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 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 a 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 Conditions
Let:
- The initial volume \( V_1 = V \)
- The initial pressure \( P_1 = P \)
- The final volume \( V_2 = \frac{V}{2} \) (since the volume is reduced to half)

### Step 3: Apply Boyle's Law
Using Boyle's Law, we can write:
\[ P_1 V_1 = P_2 V_2 \]
Substituting the known values:
\[ P \cdot V = P_2 \cdot \frac{V}{2} \]

### Step 4: Solve for Final Pressure \( P_2 \)
Now, rearranging the equation to solve for \( P_2 \):
\[ P_2 = \frac{P \cdot V}{\frac{V}{2}} \]
\[ P_2 = \frac{P \cdot V \cdot 2}{V} \]
\[ P_2 = 2P \]

### Step 5: Calculate Change in Pressure
The change in pressure \( \Delta P \) can be calculated as:
\[ \Delta P = P_2 - P_1 \]
Substituting the values we found:
\[ \Delta P = 2P - P \]
\[ \Delta P = P \]

### Conclusion
The change in pressure when the volume of the gas is reduced to half at constant temperature is equal to the initial pressure \( P \).

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