Question

(a) When a 0.235-g sample of benzoic acid is combusted in a bomb calorimeter (Figure 5.18), the temperature rises \(1.642^{\circ} \mathrm{C}\). When a \(0.265-\mathrm{g}\) 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 \(4216.44 \,\text{kJ/mol}\) and the estimated uncertainty of this value is approximately \(44.53 \,\text{kJ/mol}\).

Step-by-step solution

Calculate the heat capacity of the calorimeter

We will first use the information for the benzoic acid to determine the constant heat capacity (C) of the calorimeter. To do this, we will use the formula \(q = C\Delta T\). We have the value of the heat of combustion for benzoic acid per gram, so we can multiply it by the mass to get the heat (q): \(q_{benzoic} = 0.235 \,\text{g} \cdot 26.38 \,\text{kJ/g} = 6.1963 \,\text{kJ}\) Now we can use the heat and the temperature change of the benzoic acid to calculate the heat capacity of the calorimeter: \( C = \frac{q_{benzoic}}{\Delta T_{benzoic}} = \frac{6.1963 \,\text{kJ}}{1.642^\circ \text{C}} = 3.773 \,\text{kJ/}^\circ \text{C} \)

Calculate the heat of combustion for caffeine

Next, we will use the heat capacity we found in step 1 and the temperature change of the caffeine to calculate the heat of combustion of the caffeine. We can use the same formula for heat, \(q = C\Delta T\), and substitute the caffeine values: \(q_{caffeine} = C\Delta T_{caffeine} = 3.773 \,\text{kJ/}^\circ \text{C} \cdot 1.525^\circ \text{C} = 5.754 \,\text{kJ}\)

Calculate the heat of combustion per mole of caffeine

Now that we have the heat of combustion of the caffeine, we need to calculate the heat of combustion per mole. To do this, we will first need to calculate the molar mass of caffeine: Molar mass of caffeine: \( (8 \times 12.01) + (10 \times 1.01) + (2 \times 16.00) + (4 \times 14.01) = 194.19 \,\text{g/mol} \) Next, we'll convert the mass of caffeine to moles using the molar mass: Moles of Caffeine = \( \frac{0.265 \,\text{g}}{194.19 \,\text{g/mol}} = 0.001364 \,\text{moles} \) Now we can find the heat of combustion per mole of caffeine by dividing the heat of combustion by the number of moles: \( \frac{5.754 \,\text{kJ}}{0.001364 \,\text{moles}} = 4216.44 \,\text{kJ/mol} \) The heat of combustion per mole of caffeine at constant volume is approximately \(4216.44 \,\text{kJ/mol}\).

Calculate the estimated uncertainty

We are given uncertainties in the temperature readings and mass measurements. Using this information, we can calculate the estimated uncertainty for the heat of combustion per mole of caffeine. We know that the percent uncertainty in the heat of combustion per moles of caffeine is equal to the percent uncertainties in all individual measurements added together. Percent uncertainties: Mass of benzoic acid: \(\frac{0.001\,\text{g}}{0.235\,\text{g}} \times 100 = 0.426\%\) Mass of caffeine: \(\frac{0.001\,\text{g}}{0.265\,\text{g}} \times 100 = 0.377\%\) Temperature rise of benzoic acid: \(\frac{0.002^\circ \text{C}}{1.642^\circ \text{C}} \times 100 = 0.122\%\) Temperature rise of caffeine: \(\frac{0.002^\circ \text{C}}{1.525^\circ \text{C}} \times 100 = 0.131\%\) Total percent uncertainty: \( 0.426\% + 0.377\% + 0.122\% + 0.131\% = 1.056\% \) Finally, use the total percent uncertainty to approximate the uncertainty in the value of the heat of combustion per mole of caffeine: \( 4216.44 \,\text{kJ/mol} \times \frac{1.056\%}{100} = 44.53 \,\text{kJ/mol} \) The estimated uncertainty in the value calculated for the heat of combustion per mole of caffeine is approximately \(44.53 \,\text{kJ/mol}\).

Key concepts

Heat of Combustion

The heat of combustion tells us how much energy is released when a substance burns, typically in oxygen. This energy is often measured in kilojoules per mole (kJ/mol) and reflects the substance's potential to release energy. With benzoic acid, for example, the energy released upon combustion is a known value of 26.38 kJ/g. For caffeine, the exercise involves calculating the heat of combustion per mole.

This process involves a few steps:

• First, we calculate the heat produced by combusting a certain mass of caffeine using the formula: \(q = C \Delta T\), where \(C\) is the heat capacity and \(\Delta T\) is the temperature change.
• Next, we convert the mass of caffeine into moles, which requires knowledge of its molar mass.
• Finally, by dividing the total heat produced by the number of moles, we get the heat of combustion per mole of caffeine.

Understanding the heat of combustion helps in assessing the energy efficiency of different substances when burned.

Bomb Calorimeter

A bomb calorimeter is an apparatus used to precisely measure the heat of combustion of a substance. It isolates the reaction from its surrounding environment, ensuring that all the heat generated is accounted for internally. This is important to gain accurate measurements.

Bomb calorimeters are built to withstand the high-pressure conditions of a combustion reaction. Take note of the following when understanding a bomb calorimeter:

• The calorimeter consists of a robust container with a known heat capacity. This affects the calculations since the heat produced during combustion both increases the temperature of the sample and the water surrounding it.
• A key process step is using a known substance, such as benzoic acid, to determine the heat capacity of the bomb calorimeter at initial setup, allowing for subsequent calculations with unknown substances.
• Each measurement in a calorimeter involves careful monitoring of temperature changes to ensure data is as precise and accurate as possible.

Learning about bomb calorimeters enhances appreciation of the careful controls needed in calorimetry experiments.

Uncertainty Calculation

Uncertainty calculations in calorimetry are crucial for understanding the precision of experimental results. They provide insight into the reliability and repeatability of measurements.

Let's explore how these uncertainties can be calculated:

• Firstly, every measurement, whether mass or temperature, has an associated uncertainty. For example, the mass of caffeine may be measured with a precision of ±0.001 g.
• Slight errors in temperature readings also propagate through calculations. An uncertainty of ±0.002°C in temperature affects the final heat calculation.
• The total uncertainty for a computed value, like the heat of combustion per mole, combines all individual uncertainties. This is often calculated as a percentage of the measured values.

Ultimately, calculating uncertainty is a valuable part of any scientific experiment, as it highlights where improvements in precision can occur and guides the interpretation of results.