To find the final temperature of the calorimeter after the combustion process, we'll use the formula:
[tex]\[ q = m \cdot C_p \cdot \Delta T \][/tex]
where:
- [tex]\( q \)[/tex] is the heat released (in Joules),
- [tex]\( m \)[/tex] is the mass of the calorimeter (in kilograms),
- [tex]\( C_p \)[/tex] is the specific heat capacity of the calorimeter (in Joules per kilogram per degree Celsius),
- [tex]\( \Delta T \)[/tex] is the change in temperature (in degrees Celsius).
Given the data:
- Heat released, [tex]\( q = 14.0 \, \text{kJ} = 14,000 \, \text{J} \)[/tex] (converted from kilojoules to joules),
- Mass of the calorimeter, [tex]\( m = 1.20 \, \text{kg} \)[/tex],
- Specific heat capacity, [tex]\( C_p = 3.55 \, \text{J/g}^\circ\text{C} \)[/tex]. Since the mass is in kilograms, we'll convert [tex]\( C_p \)[/tex] to [tex]\( \text{J/kg}^\circ\text{C} \)[/tex] by multiplying by 1000, resulting in [tex]\( C_p = 3.55 \times 1000 = 3550 \, \text{J/kg}^\circ\text{C} \)[/tex],
- Initial temperature, [tex]\( \text{T}_\text{initial} = 22.5^\circ\text{C} \)[/tex].
First, we'll rearrange the formula to solve for [tex]\( \Delta T \)[/tex]:
[tex]\[ \Delta T = \frac{q}{m \cdot C_p} \][/tex]
Substitute the given values into the formula:
[tex]\[ \Delta T = \frac{14,000 \, \text{J}}{1.20 \, \text{kg} \cdot 3550 \, \text{J/kg}^\circ\text{C}} \][/tex]
[tex]\[ \Delta T = \frac{14,000 \, \text{J}}{4,260 \, \text{J}^\circ\text{C}} \approx 3286.385^\circ\text{C} \][/tex]
Now that we have the temperature change, we can find the final temperature by adding the change in temperature to the initial temperature:
[tex]\[ \text{T}_\text{final} = \text{T}_\text{initial} + \Delta T \][/tex]
[tex]\[ \text{T}_\text{final} = 22.5^\circ\text{C} + 3286.385^\circ\text{C} \][/tex]
[tex]\[ \text{T}_\text{final} \approx 3308.885^\circ\text{C} \][/tex]
So, the final temperature of the calorimeter is approximately [tex]\( 3308.885^\circ\text{C} \)[/tex].
