Explanation:
To solve this problem, we can use the ideal gas law, which relates the pressure (P), volume (V), number of moles (n), and temperature (T) of a gas:
\[ PV = nRT \]
Where:
- \( P \) is the pressure of the gas (in atmospheres),
- \( V \) is the volume of the gas (in liters),
- \( n \) is the number of moles of the gas,
- \( R \) is the ideal gas constant (0.0821 atm L / mol K),
- \( T \) is the temperature of the gas (in Kelvin).
First, we need to convert the given pressure to atmospheres, and then we can rearrange the ideal gas law to solve for temperature:
\[ T = \frac{{PV}}{{nR}} \]
Given:
- \( P = 1.0 \) atmosphere
- \( V = 85 \) L
- \( n = 3.5 \) moles
- \( R = 0.0821 \) atm L / mol K
Substituting the values into the equation:
\[ T = \frac{{(1.0 \, \text{atm})(85 \, \text{L})}}{{(3.5 \, \text{mol})(0.0821 \, \text{atm L / mol K})}} \]
\[ T = \frac{{85 \, \text{atm L}}}{{0.28735 \, \text{mol}}} \]
\[ T \approx 295.75 \, \text{K} \]
To convert the temperature from Kelvin to Celsius, we subtract 273.15:
\[ T \approx 295.75 \, \text{K} - 273.15 \, \text{K} \]
\[ T \approx 22.6 \, ^\circ \text{C} \]
So, the temperature of the balloon is approximately \( 22.6 \, ^\circ \text{C} \).
