A sample of octane [tex]\(\left( C_8 H_{18} \right)\)[/tex] with a mass of [tex]\(0.750 \, g\)[/tex] is burned in a bomb calorimeter. As a result, the temperature of the calorimeter increases from [tex]\(21.0^{\circ} C\)[/tex] to [tex]\(41.0^{\circ} C\)[/tex]. The specific heat of the calorimeter is [tex]\(1.50 \, J /\left( g \cdot{ }^{\circ} C \right)\)[/tex], and its mass is [tex]\(1.00 \, kg\)[/tex]. How much heat is released during the combustion of this sample?

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

A. [tex]\(22.5 \, kJ\)[/tex]

B. [tex]\(30.0 \, kJ\)[/tex]

C. [tex]\(31.5 \, kJ\)[/tex]

D. [tex]\(61.5 \, kJ\)[/tex]



Answer :

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]