Problem 14 - Chemistry The Molecular Science

Question

A \(100-\mathrm{W}\) light bulb is left on for \(14 \mathrm{~h} .\) Calculate how many joules are used. With electricity at \(\$ 0.09\) per \(\mathrm{kWh}\), calculate how much it costs to leave the light on for \(14 \mathrm{~h}\).

Step-by-Step Solution

Answer

The light bulb uses 5040000 J and costs $0.126 for 14 hours.

Step 1: Convert Power to Energy

First, convert the power rating of the light bulb from watts to joules. Recall that power (in watts) is energy (in joules) per second. Therefore, by multiplying power by time in seconds, we get the total energy in joules. The power of the bulb is \(100\, \mathrm{W}\) and time is \(14\, \mathrm{h}\), which must be converted to seconds: \(14\, \mathrm{h} \times 3600\, \mathrm{s/h} = 50400\, \mathrm{s}\). The total energy used is \(100\, \mathrm{W} \times 50400\, \mathrm{s} = 5040000\, \mathrm{J}\).

Step 2: Convert Joules to Kilowatt-Hours

Electricity is billed per kilowatt-hour \((\mathrm{kWh})\). Convert the total energy from joules to kilowatt-hours. Recall that \(1\, \mathrm{kWh} = 3.6 \times 10^6\, \mathrm{J}\). Therefore, the energy used in kilowatt-hours is \(\frac{5040000\, \mathrm{J}}{3.6 \times 10^6\, \mathrm{J/kWh}} = 1.4\, \mathrm{kWh}\).

Step 3: Calculate Cost of Electricity

Now that we have the energy usage in kilowatt-hours, calculate the cost. Electricity is priced at \(\\(0.09\) per \(\mathrm{kWh}\). Multiply the energy usage by the cost per \(\mathrm{kWh}\): \(1.4\, \mathrm{kWh} \times \\)0.09/\mathrm{kWh} = \$0.126\).

Key Concepts

Joules to Kilowatt-hours conversion

When considering energy usage in the context of electricity, it's essential to understand the conversion between joules and kilowatt-hours (kWh). Joules are a unit of energy used commonly in physics, while kilowatt-hours are the billing unit used by power companies. To make accurate calculations, you need to convert joules to kilowatt-hours.
Start by remembering that a joule is a relatively small measure of energy. Because of this, large energy consumptions (like those used in homes) are measured in kilowatt-hours. The relationship between these units is defined by the conversion factor:

• 1 kWh = 3.6 × 10^6 joules

To convert joules to kilowatt-hours, divide the number of joules by 3.6 million (3.6 × 10^6). For example, if a light bulb consumes 5,040,000 joules, divide 5,040,000 by 3.6 × 10^6, resulting in 1.4 kWh. This conversion is crucial for understanding how energy consumption translates into utility billing.

Electricity Cost Calculation

Calculating the cost of electricity involves determining how much energy is used and what the energy supplier charges per unit of energy. The easy way to calculate electricity cost is by using your energy consumption in kilowatt-hours and the price per kilowatt-hour charged by the utility company. Here's a straightforward process you can follow to find out the cost:

• First, convert the energy usage to kilowatt-hours if it's in another unit like joules (as explained above).
• Next, multiply this energy use in kWh by the cost rate given by your utility, usually specified in dollars per kWh.

For instance, if the energy usage is 1.4 kWh and the cost is $0.09 per kWh, multiply these to find the total cost: 1.4 kWh × $0.09/kWh = $0.126.
This method ensures accurate billing estimations and helps in managing energy expenses efficiently.

Power and Energy relationship

Understanding the relationship between power and energy is fundamental to solving related problems in physics and real-world applications. Power is defined as the rate at which energy is used or transferred. The unit of power is the watt (W), which represents one joule per second.
The basic formula connecting power (P), energy (E), and time (t) is:

• Power (P) = Energy (E) / Time (t)

Rearranging this, energy can be calculated as:

• Energy (E) = Power (P) × Time (t)

This equation shows how the amount of energy consumed depends directly on the power usage and the duration for which an appliance runs. For example, a 100-watt light bulb left on for 14 hours uses energy as calculated by multiplying 100 watts by the total time in seconds (which is 50,400 seconds), resulting in an energy usage of 5,040,000 joules. Understanding this relationship helps predict and control energy usage more effectively.