Problem 16 - 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

There are \(3.6 \times 10^6\) joules in a kilowatt-hour. The total amount of energy in kcal radiated by an adult in 24 hours is approximately 2064 kcal.

Step 1: Finding the number of joules in a kilowatt-hour

First, we need to convert kilowatt-hour into joules. We know that 1 watt is equal to 1 J/s. Therefore, 1 kilowatt is equal to 1000 J/s. So, to find the energy in joules during an hour, we have: Energy = Power × Time Energy = (1000 J/s) × (1 hour) Since 1 hour = 3600 seconds, we have: Energy = (1000 J/s) × (3600 s)

Step 2: Calculate the joules in a kilowatt-hour

Multiply the values to find the energy in joules: Energy = 1000 J/s × 3600 s Energy = \(3.6 \times 10^6\) J There are \(3.6 \times 10^6\) joules in a kilowatt-hour.

Step 3: Calculate the energy radiated by an adult in 24 hours

Now, we need to calculate the total amount of energy radiated by an adult in 24 hours. We know that they radiate heat at the same rate as a 100-watt electric incandescent lightbulb. Therefore, we can write: Power = 100 W = 100 J/s We need to calculate the energy in joules radiated by an adult in 24 hours: Energy = Power × Time Energy = (100 J/s) × (24 hours) Since 1 hour = 3600 seconds, we have: Energy = (100 J/s) × (24 × 3600 s)

Step 4: Calculate the energy in joules in 24 hours

Multiply the values to calculate the total energy in joules: Energy = 100 J/s × 24 × 3600 s Energy = \(8.64 \times 10^6\) J

Step 5: Convert the energy into kilocalories

Now, we need to convert the energy from joules to kilocalories (kcal). We know that 1 kcal is equal to 4184 J. So, we can write: Total Energy = \(8.64 \times 10^6\) J × (1 kcal / 4184 J)

Step 6: Calculate the total energy in kcal

Divide the values to find the total energy in kilocalories: Total Energy = \(8.64 \times 10^6\) J / 4184 J/kcal Total Energy ≈ 2064 kcal The total amount of energy in kcal radiated by an adult in 24 hours is approximately 2064 kcal.