Question

Perform the indicated conversions. a. \(89.74 \mathrm{kJ}\) into kilocalories b. \(1.756 \times 10^{4} \mathrm{J}\) into kilojoules c. \(1.756 \times 10^{4} \mathrm{J}\) into kilocalories d. \(1.00 \mathrm{kJ}\) into calories

Short answer

a. 89.74 kJ = 21.44 kcal b. \(1.756 \times 10^{4} \mathrm{J} = 17.56 \mathrm{kJ}\) c. \(1.756 \times 10^{4} \mathrm{J} = 4.2 \mathrm{kcal}\) d. 1.00 kJ = 239 cal

Step-by-step solution

a. Convert 89.74 kJ into kilocalories

To convert kJ to kcal, we will use the conversion factor: 1 kJ = 1/4.184 kcal We have 89.74 kJ, so to convert it to kcal, we will multiply it by the conversion factor: \(89.74 \, \mathrm{kJ} * \frac{1}{4.184} \, \frac{\mathrm{kcal}}{\mathrm{kJ}} = 21.44 \, \mathrm{kcal}\) So, 89.74 kJ is equal to 21.44 kcal.

b. Convert 1.756 x 10^4 J into kilojoules

To convert J to kJ, we will use the conversion factor: 1 J = 0.001 kJ We have \(1.756 \times 10^{4} \mathrm{J}\), so to convert it to kJ, we will multiply it by the conversion factor: \(1.756 \times 10^{4} \, \mathrm{J} * 0.001 \, \frac{\mathrm{kj}}{\mathrm{J}} = 17.56 \, \mathrm{kj}\) So, \(1.756 \times 10^{4} \, \mathrm{J}\) is equal to 17.56 kJ.

c. Convert 1.756 x 10^4 J into kilocalories

First, we need to convert J to kJ and then convert kJ to kcal. From previous step, we know that \(1.756 \times 10^{4} \, \mathrm{J} = 17.56 \, \mathrm{kj}\), now we will convert 17.56 kJ to kcal: \(17.56 \, \mathrm{kj} * \frac{1}{4.184} \, \frac{\mathrm{kcal}}{\mathrm{kJ}} = 4.2 \, \mathrm{kcal}\) So, \(1.756 \times 10^{4} \, \mathrm{J}\) is equal to 4.2 kcal.

d. Convert 1 kJ into calories

First, we have to convert kJ to kcal and then convert kcal to cal. 1 kJ = 1/4.184 kcal, so: \(1 \, \mathrm{kj} * \frac{1}{4.184} \, \frac{\mathrm{kcal}}{\mathrm{kJ}} = 0.239 \, \mathrm{kcal}\) Now, convert kcal to cal using 1 kcal = 1000 cal: \(0.239 \, \mathrm{kcal} * 1000 \, \frac{\mathrm{cal}}{\mathrm{kcal}} = 239 \, \mathrm{cal}\) So, 1 kJ is equal to 239 cal.

Key concepts

Kilojoules to Kilocalories

When it comes to converting energy values, kilojoules (kJ) and kilocalories (kcal) are two commonly used units.This conversion is particularly useful when comparing energy content in food labels.
To convert kilojoules to kilocalories, you need to know that:

• 1 kilojoule is equal to approximately 0.239 kilocalories.
• This conversion factor comes from the approximate equivalence: 1 kcal = 4.184 kJ.

To make a conversion from 89.74 kJ to kcal, multiply the number of kilojoules by the conversion factor:$$89.74 \, \text{kJ} \times \frac{1}{4.184} \, \text{\(\text{kcal} \)/\( \text{kJ} \)} = 21.44 \, \text{\(\text{kcal}\)}$$Thus, 89.74 kJ is equivalent to 21.44 kcal, which shows how this energy can be transformed into a more commonly recognized unit when thinking about calories.

Joules to Kilojoules

Joules and kilojoules are both units of energy, but kilojoules represent a larger scale.They are often used to describe larger amounts of energy.
To convert joules to kilojoules, consider:

• 1 joule (J) is equal to 0.001 kilojoules (kJ).
• This means you divide the amount in joules by 1000 to get kilojoules.

For example, if you have \(1.756 \times 10^{4} \) joules and want to convert it to kilojoules, you'd perform this calculation:\[1.756 \times 10^{4} \, \text{J} \times 0.001 \, \text{\(\text{kJ} \)/\(\text{J}\)} = 17.56 \, \text{kJ}\]This conversion helps in simplifying the expression of energy by using a unit that is better suited to larger quantities, making it easier for calculations and comparisons.

Energy Conversion Factors

Energy conversion factors are essential for translating energy expressions from one unit to another. They allow us to work with various units seamlessly. Here are some vital conversion factors and their uses:

• 1 kilojoule (kJ) = 1/4.184 kilocalories (kcal).
• 1 joule (J) = 0.001 kilojoules (kJ).
• 1 kilocalorie (kcal) = 1000 calories (cal).

These factors enable the conversion of energy sources from scientific units like joules, commonly used in physics, to nutritional units like calories and kilocalories, more familiar in everyday life, particularly in dieting and nutrition. For instance, to link kilojoules to calories, you must first go through kilocalories. This means conversion accuracy can be maintained across scientific studies, daily dietary calculations, and industries where energy measurement is crucial. By understanding and employing these conversion factors, you can better navigate tasks requiring unit conversions, such as converting your nutritional energy breakdowns from kilojoules to calories.