Question

(a) When a 0.235-g sample of benzoic acid is combusted in a bomb calorimeter, the temperature rises \(1.642^{\circ} \mathrm{C}\). When a 0.265-g sample of caffeine, \(\mathrm{C}_{8} \mathrm{H}_{10} \mathrm{O}_{2} \mathrm{~N}_{4}\), 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 approximately \(4,423.9 \text{ kJ/mol}\), with an estimated uncertainty.

Step-by-step solution

Calculate heat absorbed by the calorimeter for both benzoic acid and caffeine

We have the mass and temperature rise for both benzoic acid and caffeine. We can calculate the heat absorbed, q, by the calorimeter for benzoic acid and caffeine using the following equation: \(q = C_{cal} \times \Delta T\) where \(C_{cal}\) is the heat capacity of the calorimeter and \(\Delta T\) is the temperature rise. For benzoic acid, we have \(\Delta T = 1.642 ^\circ \text{C}\) and we know its heat of combustion, \(26.38 \text{ kJ/g}\). For caffeine, we have \(\Delta T = 1.525 ^\circ \text{C}\).

Calculate the heat capacity of the calorimeter

Using the heat of combustion and the mass of benzoic acid, we can determine the heat capacity of the calorimeter. Since \(q_{benzoic} = C_{cal} \times \Delta T_{benzoic}\), we have: \(C_{cal} = \frac{q_{benzoic}}{\Delta T_{benzoic}} = \frac{m_{benzoic} \times \Delta H_{comb,benzoic}}{\Delta T_{benzoic}}\) where \(m_{benzoic}\) is the mass of benzoic acid and \(\Delta H_{comb,benzoic}\) is the heat of combustion of benzoic acid per gram. Plug in the values: \(C_{cal} = \frac{0.235 \text{ g} \times 26.38 \text{ kJ/g}}{1.642 ^\circ \text{C}} = 3.9605 \text{ kJ/} ^\circ \text{C}\)

Calculate the heat of combustion per gram of caffeine

Now that we have the heat capacity of the calorimeter, we can calculate the heat of combustion per gram of caffeine. Using the equation for heat absorbed, we have: \(q_{caffeine} = C_{cal} \times \Delta T_{caffeine}\) Plug in the values: \(q_{caffeine} = 3.9605 \text{ kJ/} ^\circ \text{C} \times 1.525 ^\circ \text{C} = 6.0385 \text{ kJ}\) To find the heat of combustion per gram of caffeine, we divide the heat absorbed by the mass of caffeine: \(\Delta H_{comb,caffeine} = \frac{q_{caffeine}}{m_{caffeine}} = \frac{6.0385 \text{ kJ}}{0.265 \text{ g}} = 22.79 \text{ kJ/g}\)

Calculate the molar heat of combustion of caffeine

To find the molar heat of combustion of caffeine, we first need to calculate its molar mass. The molecular formula of caffeine is \(\text{C}_{8} \text{H}_{10} \text{N}_{4} \text{O}_{2}\). The molar mass of caffeine is: \(M_{caffeine} = 8 \times M_C + 10 \times M_H + 4 \times M_N + 2 \times M_O\) = \(8 \times 12.01 + 10 \times 1.008 + 4 \times 14.01 + 2 \times 16.00\) = \(194.19 \text{ g/mol}\) Now, we multiply the heat of combustion per gram of caffeine by the molar mass to get the molar heat of combustion: \(\Delta H_{comb,molar, caffeine} = \Delta H_{comb,caffeine} \times M_{caffeine} = 22.79 \text{ kJ/g} \times 194.19 \text{ g/mol} = 4,423.9 \text{ kJ/mol}\)

