Question

(a) When a 0.235 -g sample of benzoic acid is combusted in a bomb calorimeter (Figure 5.19\()\) , the temperature rises \(1.642^{\circ} \mathrm{C} .\) When a \(0.265-\mathrm{g}\) sample of caffeine, \(\mathrm{C}_{8} \mathrm{H}_{10} \mathrm{N}_{4} \mathrm{O}_{2}\) is burned, the temperature rises \(1.525^{\circ} \mathrm{C} .\) Using the value 26.38 \(\mathrm{kJ} / \mathrm{g}\) for the heat of combustion of benzoic acid, calculate the heat of combustion per mole of caffeine at constant volume. (b) Assuming that there is an uncertainty of \(0.002^{\circ} \mathrm{C}\) in each temperature reading and that the masses of samples are measured to \(0.001 \mathrm{g},\) what is the estimated uncertainty in the value calculated for the heat of combustion per mole of caffeine?

Short answer

The heat of combustion per mole of caffeine at constant volume is calculated to be 4224.97 kJ/mol. The estimated uncertainty in this value is 17.03 kJ/mol.

Step-by-step solution

(a) Calculate the heat of combustion per mole of caffeine

To calculate the heat of combustion per mole of caffeine, we must first determine the heat capacity of the bomb calorimeter. We can do this using the benzoic acid data. Given the heat of combustion of benzoic acid as 26.38 kJ/g, we can calculate the heat capacity of the calorimeter as follows: Heat capacity of calorimeter = Heat released by benzoic acid / Temperature rise due to benzoic acid combustion Heat released by benzoic acid = (0.235 g) (26.38 kJ/g) = 6.1983 kJ Temperature rise due to benzoic acid combustion = 1.642 °C Heat capacity of calorimeter = 6.1983 kJ / 1.642 °C = 3.7754 kJ/°C Now we can find the heat released by caffeine during its combustion: Heat released by caffeine = Heat capacity of calorimeter × Temperature rise due to caffeine combustion Heat released by caffeine = 3.7754 kJ/°C × 1.525 °C = 5.7596 kJ Next, we need to determine the moles of caffeine in the 0.265-g sample. The molar mass of caffeine is: C8H10N4O2 = (8 × 12.01) + (10 × 1.01) + (4 × 14.01) + (2 × 16.00) = 194.19 g/mol Moles of caffeine = 0.265 g / 194.19 g/mol = 0.001364 mol Finally, we can calculate heat of combustion per mole of caffeine: Heat of combustion per mole of caffeine = Heat released by caffeine / Moles of caffeine Heat of combustion per mole of caffeine = 5.7596 kJ / 0.001364 mol = 4224.97 kJ/mol

(b) Estimate the uncertainty in the heat of combustion per mole of caffeine

To calculate the estimated uncertainty in the heat of combustion per mole of caffeine, we can use the propagation of uncertainty formula: Relative uncertainty in heat of combustion per mole of caffeine = sqrt[(relative uncertainty in temperature)^2 + (relative uncertainty in mass)^2] Relative uncertainty in caffeine temperature reading = 0.002 °C / 1.525 °C = 0.001312 Relative uncertainty in caffeine mass reading = 0.001 g / 0.265 g = 0.003774 Relative uncertainty in heat of combustion per mole of caffeine = sqrt[(0.001312)^2 + (0.003774)^2] = 0.00403 Now, we can calculate the absolute uncertainty in the heat of combustion per mole of caffeine: Absolute uncertainty in heat of combustion per mole of caffeine = relative uncertainty × heat of combustion per mole of caffeine Absolute uncertainty in heat of combustion per mole of caffeine = 0.00403 × 4224.97 kJ/mol = 17.03 kJ/mol So, the estimated uncertainty in the value calculated for the heat of combustion per mole of caffeine is 17.03 kJ/mol.

Key concepts

Bomb Calorimeter

Understanding the use of a bomb calorimeter is crucial for grasping the concept of heat of combustion in substances like caffeine. A bomb calorimeter is a sealed container designed to measure energy release during a combustion reaction. It's sturdy enough to withstand the high pressure produced during the combustion of a sample.

To determine the heat capacity of the bomb calorimeter, which is a critical step in calorimetry, one must first burn a known quantity of a standard substance like benzoic acid. The heat released by this combustion raises the temperature of a water bath surrounding the chamber. By dividing the known heat released by the temperature change, we calculate the calorimeter's heat capacity. Once we have this information, we can measure unknown substances, such as caffeine, by observing the corresponding temperature rise when they combust within this controlled environment.

Calorimetry

Calorimetry is the process of measuring the heat of chemical reactions or physical changes as well as heat capacity. In the context of our exercise, we apply calorimetry to determine the heat of combustion of a substance—by measuring the temperature change in the calorimeter's environment. The heat of combustion is directly proportional to the temperature increase in the bomb calorimeter's water bath.

For the benzoic acid and caffeine measurements, we needed to consider the precise masses to calculate the heat released during combustion accurately. The process of calorimetry requires meticulous attention to uncertainties in measurements to ensure accurate calculation of energy changes, such as the uncertainty in temperature readings and the weighing of the samples.

Enthalpy of Combustion

The enthalpy of combustion is a critical value representing the energy released when a compound completely oxidizes to form water and carbon dioxide. It is expressed in units of energy per mole, often kilojoules per mole (kJ/mol). With reference to our exercise, we calculated the enthalpy of combustion for caffeine by establishing the energy released during its combustion and relating this to the number of moles of caffeine burned.

Improvement advice: The exercise demonstrates that the accurate determination of enthalpy involves precise measurements and considerations of uncertainty. A small error in temperature or mass could alter the final calculation significantly. The step-by-step solution concludes with the propagation of uncertainty, an indispensable method in scientific measurements, providing a deeper insight into the precision of our calculations and ensuring a more comprehensive understanding for students.

The calculation and estimation of uncertainties underscore the importance of precision in calorimetric measurements, as they play a vital role in the accuracy of the calculated enthalpy of combustion of a substance.