Answer :
Sure! Let's solve this problem step by step using the concepts from gas laws. In this case, we will use Charles's Law, which describes how gases tend to expand when heated. The law states that the volume of a gas is directly proportional to its temperature when pressure is constant.
Step-by-step Solution:
1. Understand the Problem:
- We have an initial volume of the balloon, [tex]\( V_1 = 43 \)[/tex] liters.
- The initial temperature is [tex]\( 25^\circ \text{C} \)[/tex].
- We want to find the new volume at [tex]\( -8^\circ \text{C} \)[/tex].
2. Convert Temperatures to Kelvin:
- Charles's Law uses temperatures in Kelvin. To convert Celsius to Kelvin, add 273.15 to the Celsius temperature.
- Initial temperature in Kelvin: [tex]\( 25 + 273.15 = 298.15 \)[/tex] K.
- Final temperature in Kelvin: [tex]\( -8 + 273.15 = 265.15 \)[/tex] K.
3. Apply Charles's Law:
- Charles's Law formula: [tex]\( \frac{V_1}{T_1} = \frac{V_2}{T_2} \)[/tex], where [tex]\( V_1 \)[/tex] and [tex]\( V_2 \)[/tex] are the initial and final volumes, and [tex]\( T_1 \)[/tex] and [tex]\( T_2 \)[/tex] are the initial and final temperatures in Kelvin.
- Rearranging the formula to solve for [tex]\( V_2 \)[/tex] (the final volume):
[tex]\[
V_2 = V_1 \times \frac{T_2}{T_1}
\][/tex]
4. Calculate the Final Volume:
- Plug in the known values:
[tex]\[
V_2 = 43 \, \text{L} \times \frac{265.15 \, \text{K}}{298.15 \, \text{K}}
\][/tex]
- Calculate the final volume:
[tex]\[
V_2 \approx 38.24 \, \text{L}
\][/tex]
Therefore, the volume of the balloon at [tex]\( -8^\circ \text{C} \)[/tex] is approximately [tex]\( 38.2 \)[/tex] liters.
The correct option from the choices given is 38.2 L.
Step-by-step Solution:
1. Understand the Problem:
- We have an initial volume of the balloon, [tex]\( V_1 = 43 \)[/tex] liters.
- The initial temperature is [tex]\( 25^\circ \text{C} \)[/tex].
- We want to find the new volume at [tex]\( -8^\circ \text{C} \)[/tex].
2. Convert Temperatures to Kelvin:
- Charles's Law uses temperatures in Kelvin. To convert Celsius to Kelvin, add 273.15 to the Celsius temperature.
- Initial temperature in Kelvin: [tex]\( 25 + 273.15 = 298.15 \)[/tex] K.
- Final temperature in Kelvin: [tex]\( -8 + 273.15 = 265.15 \)[/tex] K.
3. Apply Charles's Law:
- Charles's Law formula: [tex]\( \frac{V_1}{T_1} = \frac{V_2}{T_2} \)[/tex], where [tex]\( V_1 \)[/tex] and [tex]\( V_2 \)[/tex] are the initial and final volumes, and [tex]\( T_1 \)[/tex] and [tex]\( T_2 \)[/tex] are the initial and final temperatures in Kelvin.
- Rearranging the formula to solve for [tex]\( V_2 \)[/tex] (the final volume):
[tex]\[
V_2 = V_1 \times \frac{T_2}{T_1}
\][/tex]
4. Calculate the Final Volume:
- Plug in the known values:
[tex]\[
V_2 = 43 \, \text{L} \times \frac{265.15 \, \text{K}}{298.15 \, \text{K}}
\][/tex]
- Calculate the final volume:
[tex]\[
V_2 \approx 38.24 \, \text{L}
\][/tex]
Therefore, the volume of the balloon at [tex]\( -8^\circ \text{C} \)[/tex] is approximately [tex]\( 38.2 \)[/tex] liters.
The correct option from the choices given is 38.2 L.