Problem 4 - Chemistry and Chemical Reactivity

Question

A 2 -in. piece of chocolate cake with frosting provides \(1670 \mathrm{kJ}\) of energy. What is this in dietary Calories (Cal)?

Step-by-Step Solution

Answer

Approximately 399 Cal.

Step 1: Understand the Conversion Factor

Recognize that the conversion factor between kilojoules and dietary Calories is \( 1 \text{ Cal} = 4.184 \text{ kJ} \). This means that in order to convert energy in kilojoules to dietary Calories, you must divide the kilojoules by 4.184.

Step 2: Perform the Calculation

Take the total energy in kilojoules provided, which is 1670 kJ, and divide it by the conversion factor 4.184 kJ/Cal. \[\text{Dietary Calories} = \frac{1670 \text{ kJ}}{4.184 \text{ kJ/Cal}} = 399.14 \text{ Cal}.\]

Step 3: Round the Answer

Round the calculated dietary Calories to an appropriate number of significant figures. Since 1670 has three significant figures, round 399.14 to three significant figures as well.\[399.14 \approx 399 \text{ Cal}.\]

Key Concepts

Dietary Calories

Dietary Calories, often symbolized as Cal (with an uppercase 'C'), are a measure of energy that we obtain from foods such as chocolate cake. When you look at a food label, the energy content is usually given in Calories. In scientific terms, 1 dietary Calorie is equivalent to 4.184 kilojoules (kJ). This conversion is vital when calculating how much energy a particular food item contains in different units.
You might also hear the term "kilocalories" being used interchangeably with Calories. That's because 1 Calorie is technically 1 kilocalorie (1000 calories with a lowercase 'c').
This energy conversion helps in balancing dietary intake with the body’s energy needs and understanding how much energy comes from various foods. Remember, knowing how to convert between units like kilojoules and Calories is key when evaluating nutritional content.

Significant Figures

Significant figures are a crucial concept in chemistry and physics that ensure precision in our calculations. They indicate the certainty or reliability of a measurement.
In the exercise, the energy provided by the chocolate cake was given as 1670 kJ. This figure has three significant numbers (1, 6, and 7).
When performing calculations, it is important to maintain the same level of precision. In this case, we calculated the dietary Calories as 399.14, but because the original kilojoules were presented with three significant figures, the final answer must also adhere to this precision, becoming 399 Cal.

• This precision ensures that we don't imply more accuracy than our measurements allow.
• Always round your final answer to match the number of significant figures of your initial data.

Kilojoules

Kilojoules (kJ) are a unit of energy commonly used outside of the United States to quantify the energy content of foods. In the metric system, joules are the standard unit for energy, and a kilojoule is simply a multiple of a thousand joules.
When converting other units of energy like Calories, kilojoules provide a more scientific metric. A single dietary Calorie converts to exactly 4.184 kJ, which can be helpful in various calculations involving energy.
In nutrition, understanding kilojoules helps in comparing the energy content of foods when labeled in different regions. Converting between kilojoules and Calories often requires dividing or multiplying by this conversion factor, making it essential for everyone calculating energy intake and expenditure in food science and dietetics.