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 determining by what percentage the pressure of a given mass of gas should be increased to decrease its volume by 10% at constant temperature, we can use the ideal gas law, which states that:

\[ P \times V = n \times R \times T \]

Since the temperature (T) and the number of moles (n) of gas are constant, we can say that:

\[ P_1 \times V_1 = P_2 \times V_2 \]

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 1: Define the initial and final volumes
Let the initial volume \( V_1 \) be \( V \). A 10% decrease in volume means the final volume \( V_2 \) is:

\[ V_2 = V_1 - 0.1 \times V_1 = 0.9 \times V_1 = 0.9V \]

### Step 2: Set up the equation using the gas law
Using the relationship from the ideal gas law:

\[ P_1 \times V_1 = P_2 \times V_2 \]

Substituting \( V_2 \):

\[ P_1 \times V = P_2 \times (0.9V) \]

### Step 3: Simplify the equation
We can cancel \( V \) from both sides (assuming \( V \neq 0 \)):

\[ P_1 = P_2 \times 0.9 \]

### Step 4: Solve for \( P_2 \)
Rearranging the equation gives us:

\[ P_2 = \frac{P_1}{0.9} \]

### Step 5: Calculate the percentage increase in pressure
To find the percentage increase in pressure, we can express it as:

\[ \text{Percentage Increase} = \left( \frac{P_2 - P_1}{P_1} \right) \times 100 \]

Substituting \( P_2 \):

\[ \text{Percentage Increase} = \left( \frac{\frac{P_1}{0.9} - P_1}{P_1} \right) \times 100 \]

### Step 6: Simplify the expression
This simplifies to:

\[ \text{Percentage Increase} = \left( \frac{P_1 \left( \frac{1}{0.9} - 1 \right)}{P_1} \right) \times 100 \]

Cancelling \( P_1 \):

\[ \text{Percentage Increase} = \left( \frac{1}{0.9} - 1 \right) \times 100 \]

Calculating \( \frac{1}{0.9} \):

\[ \frac{1}{0.9} = 1.1111 \]

So:

\[ \text{Percentage Increase} = (1.1111 - 1) \times 100 = 0.1111 \times 100 = 11.11\% \]

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

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