Question

High-purity benzoic acid (C \(_{6} \mathrm{H}_{3} \mathrm{COOH} ; \Delta H\) for combustion \(=\) \(-3227 \mathrm{~kJ} / \mathrm{mol}\) ) is used to calibrate bomb calorimeters. A \(1.221-\mathrm{g}\) sample burns in a calorimeter that has a heat capacity of \(6.384 \mathrm{~kJ} /{ }^{\circ} \mathrm{C}\). What is the temperature change?

Short answer

The temperature change is approximately -5.06°C.

Step-by-step solution

Find the number of moles of benzoic acid

Use the molar mass of benzoic acid, which is 122.12 g/mol, to find the number of moles. Calculate it using the formula: \( \text{Number of moles} = \frac{\text{mass}}{\text{molar mass}} \)Thus, \( \text{Number of moles} = \frac{1.221 \text{ g}}{122.12 \text{ g/mol}} \approx 0.01 \text{ moles} \)

Calculate the heat released during combustion

Multiply the number of moles by the enthalpy change for combustion: \( q = \text{moles} \times \text{enthalpy of combustion} \)Thus, \( q = 0.01 \text{ moles} \times (-3227 \text{ kJ/mol}) = -32.27 \text{ kJ} \)

Use the heat capacity to find the temperature change

Rearrange the formula for heat capacity \( \text{Heat capacity} = \frac{q}{\text{change in temperature}} \) to solve for the temperature change: \( \text{change in temperature} = \frac{q}{\text{Heat capacity}} \)Then, \( \text{change in temperature} = \frac{-32.27 \text{ kJ}}{6.384 \text{ kJ/} { }^{\text{o}} \text{C}} \approx -5.06 { }^{\text{o}} \text{C} \)

Key concepts

enthalpy of combustion

Enthalpy of combustion is the heat released when one mole of a substance completely reacts with oxygen under standard conditions. It is usually measured in kilojoules per mole (kJ/mol). For high-purity benzoic acid, the enthalpy of combustion is \(-3227 \ \text{kJ/mol}\). This negative sign indicates that the reaction releases energy, making it an exothermic process.
In the exercise, we used this value to determine the total heat released when a sample of benzoic acid was burned. By knowing the sample size and its molar mass, we calculated the number of moles burned, and then multiplied this number by the given enthalpy of combustion to find the heat released.

heat capacity

Heat capacity is a property that describes how much heat is needed to change the temperature of a system by one degree Celsius. It is expressed in units of kJ/°C. The higher the heat capacity, the more heat is required to produce a given temperature change.
In our exercise, the bomb calorimeter's heat capacity is 6.384 kJ/°C. This heat capacity helps us understand how the released heat from the benzoic acid combustion affects the temperature of the calorimeter. By using this value, we can determine how much the temperature changes as the heat generated from the reaction is absorbed by the calorimeter.

temperature change

Temperature change is the difference in temperature before and after a heat exchange. The formula to find temperature change (\text{ \(\Delta \text{T}\)) involves dividing the heat (\text{q}\text{ ) by the heat capacity (\text{C}\text{):
\[ \Delta \text{T} = \frac{q \}{C} \] \}}
In our case, the heat released from burning the sample of benzoic acid was -32.27 kJ. Given that the calorimeter's heat capacity is 6.384 kJ/°C, the temperature change (\text{\text{ \Delta \text{T}}}) is found by dividing the heat by the heat capacity:
\text{\text{\text{\text{\text{ \Delta \text{T} = \()\} \frac -32.27 \text{kJ}} \left(\))\}{6.384 \text{kJ/°C}} = -5.06°C}}
This negative value indicates that the temperature of the calorimeter has decreased, corresponding to the heat energy absorbed by the system.