To determine the amount of heat produced during the combustion of hexane in a bomb calorimeter, 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 produced in Joules, [tex]\( m \)[/tex] is the mass of the calorimeter, [tex]\( C_p \)[/tex] is the specific heat capacity of the calorimeter, and [tex]\( \Delta T \)[/tex] is the change in temperature.
Let's break this down step-by-step:
1. Given Data:
- Mass of the calorimeter ([tex]\( m \)[/tex]): 1.900 kg. Since the specific heat is given in J/gK, we need to convert this mass to grams:
[tex]\[ m = 1.900 \, \text{kg} \times 1000 \frac{g}{kg} = 1900 \, \text{g} \][/tex]
- Specific heat capacity ([tex]\( C_p \)[/tex]): 3.21 J/gK
- Change in temperature ([tex]\( \Delta T \)[/tex]): 4.542 K
2. Calculate the heat produced:
Using the formula [tex]\( q = m \cdot C_p \cdot \Delta T \)[/tex]:
[tex]\[ q = 1900 \, \text{g} \times 3.21 \, \frac{J}{gK} \times 4.542 \, \text{K} \][/tex]
3. Perform the multiplication:
[tex]\[ q = 1900 \times 3.21 \times 4.542 = 27701.658 \, \text{Joules} \][/tex]
4. Convert Joules to kilojoules:
[tex]\[ q_{\text{kJ}} = \frac{27701.658 \, \text{J}}{1000} = 27.701658 \, \text{kJ} \][/tex]
5. Round to appropriate number of significant figures (based on the precision of the given data):
Given data indicates significant figures up to 4 decimal places (4.542 K), so:
[tex]\[ q_{\text{kJ}} \approx 27.7 \, \text{kJ} \][/tex]
Thus, the amount of heat produced during the combustion of hexane is approximately [tex]\( 27.7 \, \text{kJ} \)[/tex], which matches one of the provided answers. Therefore, the correct answer is:
[tex]\[ \boxed{27.7 \, \text{kJ}} \][/tex]
