Question

Convert the following numbers of calories or kilocalories into joules or kilojoules. a. 7845 cal b. \(4.55 \times 10^{4} \mathrm{cal}\) c. 62.142 kcal d. 43,024 cal

Short answer

The converted energy quantities are: a. 32,813.38 J b. \(1.90378 \times 10^{5} \mathrm{J}\) c. 259.998 kJ d. \(1.80044 \times 10^{5} \mathrm{J}\)

Step-by-step solution

Identify the conversion factors

We will use the following conversion factors: 1. 1 cal = 4.184 J 2. 1 kcal = 4.184 kJ

Convert 7845 cal to J

We will apply the conversion factor for cal to J: \( 7845 \mathrm{cal} \times \frac{4.184 \mathrm{J}}{1 \mathrm{cal}} = 32813.38 \mathrm{J} \)

Convert \(4.55 \times 10^{4} \mathrm{cal}\) to J

We will apply the conversion factor for cal to J: \( 4.55 \times 10^{4} \mathrm{cal} \times \frac{4.184 \mathrm{J}}{1 \mathrm{cal}} = 1.90378 \times 10^{5} \mathrm{J} \)

Convert 62.142 kcal to kJ

We will apply the conversion factor for kcal to kJ: \( 62.142 \mathrm{kcal} \times \frac{4.184 \mathrm{kJ}}{1 \mathrm{kcal}} = 259.998128 \mathrm{kJ} \)

Convert 43,024 cal to J

We will apply the conversion factor for cal to J: \( 43,024 \mathrm{cal} \times \frac{4.184 \mathrm{J}}{1 \mathrm{cal}} = 1.80044416 \times 10^{5} \mathrm{J} \) So, the converted energy quantities are: a. 32,813.38 J b. \(1.90378 \times 10^{5} \mathrm{J}\) c. 259.998 kJ d. \(1.80044 \times 10^{5} \mathrm{J}\)

Key concepts

Energy Conversion

Energy conversion is a fundamental concept in physical sciences that involves changing energy from one form to another. It is a key process that allows us to solve energy-related problems by using proper units and conversion methods. Whenever you are tasked with converting energy measurements, you need to familiarize yourself with specific conversion factors.

One of the most common conversions is from calories to joules. Calories are often used in food energy calculations, while joules are a more universally accepted unit in scientific calculations. To convert these, it's crucial to understand that:

• 1 calorie (cal) equals 4.184 joules (J).
• 1 kilocalorie (kcal) equals 4.184 kilojoules (kJ).

By using these conversion factors, you can interchangeably convert energy between different units to interpret or solve various energy problems more comprehensively.

Joules to Calories

Converting joules to calories is simpler than it might seem at first glance. Since 1 calorie is equivalent to 4.184 joules, to convert a measurement in joules to calories, you simply rearrange this relationship.

For example, let's say you have an energy value in joules, and you want to find out what that is in calories:

• Divide the joules by 4.184 to get the equivalent energy in calories.
  
  For example, if you have 10,000 J: \[ \frac{10000 \ J}{4.184 \ cal/J} = 2390.57 \ cal \]

Remember, calories are more often used in dietary contexts, while joules are preferred in physics and engineering, so conversion between these units might be frequent depending on the context.

Kilocalories to Kilojoules

Kilocalories to kilojoules is another important energy conversion that often comes into play, especially in nutritional science and other biological fields. Since 1 kilocalorie (commonly seen on food labels as "Calories") equals 4.184 kilojoules, you have a straightforward method for conversion.

To convert kilocalories to kilojoules:

• Simply multiply the number of kilocalories by 4.184.

For instance, imagine you have a meal that provides 500 kcal:

• The energy in kilojoules is: \[ 500 \ kcal \times 4.184 \frac{kJ}{kcal} = 2092 \ kJ \]

Understanding this conversion is useful for comparing energy information presented in nutritional contexts to that in more universal scientific contexts. This conversion is key for transporting data across different scientific disciplines.