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 3.500 -g sample of glucose is burned in a bomb calorimeter. The temperature of the calorimeter increases from 20.94 to \(24.72^{\circ} \mathrm{C}\) (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?

Short answer

(a) The total heat capacity of the calorimeter (C) can be determined using the formula C = q / ΔT. First, calculate the heat released (q) during glucose combustion: \(q = 15.57 \, kJ/g × 3.500 \, g = 54.495 \, kJ\). Next, calculate the temperature change (ΔT) of the calorimeter: ΔT = 24.72 °C - 20.94 °C = 3.78 °C. Finally, determine the total heat capacity (C): \(C = \frac{54.495 \, kJ}{3.78 \, °C} = 14.41 \, kJ/°C\). (b) To find the temperature change (ΔT') for double the glucose sample, first calculate the heat released (q') with the doubled glucose sample: q' = 2 × q = 2 × 54.495 kJ = 108.99 kJ. Then, use the total heat capacity of the calorimeter from part (a) to find the temperature change (ΔT'): \(ΔT' = \frac{108.99 \, kJ}{14.41 \, kJ/°C} = 7.56 \, °C\).

Step-by-step solution

Calculate the heat released by glucose combustion

Using the provided heat of combustion value of 15.57 kJ/g and the mass of the glucose sample (3.500 g), we can calculate the total heat released during the glucose combustion. Heat released (q) = heat of combustion × mass of glucose \(q = 15.57 \, kJ/g × 3.500 \, g\)

Calculate the temperature change of the calorimeter

We are given the initial temperature (20.94 °C) and the final temperature (24.72 °C) of the calorimeter. To find the temperature change (ΔT) of the calorimeter, we subtract the initial temperature from the final temperature. ΔT = T(final) - T(initial) ΔT = 24.72 °C - 20.94 °C

determine the total heat capacity of the calorimeter

The heat released by glucose combustion (q) equals the heat absorbed by the calorimeter. Using the formula for heat capacity (C), we can find the total heat capacity of the calorimeter. C = q / ΔT First, calculate the heat released (q) from Step 1, and the temperature change (ΔT) from Step 2. Then, plug these values into the formula to find the total heat capacity of the calorimeter.

Calculate the temperature change for double glucose sample

In this step, we will determine the temperature change (ΔT') if the glucose sample was twice as large. If the sample size is doubled, the heat released during combustion (q') will also be doubled. q' = 2 × q The heat capacity of the calorimeter remains constant. Thus, we can use the formula for heat capacity to find the temperature change for the doubled glucose sample. ΔT' = q' / C First, calculate the heat released (q') with the doubled glucose sample. Then, use the total heat capacity of the calorimeter from Step 3 to find the temperature change (ΔT').

Key concepts

Calorimetry

Calorimetry is an experimental technique used to measure the amount of heat energy released or absorbed in a chemical or physical process. This technique is based on the principle of conservation of energy, which states that energy can neither be created nor destroyed, only transferred or transformed from one form to another. In a caloric measurement, we can observe the energy changes by monitoring temperature changes in a known substance, usually referred to as the calorimeter's content.

Using the heat of combustion of a substance, such as glucose, calorimetry allows us to determine the amount of energy that is released when the substance is burned. The heat of combustion is a vital parameter here and is defined as the energy released as heat when a compound undergoes complete combustion with oxygen under standard conditions. This energy is often expressed in units of kilojoules per gram (kJ/g). In the given exercise, a bomb calorimeter, a device designed for measuring the energy of combustion, is used to ignite a sample of glucose and measure the corresponding temperature rise, allowing us to calculate the heat absorbed and thus the heat capacity of the calorimeter.

Enthalpy Changes

Enthalpy changes are central to the study of chemical thermodynamics, representing the total heat content of a system at constant pressure. The change in enthalpy (ΔH) during a reaction symbolizes the difference in heat content between the products and the reactants. In the context of a combustion reaction, such as that of glucose in the calorimetry exercise, the heat of combustion is precisely the enthalpy change of the reaction.

It's crucial to note that if the reaction emits heat, the enthalpy change is negative, indicating an exothermic process. Conversely, if the reaction absorbs heat, the enthalpy change is positive, denoting an endothermic process. In measurements such as those made in a calorimeter, the enthalpy change corresponds to the heat transferred to or from the substance in the calorimeter, and by knowing the mass of the substance and its heat of combustion, we can calculate this key thermodynamic parameter.

Bomb Calorimeter

A bomb calorimeter is a sophisticated apparatus used to determine the heat of combustion of a sample. It comprises a strong steel container (the bomb) that is filled with oxygen and placed in an insulated water bath. The sample is ignited electrically within the bomb, allowing for a rapid and complete combustion under constant volume conditions. The heat produced by the combustion is transferred to the water and the calorimeter's components, causing a measurable rise in temperature.

Because the bomb calorimeter is designed to withstand high pressure and ensure that the combustion occurs in a sealed environment, we consider that no gas expansion work is done, and thus the process is constant-volume. This is essential in measuring the internal energy change associated with the reaction, making such calorimeters particularly useful in determining precise enthalpy changes for combustion reactions. The constant-volume condition of the bomb calorimeter directly aligns with the requirement that enthalpy is measured at constant pressure, as internal energy changes are equivalent to enthalpy changes at constant volume.

Chemical Thermodynamics

Chemical thermodynamics studies the relationships between heat energy and other forms of energy in chemical processes, linking the physical properties of substances to the direction and extent of chemical reactions. In this domain, we focus on several fundamental quantities, such as enthalpy (H), entropy (S), and Gibbs free energy (G), which predict whether a process can occur spontaneously.

Through principles like Hess's law, the first and second laws of thermodynamics, and the concept of free energy, we understand that the heat of combustion measured in a calorimetry experiment is closely related to the change in Gibbs free energy under constant pressure and temperature conditions. These thermodynamic principles play significant roles in predicting reaction spontaneity and equilibrium, making the study of heat changes, like those examined by calorimetry in a bomb calorimeter, crucial in understanding the intricate dance of energy that occurs in every chemical reaction.