A sample of octane [tex]$\left( C_8H_{18} \right)$[/tex] with a mass of [tex]$0.750 \text{ g}$[/tex] is burned in a bomb calorimeter. As a result, the temperature of the calorimeter increases from [tex]$21.0^{\circ} \text{C}$[/tex] to [tex]$41.0^{\circ} \text{C}$[/tex]. The specific heat of the calorimeter is [tex]$1.50 \text{ J}/\left( \text{g} \cdot {}^{\circ} \text{C} \right)$[/tex], and its mass is [tex]$1.00 \text{ kg}$[/tex].

How much heat is released during the combustion of this sample? Use [tex]$q = mC_p\Delta T$[/tex].

A. [tex]$22.5 \text{ kJ}$[/tex]
B. [tex]$30.0 \text{ kJ}$[/tex]
C. [tex]$31.5 \text{ kJ}$[/tex]
D. [tex]$61.5 \text{ kJ}$[/tex]



Answer :

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].