Problem 42 - Chemistry: The Central Science in SI Units, Global Edition

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 light bulb. What is the total amount of energy in kcal radiated to the surroundings by an adult over a 24 h period?

Step-by-Step Solution

Answer

(a) There are \(3.6*10^6\) Joules in a kilowatt-hour. (b) An adult radiates approximately 2063 kcal of energy over a 24-hour period.

Step 1: Convert kilowatt-hour to Joules

To calculate the number of Joules in a kilowatt-hour, first, we need to convert kilowatts to watts and hours to seconds. 1 kilowatt-hour (kW-h) = 1000 watts * 3600 seconds Now we can multiply the total power by the total time to find the energy in Joules. Energy = Power × Time

Step 2: Calculate the energy in Joules

Now we can calculate the energy in Joules by using the formula: Energy = Power × Time Energy = 1000 W × 3600 s = 3,600,000 Joules So there are \(3.6*10^6\) Joules in a kilowatt-hour.

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

Now let's calculate the energy radiated by an adult over a 24 hour period. First, we will find the energy in Joules and then convert it into kilocalories (kcal). Since an adult radiates heat at the same rate as a 100-watt light bulb, we can use the power and multiply it with the 24-hour time period. Energy (Joules) = Power × Time

Step 4: Calculate the energy in Joules radiated by an adult over 24 hours

Now we need to calculate the energy in Joules over a 24-hour period. First, we will convert the 24 hours into seconds. 24 hours = 86400 seconds Energy (Joules) = Power × Time = 100 W × 86400 s = 8,640,000 Joules

Step 5: Convert energy in Joules to kilocalories (kcal)

Now we need to convert the energy from Joules to kcal using the conversion factor: 1 kcal = 4184 J Energy (kcal) = 8,640,000 Joules / 4184 Joules/kcal ≈ 2063 kcal So, an adult radiates approximately 2063 kcal of energy over a 24-hour period.

Key Concepts

Joules

The Joule is a unit of energy in the International System of Units (SI). It is defined as the amount of energy transferred when a force of one newton acts over a distance of one meter. When dealing with electrical energy, we often equate Joules to energy used per second when power is measured in watts.

• 1 Joule equals the energy transferred when 1 watt of power is used for 1 second.
• It is a versatile unit applicable to various types of energy: electrical, mechanical, thermal, etc.

Understanding Joules helps in converting between different energy units, such as kilowatt-hours or kilocalories, by using conversion factors. For instance, a kilowatt-hour contains precisely 3,600,000 Joules, as calculated based on the time-power relationship. Knowing how to convert and calculate Joules is essential in physics and engineering, where energy efficiency and consumption data often require transformations into this common unit.

Kilowatt-hour

A kilowatt-hour (kWh) is a unit of energy often used to charge for electricity consumption. It represents the energy expended by using one kilowatt of power for one hour.

• To thoroughly understand, consider that 1 kWh = 1000 watts multiplied by 3600 seconds, resulting in 3,600,000 Joules.
• Utilities use this unit to measure the electrical energy delivered to homes and businesses.

This conversion is particularly useful when calculating the cost of energy consumption over longer periods.

For example, if you are using a 100-watt appliance for 10 hours, you would consume 1 kilowatt-hour of energy. This unit simplifies the billing and understanding process for everyday energy use. Understanding kilowatt-hours is crucial in recognizing your energy consumption patterns and identifying ways to improve energy efficiency.

Kilocalories

Kilocalories (kcal) are a unit of energy often used in nutrition. This unit measures the energy content in food or the energy expenditure of physical activities.

• 1 kilocalorie is the energy needed to raise the temperature of one kilogram of water by 1°C.
• When discussing energy conversion, 1 kilocalorie equals 4184 Joules.

In the context of energy conversion across different units, we find that humans radiate energy in kilocalories similar to how appliances consume energy in kilowatt-hours.

In our example, an adult radiates approximately 2063 kcal over a 24-hour period, equivalent to the energy expenditure of various activities. This reflects the human body's metabolism and underscores how caloric consumption and expenditure are critical in managing energy balance within the body.

Power

Power is a measure of the rate at which energy is used or transferred. It is expressed in watts, where 1 watt equals 1 Joule per second.

• Power determines how quickly energy is transformed or consumed at any given moment.
• It is vital in both electrical systems and mechanical processes.

When you analyze everyday appliances or systems, their power ratings tell you how much energy they use per second. A 100-watt light bulb, for instance, uses 100 Joules of energy every second of operation. Calculating an adult's energy radiation using the concept of power over a 24-hour period allows us to visualize how much energy is naturally expended by the human body.

Understanding power is critical for energy management, optimizing appliance efficiency, and solving physics-related problems.