Problem 57 - Chemistry: The Central Science

Question

Under constant-volume conditions the heat of combustion of glucose \(\left(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\right)\) is \(15.57 \mathrm{~kJ} / \mathrm{g}\). A \(2.500-\mathrm{g}\) sample of glucose is burned in a bomb calorimeter. The temperature of the calorimeter increased from \(20.55^{\circ} \mathrm{C}\) to \(23.25^{\circ} \mathrm{C}\). (a) What is the total heat capacity of the calorimeter? (b) If the size of the glucose sample had been exactly twice as large, what would the temperature change of the calorimeter have been?

Step-by-Step Solution

Answer

The heat released by the 2.5 g glucose sample is \(q_{glucose} = 2.500 \times 15.57 \,\text{kJ/g} = 38.925 \,\text{kJ}\). The temperature change of the calorimeter is \(\Delta T = 23.25 - 20.55 = 2.7^{\circ}\text{C}\). Using these values, we can calculate the heat capacity of the calorimeter as \(C_{calorimeter} = \frac{q_{glucose}}{\Delta T} = \frac{38.925}{2.7} \approx 14.42 \,\text{kJ}/^{\circ}\text{C}\) for part (a). For part (b), if the glucose sample was twice as large, then \(\Delta T_{new} = 2 \times \Delta T = 2 \times 2.7 = 5.4^{\circ}\text{C}\), which would have been the new temperature change of the calorimeter.

Step 1: Calculate the heat released by glucose sample g

First, we need to calculate the heat released by the glucose sample when it was burned. We can do this using the heat of combustion of glucose and the mass of the sample: q_glucose = mass_glucose × heat_combustion_per_gram q_glucose = 2.500 g × 15.57 kJ/g Now, calculate the value of q_glucose.

Step 2: Calculate the temperature change of the calorimeter

Next, we need to calculate the temperature change of the calorimeter when the glucose sample was burned. We're given the initial and final temperatures: T_initial = 20.55°C T_final = 23.25°C Now, calculate the temperature change (ΔT) using the given values: ΔT = T_final - T_initial

Step 3: Calculate the heat capacity of the calorimeter

Now that we know the heat released by the glucose sample (q_glucose) and the temperature change of the calorimeter (ΔT), we can calculate the heat capacity (C_calorimeter) of the calorimeter using the equation: q_glucose = C_calorimeter × ΔT Now, rearrange the equation to solve for C_calorimeter and plug in the values calculated earlier: C_calorimeter = q_glucose/ΔT

Step 4: Predict the temperature change for a larger glucose sample

We're asked to find the temperature change of the calorimeter if we burned a glucose sample exactly twice as large as the initial one (5 g). Since the heat of combustion per gram is constant for glucose, we know that the heat released by the new sample will be twice the amount of heat released by the initial sample. Therefore, the new change in temperature can be calculated by: ΔT_new = 2 × ΔT Now, calculate the value of ΔT_new.

Key Concepts

Heat of Combustion

The heat of combustion is an important concept in thermochemistry that refers to the amount of heat that is released when a substance is burned in the presence of oxygen. It is typically measured in units of energy per mass, such as kilojoules per gram (kJ/g). This measurement tells us how much energy a particular fuel can produce, and it's essential for calculating the energy transfer during a combustion reaction.

In the context of our exercise involving glucose \(\left(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\right)\), the heat of combustion allows us to determine the total amount of heat released when a certain mass of glucose is burned. By multiplying the mass of the substance by its heat of combustion value, we can calculate the total energy (in kilojoules) released in the reaction.

Calorimetry

Calorimetry is the process of measuring the amount of heat released or absorbed during a chemical reaction, physical change, or heat capacity. To conduct these measurements, a device called a calorimeter is used. It can be as simple as an insulated container equipped with a thermometer, or as specialized as a bomb calorimeter, which is used for measuring the heat of combustion of materials.

The bomb calorimeter is particularly useful because it operates at constant volume, which is ideal for combustion reactions. During the process, the substance to be measured is placed in a strong, sealed vessel known as a 'bomb,' and is then ignited in a controlled environment. The heat released by the reaction is absorbed by the water and the calorimeter itself, causing a measurable temperature change. This result, along with knowing the heat capacity of the calorimeter, allows us to calculate the heat involved in the reaction.

Thermochemistry

Thermochemistry is a branch of chemistry that involves the study of the energy and heat associated with chemical reactions and physical transformations. It provides a quantitative connection between the heat exchanged in a system and the work done during any chemical or physical process.

A key principle of thermochemistry is that energy cannot be created or destroyed, only transformed. This principle is known as the first law of thermodynamics. In our glucose combustion example, the chemical energy stored in the glucose molecules is transformed into heat energy, which is then partly absorbed by the calorimeter. Understanding thermochemistry is essential for predicting how much energy is involved in reactions and for designing energy-efficient chemical processes.

Energy Transfer

Energy transfer in the context of calorimetry and combustion involves the movement of energy from one part of a system to another, or from one system to another system. This can occur in various forms, such as heat or work. In the case of our exercise, we focus on heat as the form of energy that gets transferred.

When glucose combusts, the energy within its chemical bonds is released and transferred to the surroundings, which includes the water and the walls of the calorimeter. Using calorimetry, we can measure this heat transfer very precisely. The amount of transferred energy is often calculated with the help of specific heat capacities and the mass of the substances involved.

Temperature Change

Temperature change is a direct outcome of heat transfer in a system and is often the measurable physical change that calorimetry relies on. It's defined as the difference between the final temperature and the initial temperature of the system. The increase in temperature within the calorimeter, due to the exothermic reaction of glucose combustion, directly relates to the amount of heat absorbed by the system.

To calculate the temperature change in a calorimetry experiment, like the one in our exercise, you subtract the initial temperature of the calorimeter from the final temperature after the combustion event. This change can then be used, along with the heat capacity of the calorimeter, to calculate the heat released by the reaction. Additionally, understanding this principle helps answer questions about scaling reactions, such as predicting the temperature change for larger samples.