By what percentage should the pressure of a given mass of a gas be increased so as to decrease its volume by 10% at a constant temperature?

Options

A. 0.05
B. `7.2`%
C. `12.5`%
D. `11.1`%

Explanation

To solve the problem of how much the pressure of a given mass of gas should be increased to decrease its volume by 10% at constant temperature, we can follow these steps:

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

### Step 2: Define the initial and final volumes
Let the initial volume \( V_1 \) be \( V \). If the volume is decreased by 10%, the final volume \( V_2 \) will be:
\[ V_2 = V - 0.1V = 0.9V \]

### Step 3: Substitute the volumes into Boyle's Law
Using Boyle's Law:
\[ P_1 V_1 = P_2 V_2 \]
Substituting \( V_1 \) and \( V_2 \):
\[ P_1 V = P_2 (0.9V) \]

### Step 4: Simplify the equation
We can cancel \( V \) from both sides (assuming \( V \neq 0 \)):
\[ P_1 = 0.9 P_2 \]

### Step 5: Solve for the final pressure \( P_2 \)
Rearranging the equation gives:
\[ P_2 = \frac{P_1}{0.9} \]

### Step 6: Calculate the percentage increase in pressure
The percentage increase in pressure can be calculated using the formula:
\[ \text{Percentage Increase} = \frac{P_2 - P_1}{P_1} \times 100\% \]
Substituting \( P_2 \):
\[ \text{Percentage Increase} = \frac{\frac{P_1}{0.9} - P_1}{P_1} \times 100\% \]

### Step 7: Simplify the percentage increase
This simplifies to:
\[ \text{Percentage Increase} = \left(\frac{1}{0.9} - 1\right) \times 100\% \]
\[ = \left(\frac{1 - 0.9}{0.9}\right) \times 100\% \]
\[ = \left(\frac{0.1}{0.9}\right) \times 100\% \]
\[ = \frac{100}{9} \approx 11.11\% \]

### Final Answer
Thus, the pressure should be increased by approximately **11.11%** to decrease the volume by 10% at constant temperature.

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