The ABC cereal company is developing a new type of breakfast cereal to compete with a rival product that they call Brand X. You are asked to compare the energy content of the two cereals to see if the new \(A B C\) product is lower in calories; so you burn \(1.00-g\) samples of the cereals in oxygen in a calorimeter with a heat capacity of \(600 . \mathrm{J} \cdot\left({ }^{\circ} \mathrm{C}\right)^{-1}\). When the Brand \(\mathrm{X}\) cereal sample burned, the temperature rose from \(300.2 \mathrm{~K}\) to \(309.0 \mathrm{~K}\). When the ABC cereal sample burned, the temperature rose from \(299.0 \mathrm{~K}\) to \(307.5 \mathrm{~K}\). (a) What is the heat output of each sample? (b) One serving of each cereal is \(30.0 \mathrm{~g}\). How would you label the packages of the two cereals to indicate the fuel value per \(30.0-\mathrm{g}\) serving in joules? in nutritional Calories (kilocalories)?

Step by step solution

01

- Find the Heat Output for Brand X

To find the heat output, use the formula: Heat Output (q) = Heat Capacity (C) × Change in Temperature (ΔT). For Brand X, ΔT is the temperature rise from 300.2 K to 309.0 K, which is 309.0 K - 300.2 K = 8.8 K. The heat capacity is given as 600 J/°C (or K). Therefore, the heat output for Brand X is calculated by multiplying the heat capacity by ΔT, which gives q = 600 J/K × 8.8 K.

02

- Find the Heat Output for ABC Cereal

Using the same formula: Heat Output (q) = Heat Capacity (C) × Change in Temperature (ΔT), and for ABC cereal, ΔT is 307.5 K - 299.0 K = 8.5 K. The heat capacity remains the same at 600 J/K, so the heat output for ABC cereal is q = 600 J/K × 8.5 K.

03

- Calculate the Heat Output per Serving for Brand X

First, calculate the heat output for 1.00 g of Brand X from Step 1. To find the energy content per 30.0 g serving, multiply the heat output for 1.00 g by 30. This results in 30 times the heat output for the 1.00 g sample.

04

- Calculate the Heat Output per Serving for ABC Cereal

Similarly, calculate the energy content per 30.0 g serving for ABC cereal. Multiply the heat output for 1.00 g of ABC cereal found in Step 2 by 30.

05

- Convert Joules to Nutritional Calories

1 nutritional Calorie (kcal) is equivalent to 4184 Joules. To convert the heat output per serving from Joules to nutritional Calories, divide the energy content per 30.0 g serving (found in Steps 3 and 4) by 4184 J/Cal.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Heat Capacity in Calorimetry

When we talk about heat capacity in the context of calorimetry, we're referring to the amount of heat energy required to raise the temperature of a substance by one degree Celsius (or one Kelvin, as the size of the degree is the same in both scales). In the provided exercise, a calorimeter with a heat capacity of 600 J/°C is used to measure the energy content of cereal samples. It means that for each degree of temperature increased in the calorimeter, 600 joules of heat energy have been absorbed.

For accurate calorimetry, it's important to know this value because it allows us to calculate the total heat released during a reaction or a process, such as the burning of cereal. In a practical sense, this information helps companies compare the energy content of their food products to ensure each serving meets certain nutritional values or marketing claims.

Energy Content of Food

The energy content of food is the amount of energy, usually measured in calories or joules, provided by consuming it. This is a crucial piece of information as it helps consumers to balance their energy intake with their energy expenditure. In the exercise, burning the cereals in a calorimeter allows the determination of the heat output, which is directly related to the food’s energy content. A 1.00-g sample is used to scale up to a typical serving size, in this case, 30.0 grams.

A lower heat output indicates fewer calories, which might be marketed as a 'low-calorie' option. The food industry uses this information to guide product development and labelling, ensuring that consumers have easy access to the data they need to make informed dietary choices.

Thermochemistry

Thermochemistry is the study of the heat energy associated with chemical reactions and physical changes. In the context of our exercise, thermochemistry relates to the process of burning the cereals, since combustion is a chemical reaction that releases energy in the form of heat. By employing a calorimeter, scientists can measure this energy release and gain insights into the thermochemical properties of the food item.

Thermochemistry principles allow us to calculate the heat output as shown in the steps provided, which is integral for understanding the energy transfer during the reaction. Understanding this heat exchange is crucial for various applications, including nutritional labelling, product development, and even environmental impact assessments of food production.

Converting Joules to Calories

The conversion of joules to calories is essential for interpreting energy content in a way that's more familiar to consumers, who typically think of food energy in terms of calories. In the exercise, we observe the conversion from joules to nutritional calories by dividing the energy content per 30.0 g serving by 4184 J/Cal. This is based on the definition of the nutritional Calorie or kilocalorie, which is equal to 1000 small calories or 4184 joules.

This conversion is important because it standardizes energy content measurements, making it easier for individuals to track their energy intake according to dietary guidance. Understanding the relationship between joules and calories can also aid in comparing the energy provided by different foods and help in calculating daily energy requirements for maintaining, gaining, or losing weight.