Complete combustion of a [tex]0.350 \, \text{g}[/tex] sample of a compound in a bomb calorimeter releases [tex]14.0 \, \text{kJ}[/tex] of heat. The bomb calorimeter has a mass of [tex]1.20 \, \text{kg}[/tex] and a specific heat capacity of [tex]3.55 \, \text{J/g} \cdot {}^{\circ} \text{C}[/tex].

If the initial temperature of the calorimeter is [tex]22.5^{\circ} \text{C}[/tex], what is its final temperature?

Use [tex]q = m C_p \Delta T[/tex].

A. [tex]19.2^{\circ} \text{C}[/tex]
B. [tex]25.8^{\circ} \text{C}[/tex]
C. [tex]34.2^{\circ} \text{C}[/tex]
D. [tex]72.3^{\circ} \text{C}[/tex]



Answer :

To determine the final temperature of the calorimeter after the complete combustion of the compound, we need to follow a series of steps, leveraging the known values and the formula [tex]\( q = m C_p \Delta T \)[/tex].

Here’s a step-by-step breakdown of the solution:

1. Identify and Define the Variables:
- Heat released ([tex]\( q \)[/tex]): [tex]\( 14.0 \)[/tex] kJ
- Mass of the calorimeter ([tex]\( m \)[/tex]): [tex]\( 1.20 \)[/tex] kg
- Specific heat capacity of the calorimeter ([tex]\( C_p \)[/tex]): [tex]\( 3.55 \)[/tex] kJ/(kg°C)
- Initial temperature ([tex]\( T_{\text{initial}} \)[/tex]): [tex]\( 22.5 \)[/tex]°C

2. Write the Formula:
The formula relating the heat, mass, specific heat capacity, and temperature change is:
[tex]\[ q = m C_p \Delta T \][/tex]

3. Isolate ΔT (Change in Temperature):
Rearrange the formula to solve for [tex]\( \Delta T \)[/tex]:
[tex]\[ \Delta T = \frac{q}{m \cdot C_p} \][/tex]

4. Substitute the Known Values:
Substitute the given values into the formula:
[tex]\[ \Delta T = \frac{14.0 \text{ kJ}}{1.20 \text{ kg} \times 3.55 \text{ kJ/(kg°C)}} \][/tex]

5. Calculate ΔT:
Perform the calculation:
[tex]\[ \Delta T = \frac{14.0}{1.20 \times 3.55} \][/tex]
[tex]\[ \Delta T \approx 3.29 \text{°C} \][/tex]

6. Determine the Final Temperature:
The final temperature ([tex]\( T_{\text{final}} \)[/tex]) is the initial temperature plus the change in temperature:
[tex]\[ T_{\text{final}} = T_{\text{initial}} + \Delta T \][/tex]
Substitute the values:
[tex]\[ T_{\text{final}} = 22.5 \text{°C} + 3.29 \text{°C} \][/tex]
[tex]\[ T_{\text{final}} \approx 25.79 \text{°C} \][/tex]

7. Select the Closest Answer:
Among the provided options, the closest value to [tex]\( 25.79 \text{°C} \)[/tex] is [tex]\( 25.8 \text{°C} \)[/tex].

Hence, the final temperature of the calorimeter is approximately [tex]\( 25.8^{\circ}C \)[/tex].