Problem 59 - 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 \(3.500-\mathrm{g}\) 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?

Step-by-Step Solution

Answer

The total heat capacity of the calorimeter is \(14.413\, \mathrm{\dfrac{kJ}{{}^{\circ}\mathrm{C}}}\). If the size of the glucose sample had been exactly twice as large, the temperature change of the calorimeter would have been \(7.56 {}^{\circ}\mathrm{C}\).

Step 1: Calculate the heat released by burning the glucose sample

To calculate the heat released by burning the glucose sample, we use the heat of combustion, \(\delta H_c\), and the mass of the glucose sample. The heat released is given by: \[ q = m \cdot \delta H_c \] Where \(q\) is the heat released, \(m\) is the mass of glucose, and \(\delta H_c\) is the heat of combustion. Now, plug in the values: \[ q = (3.500 \, \mathrm{g})\left(15.57\, \mathrm{ \dfrac{kJ}{g}} \right) = 54.495\, \mathrm{kJ}\]

Step 2:Calculate the heat absorbed by the calorimeter

Under constant volume conditions, the heat absorbed by the calorimeter is equal to the heat released by the burning of glucose. Therefore, \(q_{cal} = q\), where \(q_{cal}\) is the heat absorbed by the calorimeter. \[q_{cal} = 54.495\, \mathrm{kJ}\]

Step 3: Calculate the total heat capacity of the calorimeter

To determine the total heat capacity of the calorimeter, we can use the equation: \[q_{cal} = C_{cal} \cdot \Delta T\] Where \(C_{cal}\) is the total heat capacity of the calorimeter and \(\Delta T\) is the temperature change. We have the temperature change, so we can rearrange the equation to find \(C_{cal}\): \[ C_{cal} = \dfrac{q_{cal}}{\Delta T} \] The temperature change is given by: \[\Delta T = T_{final} - T_{initial} = 24.72{}^{\circ}\mathrm{C} - 20.94{}^{\circ}\mathrm{C} = 3.78{}^{\circ}\mathrm{C}\] Substitute the values for \(q_{cal}\) and \(\Delta T\) and find the value for \(C_{cal}\): \[C_{cal} = \dfrac{54.495\, \mathrm{kJ}}{3.78{}^{\circ}\mathrm{C}} = 14.413 \, \mathrm{\dfrac{kJ}{{}^{\circ}\mathrm{C}}}\]

Step 4: Determine the temperature change for twice the mass of glucose

For the case when the glucose sample is doubled, the heat released will also double (as the heat released is directly proportional to the mass of glucose). The doubled heat released will be: \[q_{2} = 2 \cdot q = 2 \cdot 54.495\, \mathrm{kJ} = 108.990\, \mathrm{kJ}\] Since the calorimeter absorbs all the energy, we can use the same heat capacity equation with the new heat released (q_{2}) and the unknown new temperature change \(\Delta T_2\): \[q_{2} = C_{cal} \cdot \Delta T_2\] Rearranging the equation, we have: \[\Delta T_2 = \dfrac{q_{2}}{C_{cal}}\] Substitute the values for \(q_{2}\) and \(C_{cal}\) and find the value for \(\Delta T_2\): \[\Delta T_2 = \dfrac{108.990\, \mathrm{kJ}}{14.413 \, \mathrm{\dfrac{kJ}{{}^{\circ}\mathrm{C}}}} = 7.56 {}^{\circ}\mathrm{C}\] The temperature change for the calorimeter if the size of the glucose sample had been exactly twice as large would be \(7\).56{}^{\circ}\mathrm{C}.

Key Concepts

Heat of Combustion

The heat of combustion is an essential concept in thermochemistry. It describes the total energy released as heat when a substance combusts completely with oxygen under specified conditions. For glucose, a key energy-providing sugar, its heat of combustion is 15.57 kJ per gram. This means that when 1 gram of glucose is burned, 15.57 kJ of energy is liberated as heat. Understanding this value is crucial because it helps us calculate the overall energy produced or absorbed in reactions involving glucose. In this exercise, we multiply the mass of glucose (3.5 g) by the heat of combustion to determine the heat released during combustion. This step is fundamental to finding out how much heat a calorimeter absorbs during an experiment.

