Problem 18 - Chemistry The Central Science

Question

A watt is a measure of power (the rate of energy change) equal to \(1 \mathrm{~J} / \mathrm{s}\). (a) Calculate the number of joules in a kilowatt- hour. (b) An adult person radiates heat to the surroundings at about the same rate as a 100 -watt electric incandescent lightbulb. What is the total amount of energy in kcal radiated to the surroundings by an adult in 24 hours?

Step-by-Step Solution

Answer

(a) There are 3.6 million joules in a kilowatt-hour. (b) An adult person radiates approximately 2064.81 kilocalories of energy to the surroundings in a 24-hour period.

Step 1: Convert kilowatt-hours to watt-seconds

To convert kilowatt-hours to watt-seconds, we first need to convert kilowatts to watts. There are 1000 watts in 1 kilowatt, and there are 3600 seconds in 1 hour. So, 1 kilowatt-hour is equal to 1000 watts x 3600 seconds. 1 kilowatt-hour = 1000 watts × 3600 seconds

Step 2: Convert watt-seconds to joules

Given that 1 watt is equal to 1 joule/second, we can use this relationship to convert watt-seconds to joules. Since 1 watt = 1 joule/second, 1 watt-second = 1 joule. 1 kilowatt-hour = 1000 watts × 3600 seconds × 1 joule/watt-second

Step 3: Calculate the number of joules

We can now calculate the number of joules in a kilowatt-hour. 1 kilowatt-hour = 1000 × 3600 joules 1 kilowatt-hour = 3,600,000 joules So, there are 3.6 million joules in a kilowatt-hour. #b# Part (b): Calculate Energy in kcal radiated by an Adult in 24 Hours

Step 1: Calculate energy in joules

The problem states that an adult person radiates heat at the rate of a 100-watt lightbulb. We are asked to calculate the total energy radiated in a 24-hour period. First, let's find out how many joules of energy are radiated in 24 hours: 100 watts × 24 hours × 3600 seconds/hour = 8,640,000 joules

Step 2: Convert joules to kilocalories

Now, we need to convert joules to kilocalories. 1 kilocalorie is equal to 4184 joules. Therefore, we can use this conversion factor to calculate the energy in kilocalories: 8,640,000 joules × \( \frac{1 kcal}{4184 J} \)

Step 3: Calculate the total energy in kcal

Finally, we can calculate the total energy in kilocalories: \( \frac{8,640,000}{4184} \) kcal ≈ 2064.81 kcal So, an adult person radiates approximately 2064.81 kilocalories of energy to the surroundings in a 24-hour period.

Key Concepts

Joules to Kilowatt-Hours

Energy can be measured in various units, one of which is joules. When trying to understand larger scales of energy, such as those used typically in household power consumption, kilowatt-hours (kWh) are often used. To convert from joules to kilowatt-hours, we need to know that 1 kilowatt-hour equals 3.6 million joules. This comes from the fact that 1 kilowatt is 1,000 watts, and there are 3,600 seconds in one hour. So:

• 1 kilowatt-hour = 1,000 watts × 3,600 seconds
• = 3,600,000 joules

This conversion is crucial in understanding how energy used in everyday appliances translates into everyday activities, especially when calculating electricity usage on a monthly or yearly basis.

Power Measurement

Power is the rate at which energy is used or transferred, often measured in watts. A watt is equal to one joule per second (J/s). Understanding power is essential for calculating energy consumption over time, which is calculated by multiplying the power rating (wattage) by time. For example, if an appliance has a power rating of 100 watts and it runs for 24 hours, you can calculate the total energy used by:

• 100 watts × 24 hours = 2,400 watt-hours
• = 2,400 × 3,600 seconds = 8,640,000 joules

This method of calculation can also be applied to understanding how much energy a person radiates as heat, which we'll explore in the next section.

Heat Radiation by Humans

Humans naturally radiate heat to their surroundings, similar to a lightbulb distributing light and heat. An adult person usually radiates heat at a rate comparable to 100 watts. Over a span of 24 hours, the total energy radiated by a person can be calculated.

• 100 watts × 24 hours = 2,400 watt-hours
• = 8,640,000 joules (since 1 watt = 1 joule/second)

To further understand this energy in terms of dietary energy, joules are often converted into kilocalories (kcal). Since 1 kcal = 4,184 joules, this helps in understanding energy expenditure in human terms. Converting 8,640,000 joules to kcal, we get approximately 2,064.81 kcal. This approximation tells us how much energy is used by the body in a typical day, which is vital for considerations in nutrition and daily calorie intake.