Question

Convert the following numbers of calories or kilocalories into joules and kilojoules. (Remember: kilo means 1000.) a. 75.2 kcal b. 75.2 cal c. \(1.41 \times 10^{3}\) cal d. \(1.41 \mathrm{kcal}\).

Short answer

a. 75.2 kcal = 314796.8 J = 314.7968 kJ b. 75.2 cal = 314.7648 J c. \(1.41 \times 10^{3}\) cal = \(5.8996 \times 10^{3}\) J = 5.8996 kJ d. 1.41 kcal = 5899.44 J = 5.89944 kJ

Step-by-step solution

Convert kcal to cal (if necessary)

For a. and d. the given numbers are in kcal, so first, we need to convert them to cal. To do so, we use the following conversion: 1 kcal = 1000 cal. a. 75.2 kcal 75.2 kcal * 1000 cal/kcal = 75200 cal d. 1.41 kcal 1.41 kcal * 1000 cal/kcal = 1410 cal Now we have all values in cal: - For a: 75200 cal - For b: 75.2 cal - For c: \(1.41 \times 10^{3}\) cal - For d: 1410 cal

Convert cal to J

Now, we will convert the numbers from calories to joules using the conversion 1 cal = 4.184 J. a. 75200 cal 75200 cal * 4.184 J/cal = 314796.8 J b. 75.2 cal 75.2 cal * 4.184 J/cal = 314.7648 J c. \(1.41 \times 10^{3}\) cal \(1.41 \times 10^{3}\) cal * 4.184 J/cal = \(5.8996 \times 10^{3}\) J d. 1410 cal 1410 cal * 4.184 J/cal = 5899.44 J Now we have all values in Joules(J): - For a: 314796.8 J - For b: 314.7648 J - For c: \(5.8996 \times 10^{3}\) J - For d: 5899.44 J

Convert J to kJ (if necessary)

Lastly, we will convert the Joules to kilojoules if the value is larger than 1000 J. To do so, we use the following conversion: 1 kJ = 1000 J. a. 314796.8 J 314796.8 J / 1000 J/kJ = 314.7968 kJ c. \(5.8996 \times 10^{3}\) J \(5.8996 \times 10^{3}\) J / 1000 J/kJ = 5.8996 kJ d. 5899.44 J 5899.44 J / 1000 J/kJ = 5.89944 kJ

Final answer

After all the conversions, we get the following results: a. 75.2 kcal = 314796.8 J = 314.7968 kJ b. 75.2 cal = 314.7648 J (no need to convert to kJ because the value is less than 1000 J) c. \(1.41 \times 10^{3}\) cal = \(5.8996 \times 10^{3}\) J = 5.8996 kJ d. 1.41 kcal = 5899.44 J = 5.89944 kJ

Key concepts

Calories to Joules

When converting calories to joules, it's important to remember that energy is a physical quantity that can be measured in different units. In the scientific community, joules (J) are commonly used to measure energy. However, in the context of food and nutrition, calories (cal) are often useful as they relate to energy content.

To convert from calories to joules, you will use the following conversion factor: 1 calorie is equivalent to 4.184 joules. This means if you want to convert 75.2 cal to joules, you'll compute:

• 75.2 cal * 4.184 J/cal = 314.7648 J

This straightforward multiplication allows you to easily determine how much energy something has in joules when initially measured in calories. Remember to keep track of your units. Hint: Always cancel your units to make sure you used the correct conversion factor!

Joules to Kilojoules

Sometimes, the amounts of energy you're dealing with will be very large. In such cases, expressing energy in joules might become cumbersome. You'll want to convert to kilojoules (kJ), where 1 kJ = 1000 J. This makes the numbers more manageable and easier to interpret.

For example, let’s say you have 314796.8 J from a previous calculation. Converting to kilojoules involves dividing by 1000:

• 314796.8 J / 1000 J/kJ = 314.7968 kJ

This method keeps your calculations tidy and understandable, transforming large numbers into more convenient units of kilojoules. Keep in mind that using kilojoules might not be necessary unless the value of joules exceeds 1000.

Kilocalories to Calories

In certain contexts, like nutrition, energy might be measured in kilocalories (kcal) instead of calories. A kilocalorie is equivalent to 1000 calories. This can often cause confusion, but the conversion is straightforward.

For instance, if you start off with 75.2 kcal, converting to calories is a simple multiplication:

• 75.2 kcal * 1000 cal/kcal = 75200 cal

This conversion is vital because many scientific principles and labs work with energy in calories, not kilocalories. Being comfortable with converting between these units is crucial for accurate scientific measurements and calculations.

Chemistry Problem Solving

When working with chemistry problems involving unit conversions, it's essential to apply systematic problem-solving approaches. Conversion factors are your best friends.

Follow these general steps:

• Identify what you’re asked to convert and the units you need.
• Use known conversion factors, like 1 cal = 4.184 J or 1 kcal = 1000 cal.
• Double-check unit cancellation to ensure the conversion is correct.
• Simplify your result, converting to kJ if the energy value in J is more than 1000.

This organized method ensures accurate solutions and helps avoid confusion. Practice these conversions frequently, as they're fundamental elements in chemistry, physics, and daily life applications.