Question

Assume that the nutritional content of an apple-say \(50 \mathrm{Cal}\) \((1 \mathrm{Cal}=1000 \mathrm{cal})-\) could be used to light a light bulb. For how many minutes would there be light from each of the following? (a) a 100 -watt incandescent bulb \(((1 \mathrm{~W}=1 \mathrm{~J} / \mathrm{s})\) (b) a 23 -watt compact fluorescent bulb, which provides a similar amount of light

Short answer

A 100-watt bulb can be powered for 34.87 minutes, and a 23-watt bulb can be powered for 151.59 minutes.

Step-by-step solution

Convert Calories to Joules

The first step is to convert the energy content of the apple from Calories to Joules. We know that 1 Calorie is equal to 1000 calories, and 1 calorie (small 'c') is equal to 4.184 Joules. Thus, if an apple provides 50 Calories (Cal):\[ 50 \text{ Cal} = 50 \times 1000 \text{ cal} = 50000 \text{ cal} \ 50000 \text{ cal} = 50000 \times 4.184 \text{ J} = 209200 \text{ J} \]So, the nutritional content of the apple is 209200 Joules.

Calculate the Time for a 100-Watt Bulb

To find out how long a 100-watt bulb can be powered by 209200 Joules, we use the formula that power (W) equals energy (J) divided by time (s):\[ \text{Power} = \frac{\text{Energy}}{\text{Time}} \rightarrow \text{Time} = \frac{\text{Energy}}{\text{Power}} \]For a 100-watt bulb:\[ \text{Time} = \frac{209200 \text{ J}}{100 \text{ W}} = 2092 \text{ seconds} \]Now, convert the time from seconds to minutes:\[ 2092 \text{ seconds} \times \frac{1 \text{ minute}}{60 \text{ seconds}} \approx 34.87 \text{ minutes} \]Thus, the 100-watt incandescent bulb can be powered for approximately 34.87 minutes.

Calculate the Time for a 23-Watt Bulb

Next, we calculate how long a 23-watt bulb can be powered using the same energy. Using similar calculations as before:\[ \text{Time} = \frac{209200 \text{ J}}{23 \text{ W}} = 9095.65 \text{ seconds} \]Converting this time into minutes:\[ 9095.65 \text{ seconds} \times \frac{1 \text{ minute}}{60 \text{ seconds}} \approx 151.59 \text{ minutes} \]Thus, the 23-watt compact fluorescent bulb can be powered for approximately 151.59 minutes.

Key concepts

Understanding Joules

To understand energy conversion, it's important to know what a Joule is. A Joule is a unit of energy in the International System of Units. It's defined as the amount of energy transferred when a force of one newton moves through a distance of one meter. To put it simply, it's a way to measure how much energy is used in doing work, like lighting up a bulb or powering a device.

In our example, we converted the energy from an apple's calories into Joules because Joules are more commonly used in scientific calculations. One calorie (with a small 'c') is equivalent to 4.184 Joules. So, whenever you are working with energy conversions, remember that Joules are a universal standard that can be easily converted from other units of energy.

What is a Watt?

A Watt measures power, which is the rate at which energy is used or converted. In our exercise, we dealt with light bulbs, which often range in power ratings like 23 watts or 100 watts.

One Watt is equivalent to one Joule per second. This means if a device has a power rating of 100 watts, it uses 100 Joules of energy every second. Knowing this is crucial for understanding how long a device can operate with a given amount of energy in Joules. By knowing the power in watts and the total energy, we can calculate how long the device will run, as seen in the exercise with the incandescent and compact fluorescent bulbs.

Calories and Energy Content

When it comes to food, energy is often expressed in Calories. When we talk about 50 Calories from an apple, we are actually talking about 50 kilocalories. This is because in nutritional contexts, Calories (with an uppercase 'C') are actually kilocalories, where 1 Calorie equals 1000 small calories.

Each calorie contains 4.184 Joules of energy, making food energy directly convertible into mechanical energy. In this exercise, an apple's energy content got converted to Joules to measure how much work it can do or how long it can power a light bulb. This conversion is key in understanding how food can translate to other energy forms.

Incandescent Bulbs Explained

An incandescent bulb is one of the most traditional types of electric light bulbs. It works by heating a tungsten filament until it glows, producing both light and heat. These bulbs are generally less efficient because a significant portion of the energy is lost as heat.

In our exercise, the bulb was rated at 100 watts. This means it consumes a lot of energy to produce light, which is why using the same amount of energy from our apple, it can only run for about 34.87 minutes. This inefficiency is part of the reason why these bulbs are being replaced by more energy-efficient options like compact fluorescent bulbs or LEDs.

Understanding Compact Fluorescent Bulbs

Compact fluorescent bulbs (CFLs) are more energy-efficient than traditional incandescent bulbs. They work by exciting a gas that emits ultraviolet light inside the bulb, which then interacts with a fluorescent coating to produce visible light.

In our example, the CFL was rated at 23 watts but provides a comparable amount of light to a 100-watt incandescent bulb. It uses the energy more efficiently to produce light rather than generating heat. This greater efficiency is why, using the same energy from an apple, a CFL can run for approximately 151.59 minutes, significantly longer than the incandescent bulb. CFLs help save energy and reduce electricity bills, making them a popular lighting choice.