To find the amount of heat released during the combustion of the sample of octane, we use the formula for heat transfer:
[tex]\[ q = m \cdot C_p \cdot \Delta T \][/tex]
where:
- \( q \) is the heat released,
- \( m \) is the mass of the calorimeter,
- \( C_p \) is the specific heat capacity of the calorimeter,
- \( \Delta T \) is the change in temperature.
Given:
- Mass of the calorimeter, \( m = 1.00 \text{ kg} \) (which needs to be converted to grams since specific heat is given in \( J/(g \cdot °C) \)),
- Specific heat capacity of the calorimeter, \( C_p = 1.50 \text{ J/(g} \cdot °C) \),
- Initial temperature, \( T_i = 21.0 °C \),
- Final temperature, \( T_f = 41.0 °C \).
Step 1: Convert the mass of the calorimeter to grams:
[tex]\[ 1.00 \text{ kg} = 1000.0 \text{ g} \][/tex]
Step 2: Calculate the temperature change, \( \Delta T \):
[tex]\[ \Delta T = T_f - T_i = 41.0 °C - 21.0 °C = 20.0 °C \][/tex]
Step 3: Calculate the heat released, \( q \):
[tex]\[ q = m \cdot C_p \cdot \Delta T \][/tex]
[tex]\[ q = 1000.0 \text{ g} \cdot 1.50 \text{ J/(g} \cdot °C) \cdot 20.0 °C \][/tex]
[tex]\[ q = 1000 \cdot 1.50 \cdot 20.0 \text{ J} \][/tex]
[tex]\[ q = 30000.0 \text{ J} \][/tex]
Step 4: Convert the heat released from Joules to kilojoules:
[tex]\[ q_{kJ} = \frac{q}{1000} \][/tex]
[tex]\[ q_{kJ} = \frac{30000.0 \text{ J}}{1000} \][/tex]
[tex]\[ q_{kJ} = 30.0 \text{ kJ} \][/tex]
Therefore, the amount of heat released during the combustion of the sample is \( 30.0 \text{ kJ} \).
The correct answer is [tex]\( \boxed{30.0 \text{ kJ}} \)[/tex].