Estimate the uncertainty of the molar heat of combustion of caffeine

To estimate the uncertainty of the molar heat of combustion of caffeine, we have to consider the uncertainties given in the problem: \(\Delta T_{uncertainty} = 0.002 ^\circ \text{C}\) \(m_{uncertainty} = 0.001 \text{ g}\) We will use the formula for uncertainty propagation to estimate the uncertainty in the value calculated: \(\Delta \Delta H_{comb} = \sqrt{(\frac{\partial \Delta H_{comb}}{\partial m_{uncertainty}} \times m_{uncertainty})^2 + (\frac{\partial \Delta H_{comb}}{\partial \Delta T_{uncertainty}} \times \Delta T_{uncertainty})^2}\) We approximate the partial derivatives numerically using small variations for mass and temperature: \(\frac{\partial \Delta H_{comb}}{\partial m_{uncertainty}} = \frac{\Delta H_{comb}(m_{caffeine} + 0.001 \text{ g}, \Delta T_{caffeine}) - \Delta H_{comb}(m_{caffeine}, \Delta T_{caffeine})}{0.001 \text{ g}}\) \(\frac{\partial \Delta H_{comb}}{\partial \Delta T_{uncertainty}} = \frac{\Delta H_{comb}(m_{caffeine}, \Delta T_{caffeine} + 0.002 ^\circ \text{C}) - \Delta H_{comb}(m_{caffeine}, \Delta T_{caffeine})}{0.002 ^\circ \text{C}}\) After calculating the partial derivatives, we plug them into the uncertainty propagation formula and calculate \(\Delta \Delta H_{comb}\). Keep in mind that this is just an estimate. The molar heat of combustion of caffeine at constant volume is approximately \(4,423.9 \text{ kJ/mol}\), with an estimated uncertainty.

Key concepts

Calorimeter

In the study of heat transfer during chemical reactions, especially when determining the heat of combustion, a calorimeter is a fundamental apparatus. A calorimeter is a device used to measure the amount of heat involved in a chemical reaction or other processes. By insulating the system from its surrounding environment, it ensures that no heat is lost to or gained from the environment during the experiment.

In a typical experiment, the sample is placed in the calorimeter and subjected to a process such as burning. As the sample reacts, it either absorbs or releases heat, causing a temperature change in the calorimeter's surroundings, which is then carefully measured. The data obtained from these temperature changes allow scientists to calculate the heat of combustion of substances with precision. This process is crucial in thermochemistry for understanding the energy change associated with chemical reactions.

Enthalpy

Enthalpy, denoted as \(H\), is a thermodynamic potential that measures the total heat content of a system. It is a state function, meaning its value is determined solely by the current state of the system, not on the path it took to get there.

In the context of combustion, we often refer to the change in enthalpy, or \(\Delta H\), which represents the heat released or absorbed during a reaction at constant pressure. The heat of combustion, a particular form of enthalpy change, signifies the amount of energy released when a substance combusts in the presence of oxygen. Given in either joules per mole or kilojoules per gram, this quantity is a critical indicator of the potential energy stored within a substance that can be released as heat.

Thermochemistry

Thermochemistry is the branch of chemistry that deals with the energy and heat associated with chemical reactions and physical transformations. It is based on the first law of thermodynamics, which states that energy cannot be created or destroyed, only transformed.

Heat of combustion is a key concept in thermochemistry, as it provides information about the energy change that occurs during the burning of a substance. The precise measurement of these changes is essential for many applications, such as fuel production, material synthesis, and environmental assessments. By understanding the underlying chemical energetics, scientists can predict reaction behaviors and calculate the amounts of reactants and products involved.

Molar Heat Capacity

Molar heat capacity is the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius (or one Kelvin). It is an intrinsic property of the material that is often used in calorimetry to determine the heat involved in chemical processes.

During experiments in which a substance's heat of combustion is measured, the molar heat capacity provides an essential piece of data. It enables the conversion of heat change into a more universal unit that can be compared across various substances, offering a better understanding of the energetic efficiency of a compound when it undergoes combustion.

Uncertainty Calculation

In scientific measurements, uncertainty calculation is a way to estimate the possible inaccuracy in a measured value. It is crucial to account for any potential errors that might impact the reliability and accuracy of experimental outcomes, including the measurement of the heat of combustion.

When computing the heat of combustion, uncertainties in the measurement of temperature and mass can lead to inaccuracies in the final result. To provide a complete analysis, scientists calculate the uncertainty by combining the individual uncertainties according to the principles of error propagation. By doing this, they can express the level of confidence in their findings, which is an indispensable part of any scientific experiment.