01

ON THIS PAGE

01.01

Very Small and Very Big Numbers

01.02

Logarithms

01.03

Units

01.04

Errors in Experiments

01.05

Calculating and Exporting Significant Digits

This chapter forms the foundation of all quantitative work in chemistry. In this course, we do not simply deal with abstract numbers; we deal with measured physical quantities, which consist of both a number and a unit. To describe the natural world accurately, we must be careful, precise, and systematic when performing measurements and handling data.

01.01

Very Small and Very Big Numbers

In chemistry, we study the macroscopic world by looking at microscopic entities. This means we frequently encounter incredibly large or extremely small numbers. For example:

• The Macroscopic Scale (Huge Numbers): A single $108\text{ g}$ sample of pure silver (Ag) contains approximately $6.02 \times 10^{23}$ silver atoms. If you take a single sip of water, you are holding more than $6.02 \times 10^{23}$ atoms in your mouth, and a tiny $1\text{ mg}$ crystal of table salt (NaCl) contains about $1.0 \times 10^{19}$ formula units of NaCl.
• The Microscopic Scale (Tiny Numbers): The mass of a single silver atom is approximately $1.79 \times 10^{-22}\text{ grams}$, the mass of an electron is a mere $9.11 \times 10^{-31}\text{ kg}$, and the radius of a hydrogen atom is only about $40\text{ pm}$ (picometers, or $40 \times 10^{-12}\text{ m}$).

Writing out all these zeros is cumbersome and prone to error. To resolve this, scientists use scientific notation (also known as standard notation). In this notation, we write a number in the form:
$$\mathbf{A \times 10^x}$$
where $A$ (the coefficient) is a number with exactly one non-zero digit to the left of the decimal point, and $x$ (the exponent) is an integer.

• Converting Large Numbers ($x > 0$): Move the decimal point to the left until there is only one non-zero digit to its left. The number of places moved is the positive exponent.
  • $2800 = 2.8 \times 1000 = {2.8 \times 10^3}$
  • $55,000 = {5.5 \times 10^4}$
• Converting Small Numbers ($x < 0$): Move the decimal point to the right until there is exactly one non-zero digit to its left. The number of places moved is the negative exponent.
  • $0.00035 = {3.5 \times 10^{-4}}$
  • $0.0012450 = {1.2450 \times 10^{-3}}$

Student Problem-Solving Tip: Entering Exponents on a Calculator

When using a scientific calculator to enter numbers in scientific notation (e.g., $2.8 \times 10^3$), use the EE or EXP key. This key literally stands for “times ten to the power of”.

• Correct Sequence: Enter 2.8 $\rightarrow$ press EE (or EXP) $\rightarrow$ enter 3.
• The Most Common Error: Beginners often explicitly type “$\times 10$” before hitting the exponent key. If you enter 2.8 $\rightarrow$ * $\rightarrow$ 10 $\rightarrow$ EXP $\rightarrow$ 3, the calculator multiplies $2.8 \times 10 = 28$ first, and then multiplies by $10^3$, yielding $28,000$ (which is $2.8 \times 10^4$) instead of $2,800$ ($2.8 \times 10^3$)! Always let the EE/EXP key do the “$\times 10$” work for you.