Question

Perform the indicated conversions. a. 91.74 kcal into calories b. \(1.781 \mathrm{~kJ}\) into calories c. \(4.318 \times 10^{3}\) J into kilocalories d. \(9.173 \times 10^{4}\) cal into kilojoules

Short answer

a. 91.74 kcal = 91740 cal b. 1.781 kJ = 425.311 cal c. \(4.318 \times 10^{3}\) J = 1.031291 kcal d. \(9.173 \times 10^{4}\) cal = 383.786 kJ

Step-by-step solution

Using the conversion between kcal and cal

Using the relationship between kilocalories and calories, we have 1 kcal = 1000 cal. So, to convert 91.74 kcal to calories, we can multiply it by 1000: \(91.74\ \mathrm{kcal} \times 1000\ \frac{\mathrm{cal}}{\mathrm{kcal}} = 91740\ \mathrm{cal}\) b. 1.781 kJ into calories

Using the conversion between kJ and kcal

Using the relationship between kilojoules and kilocalories, we have 1 kJ = 0.239006 kcal. So, to convert 1.781 kJ to kilocalories, we can multiply it by 0.239006: \(1.781\ \mathrm{kJ} \times 0.239006\ \frac{\mathrm{kcal}}{\mathrm{kJ}} = 0.425311 \ \mathrm{kcal}\)

Converting kcal to cal

Now that we have found the equivalent value in kilocalories, we can convert it to calories using the relationship between kilocalories and calories: \(0.425311\ \mathrm{kcal} \times 1000\ \frac{\mathrm{cal}}{\mathrm{kcal}} = 425.311\ \mathrm{cal}\) c. \(4.318 \times 10^3\) J into kilocalories

Using the conversion between J and cal

Using the relationship between Joules and calories, we have 1 J = 0.238846 cal. So, to convert \(4.318 \times 10^3\ \mathrm{J}\) to calories, we can multiply it by 0.238846: \(4.318 \times 10^3\ \mathrm{J} \times 0.238846\ \frac{\mathrm{cal}}{\mathrm{J}} = 1031.291\ \mathrm{cal}\)

Converting cal to kcal

Now that we have found the equivalent value in calories, we can convert it to kilocalories using the relationship between calories and kilocalories: \(1031.291\ \mathrm{cal} \times \frac{1\ \mathrm{kcal}}{1000\ \mathrm{cal}} = 1.031291\ \mathrm{kcal}\) d. \(9.173 \times 10^4\) cal into kilojoules

Converting cal to kcal

First, we need to convert the given calories to kilocalories using the relationship between calories and kilocalories: \((9.173 \times 10^4)\ \mathrm{cal} \times \frac{1\ \mathrm{kcal}}{1000\ \mathrm{cal}} = 91.73\ \mathrm{kcal}\)

Using the conversion between kcal and kJ

Now that we have found the equivalent value in kilocalories, we can convert it to kilojoules using the relationship between kilocalories and kilojoules: \(91.73\ \mathrm{kcal} \times 4.184\ \frac{\mathrm{kJ}}{\mathrm{kcal}} = 383.786 \ \mathrm{kJ}\)

Key concepts

Converting Calories to Kilocalories

Understanding how to convert calories to kilocalories helps you work with both small and large units of energy. The fundamental conversion rule is that 1 kilocalorie (kcal) is equal to 1000 calories (cal). Therefore, whenever you need to convert kilocalories to calories, you simply multiply the number of kilocalories by 1000. This is useful because kilocalories are commonly used in dietary and nutritional contexts, while scientific calculations may often prefer calories due to their larger quantities.

For example, let’s say you have a dietary label stating 7 kcal; converting this to calories involves multiplying by 1000:

• 7 kcal \( \times \) 1000 \( = 7000 \mathrm{cal} \)

This process helps ensure that calculations conform to different measurement needs without losing accuracy.

Converting Kilojoules to Calories

Kilojoules and calories both measure energy, but they are used in distinct contexts — kilojoules in scientific and technical environments, while calories are more common in food energy discussions. To convert kilojoules to calories, the relationship you need is that 1 kilojoule (kJ) is approximately equal to 0.239006 kilocalories. Thus, first convert kilojoules to kilocalories by multiplying with 0.239006 and then proceed to get calories by multiplying further with 1000.

For example, converting 1.5 kJ to calories involves these steps:

• First, convert kJ to kcal: \( 1.5 \mathrm{kJ} \times 0.239006 \approx 0.358509 \mathrm{kcal} \)
• Then, convert kcal to cal: \( 0.358509 \mathrm{kcal} \times 1000 \approx 358.509 \mathrm{cal} \)

This clear sequence of steps helps you seamlessly manage conversions between these two units.

Converting Joules to Kilocalories

When you need to translate joules, the international unit of energy, into kilocalories, which are often used in nutritional contexts, the key is to first convert joules to calories, then to kilocalories. The conversion rate here is that 1 joule (J) equals approximately 0.238846 calories. After obtaining the calorie equivalent, divide by 1000 to find the kilocalories.

Here’s how you do it if you have 5000 J:

• Convert J to cal: \( 5000 \mathrm{J} \times 0.238846 \approx 1194.23 \mathrm{cal} \)
• Convert cal to kcal: \( 1194.23 \mathrm{cal} / 1000 \approx 1.19423 \mathrm{kcal} \)

Such conversions are essential when transitioning between scientific studies and everyday nutritional information.

Converting Calories to Kilojoules

In many regions, particularly outside the United States, energy content in foods is often indicated in kilojoules rather than calories. To convert calories to kilojoules, you should first convert calories to kilocalories by dividing by 1000, and then multiply by 4.184, since 1 kilocalorie is equivalent to 4.184 kilojoules.

Suppose you want to convert 2000 cal to kilojoules, the steps would be:

• Convert cal to kcal: \( 2000 \mathrm{cal} / 1000 = 2 \mathrm{kcal} \)
• Convert kcal to kJ: \( 2 \mathrm{kcal} \times 4.184 \approx 8.368 \mathrm{kJ} \)

Mastering these conversions enables accurate energy measurements that are usable in both dietary information and scientific assessments.