What will be the minimum pressure required to compress 500 `dm^(3)` of air at 1 bar to 200 `dm^(3)` at `30^(@)C` ?

Explanation

To solve the problem of finding the minimum pressure required to compress 500 dm³ of air at 1 bar to 200 dm³ at 30°C, we can use Boyle's Law. Boyle's Law states that for a given mass of gas at constant temperature, the product of pressure and volume is constant. Mathematically, it can be expressed as:

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

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Initial pressure, \( P_1 = 1 \) bar
- Initial volume, \( V_1 = 500 \) dm³
- Final volume, \( V_2 = 200 \) dm³
- Final pressure, \( P_2 \) is what we need to find.

2. **Write Down Boyle's Law:**
\[ P_1 \times V_1 = P_2 \times V_2 \]

3. **Substitute the Known Values into the Equation:**
\[ 1 \, \text{bar} \times 500 \, \text{dm}^3 = P_2 \times 200 \, \text{dm}^3 \]

4. **Rearrange the Equation to Solve for \( P_2 \):**
\[ P_2 = \frac{1 \, \text{bar} \times 500 \, \text{dm}^3}{200 \, \text{dm}^3} \]

5. **Calculate \( P_2 \):**
- First, calculate the numerator:
\[ 1 \times 500 = 500 \]
- Now divide by the denominator:
\[ P_2 = \frac{500}{200} = 2.5 \, \text{bar} \]

6. **Conclusion:**
The minimum pressure required to compress 500 dm³ of air at 1 bar to 200 dm³ at 30°C is \( P_2 = 2.5 \) bar.