Question

Convert the following numbers of joules/kilojoules into kilocalories. a. \(52.18 \mathrm{kJ}\) b. \(4.298 \mathrm{J}\) c. \(5.433 \times 10^{3} \mathrm{J}\) d. \(455.9 \mathrm{kJ}\)

Short answer

a. \(52.18 kJ = 12.47 kcal\) b. \(4.298 J = 0.001027 kcal\) c. \(5.433 \times 10^{3} J = 1.299 kcal\) d. \(455.9 kJ = 108.96 kcal\)

Step-by-step solution

Identify conversion factors

We need to use two conversion factors for this exercise: 1. 1 kcal = 4.184 kJ 2. 1 kJ = 10^3 J

Convert 52.18 kJ to kcal

For this, we will use the first conversion factor: 1 kcal = 4.184 kJ. \( 52.18 kJ \times \frac{1 kcal}{4.184 kJ} = 12.47 kcal \)

Convert 4.298 J to kcal

First, we need to convert J to kJ using the second conversion factor: 1 kJ = 10^3 J. \( 4.298J \times \frac{1 kJ}{10^3 J} = 0.004298 kJ \) Next, we'll convert kJ to kcal using the first conversion factor: 1 kcal = 4.184 kJ. \( 0.004298 kJ \times \frac{1 kcal}{4.184 kJ} = 0.001027 kcal \)

Convert \(5.433 \times 10^{3} J\) to kcal

First, convert J to kJ using the second conversion factor: 1 kJ = 10^3 J. \( 5.433 \times 10^{3} J \times \frac{1 kJ}{10^3 J} = 5.433 kJ \) Next, convert kJ to kcal using the first conversion factor: 1 kcal = 4.184 kJ. \(5.433 kJ \times \frac{1 kcal}{4.184 kJ} = 1.299 kcal \)

Convert 455.9 kJ to kcal

For this, we will use the first conversion factor: 1 kcal = 4.184 kJ. \( 455.9 kJ \times \frac{1 kcal}{4.184 kJ} = 108.96 kcal \)

Results

a. 52.18 kJ = 12.47 kcal b. 4.298 J = 0.001027 kcal c. \(5.433 \times 10^{3} J\) = 1.299 kcal d. 455.9 kJ = 108.96 kcal

Key concepts

Joules to Kilocalories Conversion

Understanding the conversion from joules to kilocalories is vital for students delving into the study of energy in chemistry. This entails grasping the definition of these units and how to translate them accurately. A joule (J) is the SI unit of energy and represents the amount of work done when a force of one newton acts over a distance of one meter. Meanwhile, a kilocalorie (kcal) is often used in discussing energy in nutritional contexts and represents the energy needed to increase the temperature of one kilogram of water by one degree Celsius.

The conversion factor between joules and kilocalories is based on the specific heat capacity of water, where 1 kcal is equal to 4,184 joules. To convert joules to kilocalories, you divide the number of joules by 4,184.
For example, to convert 4,298 J to kilocalories:
\[ 4,298 J \times \frac{1 kcal}{4,184 J} = 1.028 \times 10^{-3} kcal \] Simplicity is key for comprehension: remember that you're shifting from a smaller unit (joules) to a larger unit (kilocalories), which means the resultant number should be smaller than the original.

Energy Units in Chemistry

In chemistry, energy is a fundamental concept, often observed in different forms such as thermal, chemical, and electrical energy. It is essential to understand the units used to measure energy to grasp the principles governing chemical reactions and processes. The most common units for energy in chemistry are joules, calories, kilojoules, kilocalories, and the electronvolt (eV).

• Joule (J): The official SI unit for energy, used broadly in scientific contexts.
• Kilojoule (kJ): Equal to 1,000 joules, more convenient for larger amounts of energy.
• Calorie (cal): Historically used for energy in the context of foods. 1 cal is the energy needed to raise the temperature of 1 gram of water by 1 degree Celsius.
• Kilocalorie (kcal): Often referred to as 'Calorie' with a capital 'C' in dietary contexts and equals 1,000 calories.
• Electronvolt (eV): Utilized primarily in physics to describe energies on the atomic or subatomic scale, equal to the energy gained by a single electron when accelerated through a potential difference of one volt.

Converting between these units is critical for analytical calculations and understanding chemical energy exchanges. Using conversion factors like the ones provided in the exercise, students can easily switch between units to express energy in the most appropriate context.

Kilocalories to Kilojoules

When converting from kilocalories to kilojoules, it's important to understand that you're transitioning from one unit that represents larger amounts of energy (kcal) to another that's smaller (kJ), meaning the result will be a larger numerical value. As highlighted before, the conversion factor is that 1 kcal is equivalent to 4.184 kJ, stemming from the specific heat capacity of water.

Here's a simple formula example to convert 1.299 kcal to kilojoules: \[ 1.299 kcal \times 4.184 kJ/kcal = 5.433 kJ \] This basic arithmetic involves multiplication instead of division, as we are scaling up from kilocalories to kilojoules.
It is good practice to encourage students to acknowledge the direct proportional relationship between these units; as the number of kilocalories increases, so does the number of kilojoules. Having a firm grasp on these conversion techniques not only aids in solving textbook problems but also in real-world applications where energy measurements are essential.