Problem 90 - Algebra A Combined Function

Question

In chemistry, Avogadro's number is the number of atoms in one mole of an element. Avogadro's number is \(6.02214199 \times 10^{23}\). Write this number in standard form. (Source: National Institute of Standards and Technology)

Step-by-Step Solution

Answer

Avogadro's number in standard form is 602,214,199,000,000,000,000,000.

Step 1: Understanding Scientific Notation

Scientific notation expresses numbers as a coefficient and a power of ten. In the given problem, Avogadro's number is already in scientific notation as \(6.02214199 \times 10^{23}\), where \(6.02214199\) is the coefficient and \(10^{23}\) is the power of ten.

Step 2: Convert Scientific Notation to Standard Form

Standard form is a way of writing numbers without using exponents. To convert \(6.02214199 \times 10^{23}\) to standard form, we move the decimal point in the number \(6.02214199\) 23 places to the right because the exponent is positive.

Step 3: Calculate the Resulting Number

When we move the decimal 23 places to the right, we add zeros after the digits of \(6.02214199\) to account for all places moved. This yields the number:\[602,214,199,000,000,000,000,000.\]

Key Concepts

Avogadro's Number

Avogadro's Number is a fundamental constant in chemistry. It tells us the number of atoms, molecules, or particles in one mole of a substance. This number is immensely large due to the tiny size of atoms and molecules, making it very important for scientific calculations. The exact value is approximately 6.02214199 multiplied by 10 to the power of 23, or in scientific terms expressed as \(6.02214199 \times 10^{23}\). Understanding Avogadro's Number allows chemists to convert between atoms and moles, which is crucial when balancing chemical equations and analyzing reactions. Understanding this number is a key aspect in fields such as stoichiometry which require a link between the atomic scale and macroscopic chemical amounts.

Standard Form

Standard form is a way of writing down very large or very small numbers easily. It's simply a way of expressing numbers without using exponents. When converting from scientific notation to standard form, there's one main task: move the decimal point to the right if the exponent is positive and to the left if it is negative. For the case of Avogadro's number, expressed as \(6.02214199 \times 10^{23}\), we move the decimal 23 places to the right. Hence, the number becomes very large. In standard form, it becomes 612,214,199,000,000,000,000,000, an extremely large number that can be challenging to comprehend without scientific notation.

Exponents

Exponents are used to express repeated multiplication of a base number. In the context of scientific notation, they represent how many times the base number 10 is multiplied by itself. A positive exponent moves the decimal point to the right, expanding the number, while a negative exponent moves the decimal point to the left, making the number smaller. Understanding exponents is crucial when dealing with scientific notation because they determine the scale of the number. In Avogadro's Number, the exponent 23 indicates that we multiply the coefficient 6.02214199 by 10, twenty-three times over, leading to a substantial increase in size.

Decimal Placement

Decimal placement refers to the position of the decimal point in a number, which indicates its value. In standard form conversion, this involves moving the decimal point to its correct place according to the exponent. When converting Avogadro's number from scientific notation to standard form, the decimal is shifted 23 places to the right. Each movement to the right increases the number by a power of ten. This results in adding zeros to fill in the missing places, turning it from the compact form \(6.02214199 \times 10^{23}\) to the full version \(6,022,141,990,000,000,000,000\). Understanding how and where to move the decimal is essential to accurately converting numbers between forms.