Problem 42 - Chemistry The Central Science In Si Units

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) 1 kWh = 3,600,000 J. (b) An adult radiates ~2064 kcal over 24 hours.

Step 1: Convert Kilowatt-Hour to Joules

A kilowatt-hour is a unit of energy. It represents the energy consumed by one kilowatt over the period of one hour. Since 1 watt is equivalent to 1 joule/second, we need to convert kilowatts to watts and hours to seconds. First, convert kilowatts to watts: \[ 1 \text{ kW} = 1000 \text{ W} \].Next, convert one hour to seconds:\[ 1 \text{ hour} = 3600 \text{ seconds} \].Now, calculate the joules in 1 kilowatt-hour:\[ 1000 \text{ W} \times 3600 \text{ s} = 3,600,000 \text{ J} \].Therefore, 1 kilowatt-hour equals 3,600,000 joules.

Step 2: Calculate Energy Radiated by a Person in Joules

An adult radiates energy at a rate of 100 watts. This means they use 100 joules per second. To find the total energy radiated over 24 hours, convert 24 hours to seconds:\[ 24 \text{ hours} = 24 \times 3600 \text{ seconds} = 86,400 \text{ seconds} \].Then, calculate the total energy radiated:\[ 100 \text{ J/s} \times 86,400 \text{ s} = 8,640,000 \text{ J} \].Thus, over 24 hours, an adult radiates 8,640,000 joules.

Step 3: Convert Joules to Kilocalories

To convert joules to kilocalories, use the conversion factor where 1 calorie is approximately 4.184 joules. Therefore, 1 kilocalorie is 4184 joules. Calculate the number of kilocalories radiated by the adult:\[ \frac{8,640,000 \text{ J}}{4184 \text{ J/kcal}} \approx 2064 \text{ kcal} \].Thus, an adult radiates approximately 2064 kilocalories over 24 hours.

Key Concepts

Joules

Joules are a fundamental unit of energy in the International System of Units (SI). They are used to quantify different forms of energy, including kinetic, potential, and thermal energy. One joule is the energy transferred when a force of one newton moves an object one meter. This can also be understood as the energy exerted or required to keep a 1-watt bulb illuminated for one second.

In calculations, joules often appear when converting other units of energy. For example, when converting kilowatt-hours into joules, we utilize the fact that:\[ \text{1 kWh} = 1000 \text{ W} \times 3600 \text{ s} = 3,600,000 \text{ J} \]

This calculation shows us how joules can bridge the gap between different energy measurements, making them an essential concept for studying energy conversion.

Kilocalories

Kilocalories (kcal) are a unit of measurement commonly used to describe the amount of energy in food. While a calorie measures the energy necessary to increase the temperature of 1 gram of water by 1 degree Celsius, a kilocalorie equates to 1000 of these smaller units, offering a more practical scope for our daily energy needs.

In scientific contexts, kilocalories are often converted from joules using the conversion factor 1 kcal = 4184 J. This conversion is particularly handy in understanding energy expenditures of the human body. For instance, to determine the energy radiated by an adult over a 24-hour period in kilocalories, you convert the total joules radiated, calculated as 8,640,000 J, using the formula:\[ \text{Energy in kcal} = \frac{8,640,000 \text{ J}}{4184 \text{ J/kcal}} \approx 2064 \text{ kcal} \]

The concept of kilocalories allows us to visualize and manage our energy intake and consumption more appropriately.

Kilowatt-hour

A kilowatt-hour (kWh) is a unit of energy that is commonly used by electric utilities to measure the amount of energy consumed by appliances and devices. It measures the energy used by consuming power at a rate of one kilowatt over a period of one hour.

To visualize: imagine a 1000-watt heater running continuously for one hour; this consumes exactly 1 kWh of energy. We can convert these kilowatt-hours into joules for compatibility with other energy measures. As calculated in the given exercise, 1 kWh equals 3,600,000 J. This conversion stems from multiplying 1000 watts by the number of seconds in an hour (3600 seconds): \[ 1 \text{ kWh} = 1000 \text{ W} \times 3600 \text{ s} = 3,600,000 \text{ J} \]

This unit helps individuals understand their electricity bills and manage their energy usage efficiently.