At a constant temperature a gas is initially at 2 atm pressure. To compress it to 1/8th of its initial volume, pressure to be applied is

Options

A. 4atm
B. 8atm
C. 12atm
D. 16atm

Answer

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 mathematically 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-by-Step Solution:

1. **Identify the Given Values**:
- Initial pressure, \( P_1 = 2 \) atm
- The gas is compressed to \( \frac{1}{8} \) of its initial volume, which means \( V_2 = \frac{1}{8} V_1 \).

2. **Set Up the Boyle's Law Equation**:
- According to Boyle's Law:
\[ P_1 V_1 = P_2 V_2 \]

3. **Substitute the Known Values**:
- We know that \( V_2 = \frac{1}{8} V_1 \). Substitute this into the equation:
\[ 2 \, \text{atm} \cdot V_1 = P_2 \cdot \left(\frac{1}{8} V_1\right) \]

4. **Simplify the Equation**:
- Cancel \( V_1 \) from both sides (assuming \( V_1 \neq 0 \)):
\[ 2 = P_2 \cdot \frac{1}{8} \]

5. **Solve for \( P_2 \)**:
- Multiply both sides by 8 to isolate \( P_2 \):
\[ P_2 = 2 \cdot 8 = 16 \, \text{atm} \]

6. **Conclusion**:
- The pressure to be applied to compress the gas to \( \frac{1}{8} \) of its initial volume is \( 16 \) atm.

### Final Answer:
The pressure to be applied is **16 atm**.

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