To determine the amount of heat released during the combustion of the octane sample, we can use the given formula:
[tex]\[ q = m C_p \Delta T \][/tex]
Step-by-Step Solution:
1. Identify the variables and given information:
- Mass of the calorimeter ([tex]\(m\)[/tex]): [tex]\(1.00 \, \text{kg}\)[/tex]
- Specific heat of the calorimeter ([tex]\(C_p\)[/tex]): [tex]\(1.50 \, \text{J/(g·°C)}\)[/tex]
- Initial temperature ([tex]\(T_i\)[/tex]): [tex]\(21.0 \, \text{°C}\)[/tex]
- Final temperature ([tex]\(T_f\)[/tex]): [tex]\(41.0 \, \text{°C}\)[/tex]
- Temperature change ([tex]\(\Delta T\)[/tex]): [tex]\(T_f - T_i = 41.0 \, \text{°C} - 21.0 \, \text{°C} = 20.0 \, \text{°C}\)[/tex]
2. Convert the mass of the calorimeter from kilograms to grams:
[tex]\[ 1.00 \, \text{kg} \times 1000 \, \text{g/kg} = 1000 \, \text{g} \][/tex]
3. Calculate the heat ([tex]\(q\)[/tex]) released using the formula:
[tex]\[ q = m C_p \Delta T \][/tex]
Substituting the values we have:
[tex]\[ q = (1000 \, \text{g}) \times (1.50 \, \text{J/(g·°C)}) \times (20.0 \, \text{°C}) \][/tex]
4. Perform the multiplication to get the heat in joules:
[tex]\[ q = 1000 \, \text{g} \times 1.50 \, \text{J/(g·°C)} \times 20.0 \, \text{°C} \][/tex]
[tex]\[ q = 1000 \times 1.50 \times 20.0 \][/tex]
[tex]\[ q = 30000 \, \text{J} \][/tex]
5. Convert the heat from joules to kilojoules:
[tex]\[ q = \frac{30000 \, \text{J}}{1000 \, \text{J/kJ}} = 30.0 \, \text{kJ} \][/tex]
Thus, the amount of heat released during the combustion of the octane sample is [tex]\(30.0 \, \text{kJ}\)[/tex].
The correct answer is:
[tex]\[ \boxed{30.0 \, \text{kJ}} \][/tex]
