Sure, let's walk through the solution step-by-step.
To solve this problem, we use Charles's Law, which states that for a given quantity of gas at constant pressure, the volume is directly proportional to the temperature (measured in Kelvin). Mathematically, this is expressed as:
[tex]\[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \][/tex]
Given:
- Initial volume ([tex]\(V_1\)[/tex]) = 2.75 liters
- Initial temperature ([tex]\(T_1\)[/tex]) = 295 Kelvin
- Final volume ([tex]\(V_2\)[/tex]) = 5.00 liters
We need to find the final temperature ([tex]\(T_2\)[/tex]). Rearrange the equation to solve for [tex]\(T_2\)[/tex]:
[tex]\[ T_2 = \frac{V_2 \cdot T_1}{V_1} \][/tex]
Substitute the given values into the equation:
[tex]\[ T_2 = \frac{5.00 \, \text{L} \cdot 295 \, \text{K}}{2.75 \, \text{L}} \][/tex]
Now, calculate the value step-by-step:
1. Multiply [tex]\(5.00 \, \text{L}\)[/tex] by [tex]\(295 \, \text{K}\)[/tex]:
[tex]\[ 5.00 \times 295 = 1475 \][/tex]
2. Divide the result by [tex]\(2.75 \, \text{L}\)[/tex]:
[tex]\[ \frac{1475}{2.75} \approx 536.3636363636364 \, \text{K} \][/tex]
According to significant figure rules, since all given values ([tex]\(V_1\)[/tex], [tex]\(T_1\)[/tex], and [tex]\(V_2\)[/tex]) have 3 significant figures, the final temperature should also be rounded to 3 significant figures:
[tex]\[ T_2 \approx 536 \, \text{K} \][/tex]
Therefore, to increase the volume of the balloon to 5.00 liters, you would have to heat it to approximately 536 Kelvin.
