Question

If a stack of 500 sheets of paper is \(4.60 \mathrm{~cm}\) high, what will be the height, in meters, of a stack of Avogadro's number of sheets of paper?

Short answer

The height is \(5.54 \times 10^{19}\) meters.

Step-by-step solution

Understanding the problem

Identify the given information and the quantity to be found. Given: 500 sheets of paper have a height of 4.60 cm. Find: Height of Avogadro's number of sheets in meters.

Find the height of one sheet

First, find the height of one sheet by dividing the total height of 500 sheets by 500. \[\text{Height of one sheet} = \frac{4.60 \text{ cm}}{500} = 0.0092 \text{ cm}\]

Use Avogadro's number

Avogadro's number (\(6.022 \times 10^{23}\)) represents the number of sheets in our calculation. Multiply the height of one sheet by Avogadro's number. \[\text{Total height} = 0.0092 \text{ cm/sheet} \times 6.022 \times 10^{23} \text{ sheets}\]

Calculate the height in cm

Perform the multiplication to get the height in cm. \[\text{Total height} = 0.0092 \text{ cm} \times 6.022 \times 10^{23} = 5.54 \times 10^{21} \text{ cm}\]

Convert cm to meters

Convert the height from cm to meters by dividing by 100 (since 1 m = 100 cm). \[\text{Total height in meters} = \frac{5.54 \times 10^{21} \text{ cm}}{100} = 5.54 \times 10^{19} \text{ m}\]

Key concepts

Unit Conversion

Unit conversion is a fundamental skill in science and everyday life. It's the process of changing a measurement from one unit to another. For example, we converted the height of a paper stack from centimeters to meters in this exercise.
Unit conversion involves knowing the relationship between units, such as:

• 1 meter (m) = 100 centimeters (cm)
• 1 kilogram (kg) = 1000 grams (g)
• 1 hour = 60 minutes

To convert between units, use multiplication or division by the conversion factor. In this exercise, we divided the height in cm by 100 to convert to meters since 1 m = 100 cm. Understanding these relationships, and practicing them often, makes unit conversion second nature.

Avogadro's Number

Avogadro's number is a big concept in chemistry and physics. It represents the number of particles (like atoms or molecules) in one mole of a substance. Avogadro's number is approximately \(6.022 \times 10^{23}\).
This gigantic number helps us deal with particles on a scale that we can work with using standard units. In our exercise, understanding Avogadro's number helped us figure out the height of an astronomical number of paper sheets.
By knowing the height of one sheet and multiplying it by Avogadro's number, we calculated the total height. For students, recognizing that this number is used to relate macroscopic measurements to microscopic scales is crucial for mastering concepts in chemistry and beyond.

Scientific Notation

Scientific notation is a way to express very large or very small numbers. It makes it easier to read, write, and work with these numbers. A number in scientific notation looks like \(a \times 10^n\), where \(a\) is a number between 1 and 10, and \(n\) is an integer.
For instance, Avogadro's number \(6.022 \times 10^{23}\) is written in scientific notation. This format lets us handle the enormous magnitude simply.

• An example: 300 can be written as \(3 \times 10^2\).
• Another example: 0.005 can be expressed as \(5 \times 10^{-3}\).

In our exercise, we used scientific notation to represent the final height in a manageable form as \(5.54 \times 10^{19} \, \text{meters}\). Learning to use and understand scientific notation is essential for anyone working in scientific fields.

Height Calculation

Height calculation involves some basic math to determine the total height, given certain parameters. In this exercise, we started with the height of 500 sheets of paper totalling 4.60 cm.
Here's a quick step-by-step breakdown:

• Determine the height of one sheet by dividing the total height (4.60 cm) by the number of sheets (500). The result was 0.0092 cm per sheet.
• To find the height of Avogadro's number of sheets, multiply the height of one sheet by Avogadro's number \(6.022 \times 10^{23}\).
• This gave us a very large number in centimeters, so we converted to meters by dividing by 100.

This exercise showed how breaking down a problem into smaller steps, and applying basic arithmetic, leads to a solution. Practicing these fundamentals bolsters your problem-solving skills.