Problem 21 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

A slice of banana weighing \(2.7 \mathrm{g}\) was burned in oxygen in a bomb calorimeter and produced a temperature rise of \(3.05 \mathrm{K}\) In the same calorimeter, the combustion of \(0.316 \mathrm{g}\) of benzoic acid produced a temperature rise of \(3.24 \mathrm{K}\). Ac U for benzoic acid is \(-3251 \mathrm{kJmol}^{-1}\). If the average mass of a banana is \(125 \mathrm{g}\), how much energy in (a) \(\mathrm{kJ}\) and (b) calories (kcal) can be obtained on average from a banana? (1 calorie \(=4.18 \mathrm{kJ}\) ) (Section 13.5 )

Step-by-Step Solution

Answer

The whole banana yields approximately -366.67 kJ or -87.69 kcal.

Step 1: Calculate Heat Capacity of the Calorimeter

First, determine the heat capacity of the calorimeter using the data from benzoic acid. The equation for heat transfer is: \[ q = C_{cal} \times \Delta T \]where \( q \) is the heat produced by the benzoic acid, \( C_{cal} \) is the heat capacity of the calorimeter, and \( \Delta T \) is the temperature change. Given:- Mass of benzoic acid = 0.316 g- \( \Delta T = 3.24 \mathrm{K} \)- Molar heat of combustion for benzoic acid \( = -3251 \mathrm{kJmol}^{-1} \)- Molar mass of benzoic acid \( \approx 122 \mathrm{g/mol} \)Calculate \( q \) for benzoic acid:\[ q = \frac{0.316 \text{ g}}{122 \text{ g/mol}} \times (-3251 \text{ kJ/mol}) = -8.41 \text{ kJ} \]Now, calculate \( C_{cal} \):\[ C_{cal} = \frac{-8.41 \text{ kJ}}{3.24 \text{ K}} \approx -2.595 \text{ kJ/K} \]

Step 2: Calculate the Energy from the Banana Slice

Using the heat capacity of the calorimeter, calculate the energy released from the banana slice.Given:- Mass of banana slice = 2.7 g- \( \Delta T = 3.05 \mathrm{K} \)- \( C_{cal} = -2.595 \text{ kJ/K} \)Use the calorimeter formula:\[ q_{banana} = C_{cal} \times \Delta T = -2.595 \text{ kJ/K} \times 3.05 \text{ K} = -7.92 \text{ kJ} \]

Step 3: Scale Energy to the Whole Banana

Now scale the energy calculated from the slice to an entire banana.Given:- Energy from one slice \( q_{slice} = -7.92 \text{ kJ} \)- Slice mass = 2.7 g- Average banana mass = 125 gCalculate the energy for the whole banana:\[ q_{whole} = -7.92 \text{ kJ} \times \frac{125 ext{ g}}{2.7 ext{ g}} = -366.67 \text{ kJ} \]

Step 4: Convert Energy to Calories

Convert the energy obtained for the entire banana from kJ to calories.Given conversion factor:1 calorie = 4.18 kJThus, the energy in calories is:\[ \text{Calories} = \frac{-366.67 \text{ kJ}}{4.18 \text{ kJ/cal}} \approx -87.69 \text{ kcal} \]

Key Concepts

Heat Capacity

Heat capacity is a fundamental concept in calorimetry, and it refers to the amount of heat needed to raise the temperature of a system by one degree Celsius or Kelvin. In any calorimetric experiment, understanding heat capacity is crucial to determining how much energy a substance releases or absorbs.

• Heat capacity is typically denoted by the symbol \( C \).
• In this context, heat capacity refers to the calorimeter's ability to absorb heat during the reactions.

For instance, if a system has a high heat capacity, it will require more heat to increase its temperature compared to a system with a low heat capacity. In a bomb calorimeter, the heat absorbed is used to calculate the energy produced in combustion reactions.
Once you know the heat capacity of the calorimeter, you can use it to estimate the energy from other reactions by observing the temperature changes they cause.

Bomb Calorimeter

The bomb calorimeter is a specialized device used for measuring the heat of combustion of a substance. It is so named because the sample is placed in a sealed "bomb," which is then filled with oxygen to ensure complete combustion. Here are some of its key features:

• It's insulated to prevent heat loss to the surroundings, ensuring accurate measurements.
• The calorimeter measures changes in water temperature, from which the energy of combustion can be inferred.

Using a bomb calorimeter, scientists can measure the energy change of combustion processes very precisely. When a substance like a banana slice is combusted in a bomb calorimeter, the resulting temperature change can be used to calculate the heat released, helping us understand the energy content of the food.

Energy Conversion

Energy conversion involves changing energy from one form to another. In calorimetry, we often convert chemical energy into thermal energy, which is then measured. For instance:

• Combustion in a calorimeter converts the chemical energy in food to thermal energy.
• The amount of thermal energy generated is calculated using the calorimeter's temperature change.

In our context, energy conversion helps translate the heat of combustion into comprehensible energy values, like kilojoules and calories. We use conversion factors to express energy in different units: 1 calorie equals 4.18 kJ. This allows for easy comparison of energy content across various studies and applications.

Combustion

Combustion refers to the chemical reaction where a fuel reacts with oxygen to produce heat and light. In calorimetry, it is the basis of measuring the energy content of entities such as food. During the process:

• Substances react with oxygen to release energy.
• The energy produced is often measured in a calorimeter by the resulting temperature increase.

In our exercise, the combustion of a banana slice in a bomb calorimeter allows us to understand how much energy a banana can provide. By calculating the heat released from a known amount of substance, we can determine the energy content on a larger scale, such as that of a whole banana. Understanding combustion is pivotal in fields like nutrition and energy engineering, where calorific values determine dietary and fuel efficiencies.