To determine the final temperature of the bomb calorimeter after the combustion process, we need to use the formula for heat transfer:
[tex]\[ q = m \cdot C_p \cdot \Delta T \][/tex]
where:
- [tex]\( q \)[/tex] is the heat absorbed or released,
- [tex]\( m \)[/tex] is the mass of the calorimeter,
- [tex]\( C_p \)[/tex] is the specific heat capacity,
- [tex]\( \Delta T \)[/tex] is the change in temperature.
From the problem, we know:
- The heat released by the combustion ([tex]\( q \)[/tex]) is [tex]\( 24.0 \)[/tex] kJ (or [tex]\( 24,000 \)[/tex] J after converting from kJ to J),
- The mass of the calorimeter ([tex]\( m \)[/tex]) is [tex]\( 1.30 \)[/tex] kg (or [tex]\( 1300 \)[/tex] g after converting from kg to g),
- The specific heat capacity ([tex]\( C_p \)[/tex]) is [tex]\( 3.41 \)[/tex] J/(g·°C),
- The initial temperature ([tex]\( T_{\text{initial}} \)[/tex]) is [tex]\( 25.5^{\circ} \)[/tex]C.
First, we solve for the change in temperature ([tex]\( \Delta T \)[/tex]) using the formula rearranged to isolate [tex]\( \Delta T \)[/tex]:
[tex]\[ \Delta T = \frac{q}{m \cdot C_p} \][/tex]
Substitute in the known values:
[tex]\[ \Delta T = \frac{24,000 \, \text{J}}{1300 \, \text{g} \cdot 3.41 \, \frac{\text{J}}{\text{g} \cdot ^{\circ} \text{C}}} \][/tex]
[tex]\[ \Delta T = \frac{24,000}{4433} \][/tex]
[tex]\[ \Delta T \approx 5.41^{\circ} \text{C} \][/tex]
Next, we find the final temperature ([tex]\( T_{\text{final}} \)[/tex]) by adding the change in temperature to the initial temperature:
[tex]\[ T_{\text{final}} = T_{\text{initial}} + \Delta T \][/tex]
[tex]\[ T_{\text{final}} = 25.5^{\circ} \text{C} + 5.41^{\circ} \text{C} \][/tex]
[tex]\[ T_{\text{final}} \approx 30.91^{\circ} \text{C} \][/tex]
Thus, the final temperature of the calorimeter is [tex]\( \boxed{30.9^{\circ} \text{C}} \)[/tex].
