Sure, let's solve the problem step-by-step.
### Step 1: Identify the known variables
- Initial volume, [tex]\( V_1 = 2.0 \)[/tex] liters
- Initial temperature, [tex]\( T_1 = 25^\circ \text{C} \)[/tex]
- Final temperature, [tex]\( T_2 = 35^\circ \text{C} \)[/tex]
- Constant pressure, [tex]\( P = 1.0 \)[/tex] atm
### Step 2: Convert the temperatures to Kelvin
The temperature in Kelvin (K) is given by the formula [tex]\( T (\text{K}) = T (\text{^\circ C}) + 273.15 \)[/tex].
1. Initial temperature in Kelvin:
[tex]\[
T_1 = 25^\circ \text{C} + 273.15 = 298.15 \text{ K}
\][/tex]
2. Final temperature in Kelvin:
[tex]\[
T_2 = 35^\circ \text{C} + 273.15 = 308.15 \text{ K}
\][/tex]
### Step 3: Use the ideal gas law formula [tex]\( \frac{V_1}{T_1} = \frac{V_2}{T_2} \)[/tex] to find the final volume
Rearrange the equation to solve for [tex]\( V_2 \)[/tex]:
[tex]\[
V_2 = V_1 \cdot \frac{T_2}{T_1}
\][/tex]
### Step 4: Substitute the known values into the equation
[tex]\[
V_2 = 2.0 \text{ L} \cdot \frac{308.15 \text{ K}}{298.15 \text{ K}}
\][/tex]
### Step 5: Calculate the final volume [tex]\( V_2 \)[/tex]
Substitute the values and compute:
[tex]\[
V_2 = 2.0 \text{ L} \cdot 1.0335401643468055 \approx 2.067080328693611 \text{ L}
\][/tex]
### Final Answer:
The final volume of the balloon when the temperature is increased to [tex]\( 35^\circ \text{C} \)[/tex] while maintaining constant pressure is approximately [tex]\( 2.067 \text{ L} \)[/tex].
