A gas is initially at 1 atm pressure. To compress it to 1/4 th of its initial volume, pressure to be applied is: 1 atm, 2 atm, 3 atm, 1/4 atm

Options

A. 1 atm
B. 2 atm
C. 4 atm
D. 1/4 atm.

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. Mathematically, this is 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 Initial Conditions:**
- Initial pressure, \( P_1 = 1 \, \text{atm} \)
- Let the initial volume be \( V_1 = V \).

2. **Determine the Final Volume:**
- The gas is compressed to one-fourth of its initial volume, so:
\[ V_2 = \frac{V_1}{4} = \frac{V}{4} \]

3. **Apply Boyle's Law:**
- According to Boyle's Law:
\[ P_1 V_1 = P_2 V_2 \]
- Substituting the known values:
\[ 1 \, \text{atm} \cdot V = P_2 \cdot \left(\frac{V}{4}\right) \]

4. **Rearranging the Equation:**
- To find \( P_2 \), we can rearrange the equation:
\[ P_2 = \frac{1 \, \text{atm} \cdot V}{\frac{V}{4}} \]

5. **Simplifying the Expression:**
- The \( V \) terms cancel out:
\[ P_2 = 1 \, \text{atm} \cdot 4 = 4 \, \text{atm} \]

6. **Conclusion:**
- The pressure that needs to be applied to compress the gas to one-fourth of its initial volume is \( 4 \, \text{atm} \).

### Final Answer:
The pressure to be applied is **4 atm**.

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