A gaseous system has a volume of `580 cm^(3)` at a certain pressure. If its pressure in increased by 0.96 atm, its volume becomes `100 cm^(3)`. Determine the pressure of the system

Explanation

To determine the pressure of the gaseous system, we can use Boyle's Law, which states that for a given amount of gas at constant temperature, the product of pressure and volume is constant. This can be expressed mathematically as:

\[ P_1 V_1 = P_2 V_2 \]

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-by-Step Solution:

1. **Identify the known values:**
- Initial volume, \( V_1 = 580 \, \text{cm}^3 \)
- Final volume, \( V_2 = 100 \, \text{cm}^3 \)
- Increase in pressure, \( \Delta P = 0.96 \, \text{atm} \)

2. **Express the final pressure in terms of the initial pressure:**
- The final pressure \( P_2 \) can be expressed as:
\[ P_2 = P_1 + 0.96 \, \text{atm} \]

3. **Substitute the known values into Boyle's Law:**
- According to Boyle's Law:
\[ P_1 V_1 = P_2 V_2 \]
- Substitute \( P_2 \):
\[ P_1 \cdot 580 = (P_1 + 0.96) \cdot 100 \]

4. **Expand and rearrange the equation:**
- Distributing \( 100 \) on the right side:
\[ P_1 \cdot 580 = P_1 \cdot 100 + 0.96 \cdot 100 \]
- This simplifies to:
\[ 580 P_1 = 100 P_1 + 96 \]

5. **Combine like terms:**
- Rearranging gives:
\[ 580 P_1 - 100 P_1 = 96 \]
\[ 480 P_1 = 96 \]

6. **Solve for \( P_1 \):**
- Dividing both sides by \( 480 \):
\[ P_1 = \frac{96}{480} \]
\[ P_1 = 0.2 \, \text{atm} \]

### Final Answer:
The pressure of the system is \( P_1 = 0.2 \, \text{atm} \).