \) = initial pressure
- \( V_1 \) = initial volume
- \( P_2 \) = final pressure
- \( V_2 \) = final volume

### Step 2: Define Initial and Final Conditions
Let:
- Initial volume \( V_1 = V \)
- Final volume \( V_2 = \frac{V}{2} \) (since the volume is reduced to half)
- Initial pressure \( P_1 \) (unknown)
- Final pressure \( P_2 \) (unknown)

### Step 3: Substitute Values into Boyle's Law
Substituting the known values into Boyle's Law:
\[ P_1 \cdot V = P_2 \cdot \left(\frac{V}{2}\right) \]

### Step 4: Simplify the Equation
Rearranging the equation gives:
\[ P_1 \cdot V = \frac{P_2 \cdot V}{2} \]
To eliminate \( V \) (assuming \( V \neq 0 \)), we can divide both sides by \( V \):
\[ P_1 = \frac{P_2}{2} \]

### Step 5: Solve for Final Pressure
From the equation \( P_1 = \frac{P_2}{2} \), we can express \( P_2 \) in terms of \( P_1 \):
\[ P_2 = 2P_1 \]

### Step 6: Conclusion
This means that when the volume is reduced to half at constant temperature, the final pressure \( P_2 \) is double the initial pressure \( P_1 \). Therefore, the change in pressure is that it has doubled.

### Summary
- Initial pressure \( P_1 \)
- Final pressure \( P_2 = 2P_1 \)
- Change in pressure: The pressure has doubled.

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