Problem 42 - Chemistry The Central Science

Question

A watt is a measure of power (the rate of energy change) equal to 1 \(\mathrm{Js.}(\mathbf{a})\) 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 3,600,000 joules in a kilowatt-hour. (b) An adult person radiates approximately 2064 kilocalories of energy to the surroundings over a 24-hour period.

Step 1: (a) Convert kilowatt-hour to joules

First, we need to understand the relationships between different units of measurement. A watt (W) is equal to 1 joule per second (J/s), and a kilowatt (kW) is equal to 1000 watts. An hour (h) is equal to 3600 seconds (s). So, we need to convert from kilowatt-hours (kWh) to joules (J) using these relationships. 1 kWh = 1 kW * 1 h Now, we can convert kilowatts to watts and hours to seconds: 1 kWh = (1 * 1000) W * (1 * 3600) s 1 kWh = 1000 J/s * 3600 s The seconds (s) cancel out: 1 kWh = 1000 J * 3600 1 kWh = 3,600,000 J So, there are 3,600,000 joules in a kilowatt-hour.

Step 2: (b) Calculate energy radiated by an adult in 24 hours

We are given that an adult person radiates heat at the same rate as a 100-watt light bulb. We need to find the total amount of energy in kilocalories (kcal) radiated by an adult over a 24-hour period. First, let's find the total amount of energy in joules. We know that 1 watt is equal to 1 Joule/second, so: 100 W * 1 J/s = 100 J/s Now, we will multiply the rate of energy (100 J/s) by the total time in seconds (24 hours): 100 J/s * 24 h * 3600 s/h = 8,640,000 J Now, we need to convert this energy in joules to kilocalories. We know that 1 kcal is equal to 4,184 J, so: 8,640,000 J * (1 kcal / 4,184 J) ≈ 2064 kcal So, an adult person radiates approximately 2064 kilocalories of energy to the surroundings over a 24-hour period.

Key Concepts

Power and Energy Units

Understanding the basics of power and energy units is essential in the realm of physics, as well as in everyday life when dealing with electricity bills and appliance usage. Power is the rate at which energy is transferred or converted and it's measured in watts (W). One watt is defined as the transfer of one joule of energy per second. But what are joules? A joule (J) is a unit of energy in the International System of Units (SI). It's the energy transferred to an object when a force of one newton acts on that object in the direction of the force's motion over a distance of one meter.

This may sound technical, but in simpler terms, if you used a one-watt appliance for one second, it would use one joule of energy. Larger energy consumptions are often measured in kilowatt-hours (kWh), which is what electric companies use to bill households. One kilowatt is equivalent to 1,000 watts, and when this power is used for an hour, it comprises a kilowatt-hour. Understanding how to convert between these units is vital for calculations regarding energy consumption and efficiency.

Kilowatt-hour to Joules Conversion

To carry out energy conversion calculations like converting a kilowatt-hour to joules, one must grasp the relationship between different units of power and energy. As reviewed, a kilowatt (kW) is 1,000 watts (W), and a watt is one joule per second (J/s). An hour contains 3,600 seconds, so when we want to convert kilowatt-hours to joules, we multiply the number of kilowatts by the number of seconds in an hour.

Using the equation \(1\ \text{kWh} = 1\ \text{kW} \times 1\ \text{h}\), and converting kilowatts to watts and hours to seconds, we get \(1\ \text{kWh} = 1,000\ \text{W} \times 3,600\ \text{s}\), simplifying to \(1\ \text{kWh} = 3,600,000\ \text{J}\). Therefore, one kilowatt-hour equals 3.6 million joules. This conversion is particularly useful when translating electrical energy usage into a more tangible concept of energy, as seen in physics problems and everyday applications like understanding your energy bill.

Human Body Heat Radiation

The human body is like a power plant that continuously radiates heat, much like how a light bulb emits light. This concept of human body heat radiation can be illustrated by comparing it to an incandescent light bulb's power usage. A standard 100-watt bulb, which was once common in households, radiates energy at a rate of 100 joules per second, as does the human body at rest.

To calculate the total energy radiated by an adult in a day, we can use the power emitted (100 watts) multiplied by the number of seconds in 24 hours (\(24 \times 3,600\)). This translates to a total of 8,640,000 joules. Since energy is not always measured in joules, converting it to kilocalories – the unit often used in dietary contexts – provides a different perspective. With 1 kcal equal to 4,184 joules, the total heat radiated by a human in 24 hours is approximately 2064 kcal. Understanding this energy output is not only intriguing from a scientific standpoint but also provides insight into the metabolic and physiological processes of the human body.