Heat Capacity

Heat capacity, particularly in the context of a calorimeter, is the amount of heat needed to raise the temperature of an object by 1 degree Celsius. Knowing the heat capacity of a calorimeter allows us to understand how well it absorbs heat energy from chemical reactions or physical processes.The total heat capacity of the calorimeter in this exercise was calculated as 14.413 kJ per degree Celsius using the formula:\[C_{cal} = \frac{q_{cal}}{\Delta T}\]Where:- \(q_{cal}\) is the heat absorbed by the calorimeter (54.495 kJ)- \(\Delta T\) is the temperature change (3.78 °C)This value tells us how much heat the calorimeter inside a laboratory setup can absorb for each degree of temperature change. Understanding this helps in designing experiments and interpreting results from experiments involving calorimeters.

Temperature Change

Temperature change is a vital measure in calorimetry experiments as it indicates the amount of heat energy absorbed or released. In these settings, the difference in initial and final temperatures of the system provides insights into the energetic changes taking place.In our glucose combustion example, the calorimeter temperature rose from 20.94°C to 24.72°C, showing a temperature change (\(\Delta T\)) of 3.78°C. This measure, when combined with the calorimeter's heat capacity, allows us to calculate the total heat absorbed by the calorimeter and reflects the amount of heat released by burning the glucose sample.This relation is expressed mathematically as:\[\Delta T = T_{final} - T_{initial}\]Thus, knowing the temperature change is key to quantifying the heat exchange in calorimetric studies.

Bomb Calorimeter

The bomb calorimeter is a precise device used in thermochemistry to measure the heat of combustion of substances. It operates under constant volume conditions ensuring accurate and consistent readings by containing the sample in a strong metallic chamber. In these setups, the substance combusts completely, and the calorimeter measures the subsequent temperature rise. It's critical in academic and research settings because it allows for determining the energy content of food, fuels, and other materials through this rigorous physical measurement. In our example, the bomb calorimeter accurately recorded the full energy exchange from the complete combustion of glucose. Its design minimizes heat loss to the surroundings, providing an enclosed environment for precise calorimetric analysis.

Thermochemistry

Thermochemistry is the branch of chemistry focused on studying heat change during chemical reactions and physical transformations. It encompasses the principles of energy conservation and transfer, crucial for understanding real-world chemical processes. The reaction of glucose combustion is a classic example demonstrating these principles. As glucose burns, chemical bonds break or form, resulting in an extensive heat release calculated using its heat of combustion.

• It provides insight into energy flow, demonstrating how chemical energy converts to heat.
• It underlines the importance of understanding enthalpy changes during reactions.

Through experiments with calorimeters in thermochemistry, we gain a quantitative grasp of these energetic changes, enhancing our comprehension of energy use in industrial and laboratory settings.

Glucose Combustion

The combustion of glucose is a fundamental biochemical reaction that releases substantial energy, vital for both industrial and biological processes. This reaction involves the molecular breakdown of glucose (\(\mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_6\)) in the presence of oxygen to produce carbon dioxide, water, and energy.In a mathematical form, this energy release can be represented by:\[\mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_6 + 6\; \mathrm{O}_2 \rightarrow 6\, \mathrm{CO}_2 + 6\, \mathrm{H}_2\mathrm{O} + \text{energy}\]This process is notably exothermic, signifying that it releases heat, captured using calorimeters to study the energetic profile of glucose combustion.

• This principle is used to understand cellular respiration—the metabolic process by which cells harness energy.
• It provides insight into calorie measurements in food industry applications.

Overall, studying glucose combustion illuminates how effectively glucose can provide energy, forming a basis for many bioenergetic and thermochemical analyses.