Question

Boyles law formula, relationship, example problem

Answer 1

### Boyle's Law Formula

Boyle's Law states that the pressure of a given mass of gas is inversely proportional to its volume when the temperature is held constant. The formula 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

### Relationship

From Boyle's Law, we can derive the relationship:

- If the volume of a gas decreases (\( V \downarrow \)), the pressure increases (\( P \uparrow \)), assuming temperature remains constant.
- If the volume of a gas increases (\( V \uparrow \)), the pressure decreases (\( P \downarrow \)), assuming temperature remains constant.

This means that the product of pressure and volume for a fixed amount of gas at constant temperature is a constant value.

### Example Problem

**Problem:** A balloon has a volume of 2.0 L at a pressure of 1.0 atm. If the gas is compressed to a volume of 1.0 L, what will the new pressure be?

**Given:**
- \( V_1 = 2.0 \, \text{L} \)
- \( P_1 = 1.0 \, \text{atm} \)
- \( V_2 = 1.0 \, \text{L} \)

**Find:**
- \( P_2 \)

**Using Boyle’s Law, we can rearrange the formula to solve for \( P_2 \):**

\[
P_2 = \frac{P_1 V_1}{V_2}
\]

**Substituting the known values:**

\[
P_2 = \frac{1.0 \, \text{atm} \times 2.0 \, \text{L}}{1.0 \, \text{L}}
\]

\[
P_2 = \frac{2.0 \, \text{atm} \cdot \text{L}}{1.0 \, \text{L}} = 2.0 \, \text{atm}
\]

### Conclusion

Thus, if the balloon is compressed to a volume of 1.0 L, the new pressure will be **2.0 atm**.