Question

An average person consumes about \(2.0 \times 10^{3} \mathrm{kcal}\) of food energy per day. How many kilowatt-hours of energy are consumed? How long could you light a 40-W light bulb with that energy?

Short answer

The energy consumed is 8.368 kWh, and it could light a 40-W bulb for approximately 209.2 hours.

Step-by-step solution

Convert food energy from kilocalories to kilojoules

Since 1 kcal is equivalent to 4.184 kJ, the conversion from kilocalories to kilojoules is done by multiplying the energy in kilocalories with the conversion factor. Thus, the energy in kilojoules (kJ) is given by: \(2.0 \times 10^{3} \mathrm{kcal} \times 4.184 \mathrm{kJ/kcal}\).

Convert the energy from kilojoules to kilowatt-hours

To convert the energy to kilowatt-hours, use the fact that 1 kJ equals 0.000277778 kWh. Multiply the energy in kilojoules by the conversion factor to get the energy in kilowatt-hours: \((2.0 \times 10^{3} \times 4.184) \mathrm{kJ} \times 0.000277778 \mathrm{kWh/kJ}\).

Calculate the duration for lighting a 40-W bulb

With the energy in kilowatt-hours calculated in Step 2, divide this by the power of the light bulb to find the duration for which it can be lit. Use the formula: \(\text{Duration (hours)} = \frac{\text{Energy (kWh)}}{\text{Power (kW)}}\). Since the light bulb is 40 W, which is 0.04 kW, the formula becomes \(\text{Duration} = \frac{\text{Energy (kWh)}}{0.04}\).

Key concepts

Calorie to Joule Conversion

Understanding the conversion between calories and joules is essential for grasping energy-related concepts in chemistry. A calorie (cal) is a unit of energy that was originally defined as the amount of heat needed to raise the temperature of one gram of water by one degree Celsius. The joule (J), on the other hand, is the SI unit of energy, widely used in science.

To convert calories to joules, we use the conversion factor: 1 calorie is equivalent to 4.184 joules. So, when you have an energy value in calories, you simply multiply by 4.184 to get the equivalent energy in joules. For example, to convert 10 calories to joules, you would calculate:
\(10 \text{ cal} \times 4.184 \text{ J/cal} = 41.84 \text{ J}\).

This conversion is fundamental in various scientific fields, particularly when it comes to nutritional energy content of foods, and it's widely utilized in chemistry and physics for energy conversion calculations.

Kilojoules to Kilowatt-hours Conversion

When dealing with larger energy scales, such as in food consumption or electricity usage, it is common to use units like kilojoules (kJ) and kilowatt-hours (kWh). However, these units serve different applications, and it is important to understand their relationship.

A kilojoule is a unit of energy equal to 1,000 joules, which is often used to express nutritional energy values. A kilowatt-hour, typically used for electrical energy measurement, is the amount of energy equivalent to a power of one kilowatt running for one hour. To convert kilojoules to kilowatt-hours, the conversion factor is: 1 kJ = 0.000277778 kWh.

As an example, to convert 500 kJ to kWh, you would perform the following calculation:
\(500 \text{ kJ} \times 0.000277778 \text{ kWh/kJ} = 0.138889 \text{ kWh}\).

This conversion is widely applicable in everyday scenarios like understanding your own energy consumption or calculating the energy efficiency of appliances.

Energy Consumption Calculations

Energy consumption calculations are crucial to a variety of disciplines, from environmental science to engineering and economics. These calculations involve determining the amount of energy used by an appliance or system over a certain period. One common application is assessing the energy used by a light bulb, as illustrated by the example provided.

Here is how to calculate the energy consumption:

• Determine the power rating of the device (in watts or kW).
• Calculate the amount of energy (in kWh) based on the time the device is running.
• Use the formula \(Energy \text{ (kWh)} = Power \text{ (kW)} \times Time \text{ (h)}\).

For example, if a 40-Watt light bulb is used for 10 hours, the calculation would be:
\(0.04 \text{ kW} \times 10 \text{ h} = 0.4 \text{ kWh}\).

Therefore, by knowing the duration of use and the power of an appliance, one can convert these values into the total energy consumption. This is frequently used to help conserve resources, reduce costs, and minimize environmental impact.