Problem 58 - College Algebra with Corequisite Support

Question

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 602214130000000000000000.

Step 1: Understand Scientific Notation

Scientific notation is a way to express very large or very small numbers. It has the form \( a \times 10^n \), where \( a \) is a coefficient that is typically between 1 and 10, and \( n \) is an integer.

Step 2: Identify the Coefficient and Exponent

In Avogadro's constant, \(6.0221413 \times 10^{23}\), the number \(6.0221413\) is the coefficient and \(23\) is the exponent.

Step 3: Convert to Standard Notation

To convert from scientific notation to standard notation, multiply the coefficient by \(10\) raised to the power of the exponent. Essentially, shift the decimal point of the coefficient 23 places to the right.

Step 4: Perform the Calculation

Write the number \(6.0221413\) and move the decimal point 23 places to the right. Since \(6.0221413\) moved 23 places to the right becomes \(602214130000000000000000\), therefore, Avogadro's constant in standard notation is \(602214130000000000000000\).

Key Concepts

Scientific Notation

Scientific notation is a powerful method of simplifying the representation of very large or very small numbers. For instance, Avogadro's constant is quite large and can be difficult to express in full without scientific notation.

• It involves writing numbers in the form of \(a \times 10^n\),
• where \(a\) is the significant figure (or coefficient) between 1 and 10, and \(n\) is an integer indicating the power of 10.

This notation helps in handling numbers that are not easily manageable in their full form. For example, Avogadro's constant is written as \(6.0221413 \times 10^{23}\). This implies that if you multiply 6.0221413 by the number 10 raised to the 23rd power, you get the actual value represented. It's an efficient way to work with numbers in sciences like chemistry and physics.

Standard Notation

Standard notation is the usual way of writing numbers, where all digits are explicitly shown without any expressions or exponents. When you express Avogadro's constant in standard notation, it requires writing out the entire number

Converting Scientific to Standard

To convert a number from scientific to standard notation, you directly perform the calculation implied by the scientific notation.

• For Avogadro's constant \(6.0221413 \times 10^{23}\),
• shift the decimal point in \(6.0221413\) twenty-three places to the right.

This results in the number \(602214130000000000000000\), illustrating how large Avogadro's number truly is. Such conversions are crucial in practical calculations, especially when needing precise or exact values.

Moles in Chemistry

Moles are a fundamental unit in chemistry that measure the amount of a substance. This concept is critical when dealing with chemical reactions and compositions, as it allows for a consistent method of quantifying material quantities.

• One mole of any substance contains exactly Avogadro's number of molecules or atoms, which is \(6.0221413 \times 10^{23}\).
• This vast number helps us relate masses of substances to measurable numbers of molecules or ions.

By using moles, chemists can calculate the amounts of reactants required or products formed in a chemical reaction with great accuracy. The mole bridges the gap between the atomic scale and the macroscopic amounts used in laboratories, making it an indispensable tool in chemistry education and practice.