The volume of a given mass of a gas in a vessel is reduced to half at constant temperature what is the change in pressure?

Explanation

To solve the problem, we will use Boyle's Law, which states that for a given mass of gas at constant temperature, the product of pressure and volume is constant. This can be expressed mathematically as:

\[ P_1 V_1 = P_2 V_2 \]

### Step-by-Step Solution

1. **Identify Initial Conditions**:
Let the initial pressure be \( P_0 \) and the initial volume be \( V_0 \).

2. **Determine Final Volume**:
According to the problem, the volume is reduced to half. Therefore, the final volume \( V_2 \) can be expressed as:
\[ V_2 = \frac{V_0}{2} \]

3. **Apply Boyle's Law**:
Using Boyle's Law, we can write:
\[ P_0 V_0 = P_2 V_2 \]

4. **Substitute Final Volume**:
Substitute \( V_2 \) into the equation:
\[ P_0 V_0 = P_2 \left(\frac{V_0}{2}\right) \]

5. **Simplify the Equation**:
To isolate \( P_2 \), we can rearrange the equation:
\[ P_2 = \frac{P_0 V_0}{\frac{V_0}{2}} = \frac{P_0 V_0 \cdot 2}{V_0} \]
Here, \( V_0 \) cancels out:
\[ P_2 = 2 P_0 \]

6. **Conclusion**:
The final pressure \( P_2 \) is twice the initial pressure \( P_0 \). Therefore, we conclude that if the volume is reduced to half at constant temperature, the pressure will double.

### Final Answer:
The change in pressure is that it doubles, i.e., \( P_2 = 2 P_0 \).

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