A certain gas at atmospheric pressure is compressed adiabatically so that its volume becomes half of its original volume. Calculate the resulting pressure in `Nm^(-2)` . Given `gamma` for air = 1.4.

Answer

To solve the problem of finding the resulting pressure when a gas is compressed adiabatically to half of its original volume, we can use the formula derived from the adiabatic process for an ideal gas. The formula is given by:

\[ P_2 = P_1 \left( \frac{V_1}{V_2} \right)^\gamma \]

Where:
- \( P_1 \) is the initial pressure,
- \( V_1 \) is the initial volume,
- \( V_2 \) is the final volume,
- \( \gamma \) (gamma) is the heat capacity ratio (given as 1.4 for air).

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Initial pressure \( P_1 = 1 \text{ atm} = 1.013 \times 10^5 \, \text{N/m}^2 \)
- The volume is compressed to half, so \( V_2 = \frac{V_1}{2} \).
- Therefore, \( \frac{V_1}{V_2} = 2 \).
- \( \gamma = 1.4 \).

2. **Substitute the Values into the Formula:**
\[ P_2 = P_1 \left( \frac{V_1}{V_2} \right)^\gamma \]
Substituting the known values:
\[ P_2 = 1.013 \times 10^5 \left( 2 \right)^{1.4} \]

3. **Calculate \( 2^{1.4} \):**
\[ 2^{1.4} \approx 2.639 \]

4. **Calculate \( P_2 \):**
\[ P_2 = 1.013 \times 10^5 \times 2.639 \]
\[ P_2 \approx 2.674 \times 10^5 \, \text{N/m}^2 \]

5. **Final Result:**
The resulting pressure after adiabatic compression is approximately:
\[ P_2 \approx 2.674 \times 10^5 \, \text{N/m}^2 \]