The pressure of a given mass of a gas contained in a vessel at constant temperature is reduced to half. Calculate the change in volume of the gas.

Explanation

To solve the problem of how the volume of a gas changes when its pressure 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 a constant. Mathematically, this is expressed as:
\[ P_1 V_1 = P_2 V_2 \]
where:
- \( P_1 \) is the initial pressure,
- \( V_1 \) is the initial volume,
- \( P_2 \) is the final pressure,
- \( V_2 \) is the final volume.

### Step 2: Define Initial Conditions
Let’s denote the initial pressure as \( P \) and the initial volume as \( V \):
- \( P_1 = P \)
- \( V_1 = V \)

### Step 3: Define Final Conditions
According to the problem, the pressure is reduced to half:
- \( P_2 = \frac{P}{2} \)

### Step 4: Apply Boyle's Law
Now, substituting the known values into Boyle's Law:
\[ P_1 V_1 = P_2 V_2 \]
Substituting the values:
\[ P \cdot V = \left(\frac{P}{2}\right) \cdot V_2 \]

### Step 5: Solve for Final Volume
To find \( V_2 \), we can rearrange the equation:
\[ P \cdot V = \frac{P}{2} \cdot V_2 \]
Dividing both sides by \( P \) (assuming \( P \neq 0 \)):
\[ V = \frac{1}{2} V_2 \]
Now, multiplying both sides by 2:
\[ 2V = V_2 \]

### Step 6: Conclusion
Thus, the final volume \( V_2 \) is:
\[ V_2 = 2V \]
This indicates that the volume of the gas doubles when the pressure is reduced to half.

### Step 7: Calculate Change in Volume
The change in volume (\( \Delta V \)) can be calculated as:
\[ \Delta V = V_2 - V_1 = 2V - V = V \]

### Final Answer
The change in volume of the gas is \( V \).

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