To solve the problem, we'll use Boyle's Law, which states that the product of the initial pressure and volume of a gas is equal to the product of the final pressure and volume of the gas, given the temperature remains constant. The formula for Boyle's Law is:
[tex]\[ P_1 \times V_1 = P_2 \times V_2 \][/tex]
Where:
- [tex]\( P_1 \)[/tex] is the initial pressure
- [tex]\( V_1 \)[/tex] is the initial volume
- [tex]\( P_2 \)[/tex] is the final pressure
- [tex]\( V_2 \)[/tex] is the final volume
We are given:
- [tex]\( P_1 = 1550.50 \)[/tex] atm
- [tex]\( V_1 = 2.40 \)[/tex] mL
- [tex]\( P_2 = 1.49 \)[/tex] atm
We need to find the final volume [tex]\( V_2 \)[/tex]. Rearranging the formula for [tex]\( V_2 \)[/tex]:
[tex]\[ V_2 = \frac{P_1 \times V_1}{P_2} \][/tex]
Substituting the given values into the formula:
[tex]\[ V_2 = \frac{1550.50 \, \text{atm} \times 2.40 \, \text{mL}}{1.49 \, \text{atm}} \][/tex]
First, calculate the numerator:
[tex]\[ 1550.50 \times 2.40 = 3721.2 \, \text{mL} \cdot \text{atm} \][/tex]
Then, divide by the final pressure:
[tex]\[ V_2 = \frac{3721.2 \, \text{mL} \cdot \text{atm}}{1.49 \, \text{atm}} \][/tex]
[tex]\[ V_2 \approx 2497.45 \, \text{mL} \][/tex]
Hence, the new volume of the balloon, after rising and experiencing the pressure change, is approximately [tex]\( 2497.45 \)[/tex] mL.
