Problem 58 - College Algebra

Question

For the following exercises, simplify the given expression. Write answers with positive exponents.Avogadro's constant is used to calculate the number of particles in a mole. A mole is a basic unit in chemistry to measure the amount of a substance. The constant is \(6.0221413 \times 10^{23} .\) Write Avogadro's constant in standard notation.

Step-by-Step Solution

Answer

Avogadro's constant in standard notation is 602,214,130,000,000,000,000,000.

Step 1: Understanding Scientific Notation

Scientific notation expresses numbers as a product of a number (between 1 and 10) and a power of ten. Here, Avogadro's constant is given as \(6.0221413 \times 10^{23}\). This indicates \(6.0221413\) is multiplied by \(10^{23}\), representing a very large number.

Step 2: Converting to Standard Notation

To convert from scientific to standard notation, move the decimal point to the right by the number indicated by the exponent of 10. In this case, the exponent is 23, so move the decimal 23 places to the right.

Step 3: Applying the Conversion

Start with 6.0221413 and move the decimal 23 places to the right. This means adding zeros to extend the number until it reaches a total of 24 digits (since the original number 6.0221413 has 8 digits, including the decimal).

Step 4: Final Expression in Standard Notation

After moving the decimal point 23 places to the right, the resulting number is 602,214,130,000,000,000,000,000. This is Avogadro's constant in standard notation.

Key Concepts

Scientific Notation

Scientific notation is a method of expressing very large or very small numbers in a more convenient form. This can be particularly useful in scientific fields like chemistry and physics, where such numbers frequently occur. The format involves expressing a number as the product of two parts:

• A coefficient: A number between 1 and 10.
• A power of ten: This shows how many places the decimal has moved.

This way, Avogadro's constant, which is a very large number, is written as \(6.0221413 \times 10^{23}\). The number after the multiplication sign indicates how many times you move the decimal point to the right. Here, 23 shows that to convert it into standard notation, you will move the decimal 23 places to the right.

Standard Notation

Standard notation is the regular way of writing numbers, without using exponents or any special format. It is simply the number as you would see it traditionally, with digits and no extra symbols. In the case of Avogadro's constant, converting from scientific to standard notation involves moving the decimal point 23 places to the right.

During this process, we start with the number 6.0221413 and shift the decimal to make it a whole number. Since the original number only contains 8 digits, additional zeros will be needed to fill in any gaps as you count 23 places. The final result provides a clearer picture of the large number, shown as 602,214,130,000,000,000,000,000 in standard notation.

Mole Calculation

The mole is a unit of measure in chemistry that helps to quantify the amount of substance. It is crucial for understanding the scale of atoms and molecules, which are usually present in very large numbers.

Avogadro's constant, \(6.0221413 \times 10^{23}\), is a key figure in mole calculations. This number represents the number of atoms, molecules, or particles in one mole of a substance. Basically, when you have a mole of carbon, you have approximately 602,214,130,000,000,000,000,000 carbon atoms.

• This helps chemists to work with masses rather than counting individual atoms.
• It simplifies the calculations in reactions, leading to a more macroscopic view of reacting substances.

Understanding mole calculations is fundamental in making accurate predictions and efficient use of substances in chemical reactions.