To solve the problem of finding the total heat capacity of the calorimeter, let's go through the step-by-step calculations:
1. Understanding the Given Information:
- The benzoic acid sample released [tex]\( 31.66 \, \text{kJ} \)[/tex] of energy.
- The calorimeter absorbed the same amount of energy, [tex]\( 31.66 \, \text{kJ} \)[/tex].
- The initial temperature of the calorimeter was [tex]\( 22.45^\circ \text{C} \)[/tex].
- The final temperature of the calorimeter was [tex]\( 26.10^\circ \text{C} \)[/tex].
2. Finding the Temperature Change ([tex]\(\Delta T\)[/tex]):
- [tex]\(\Delta T\)[/tex] is the change in temperature of the calorimeter.
- [tex]\(\Delta T = \text{final temperature} - \text{initial temperature} = 26.10^\circ \text{C} - 22.45^\circ \text{C}\)[/tex].
3. Calculate the Temperature Change:
[tex]\[
\Delta T = 26.10^\circ \text{C} - 22.45^\circ \text{C} = 3.65^\circ \text{C}
\][/tex]
4. Using the Heat Capacity Formula:
- The formula for the total heat capacity ([tex]\(C_{\text{cal}}\)[/tex]) of the calorimeter is:
[tex]\[
C_{\text{cal}} = \frac{\text{Energy released}}{\Delta T}
\][/tex]
- Substituting the values we have:
[tex]\[
C_{\text{cal}} = \frac{31.66 \, \text{kJ}}{3.65^\circ \text{C}}
\][/tex]
5. Calculating the Total Heat Capacity:
[tex]\[
C_{\text{cal}} = \frac{31.66 \, \text{kJ}}{3.65^\circ \text{C}} \approx 8.674 \, \text{kJ} / ^\circ \text{C}
\][/tex]
6. Final Answer:
The total heat capacity of the calorimeter is approximately [tex]\( 8.674 \, \text{kJ} / ^\circ \text{C} \)[/tex].
So, in conclusion, the total heat capacity of the calorimeter, based on the given information and calculations, is [tex]\( 8.674 \, \text{kJ} / ^\circ \text{C} \)[/tex].
