Measurement and Problem Solving: Scientific Notation and Significant Figures

Measurement and Problem Solving

Introduction

This chapter introduces foundational concepts in measurement and problem solving in chemistry, focusing on scientific notation and significant figures. These concepts are essential for expressing and interpreting numerical data accurately in scientific contexts.

Scientific Notation: Writing Large and Small Numbers

Purpose and Definition

• Scientific notation is a technique used to express very large or very small numbers in a more compact and manageable form.

• This notation is especially useful in chemistry, where measurements can span many orders of magnitude.

Structure of Scientific Notation

• Any number in scientific notation has two parts:

  • Decimal part: A number usually between 1 and 10.

  • Exponential part: 10 raised to an exponent, n.

• General form: $a \times 10^{n}$, where $1 \leq a < 10$ and $n$ is an integer.

Examples

• Distance from Earth to Sun: 93,000,000 miles = $9.3 \times 10^{7}$ miles

• Mass of a single carbon atom: 0.0000000000000000000000199 g = $1.99 \times 10^{-23}$ g

Positive and Negative Exponents

• Positive exponent (n > 0): Indicates a large number; move the decimal point to the left.

• Negative exponent (n < 0): Indicates a small number; move the decimal point to the right.

Mathematical Meaning

• $10^{0} = 1$

• $10^{1} = 10$

• $10^{2} = 100$

• $10^{3} = 1000$

• $10^{-1} = 0.1$

• $10^{-2} = 0.01$

• $10^{-3} = 0.001$

How to Express a Number in Scientific Notation

1. Move the decimal point to obtain a number between 1 and 10.

2. Multiply that number (the decimal part) by 10 raised to the number of places the decimal point was moved.

3. If the decimal point is moved to the left, the exponent is positive (LIP: Left Is Positive).

4. If the decimal point is moved to the right, the exponent is negative.

Examples

• 5,983 = $5.983 \times 10^{3}$ (Move 3 places left, n > 0)

• 0.00034 = $3.4 \times 10^{-4}$ (Move 4 places right, n < 0)

• 68,000,000 = $6.8 \times 10^{7}$ (Move 7 places left, n > 0)

• 0.000000632 = $6.32 \times 10^{-7}$ (Move 7 places right, n < 0)

Significant Figures: Writing Numbers to Reflect Precision

Definition and Importance

• Significant figures (sig figs) are the digits in a measurement that are known with certainty plus one digit that is estimated.

• They reflect the precision of a measured quantity and are crucial for reporting scientific data accurately.

Rules for Counting Significant Figures

• All nonzero digits are significant.

• Captive zeros (zeros between nonzero digits) are significant. Example: 3005 (4 sig figs)

• Trailing zeros (zeros at the right end of a number):

  • If after a decimal point, they are significant. Example: 16.00 (4 sig figs)

  • If before a decimal point and after a nonzero digit, they are significant. Example: 50.00 (4 sig figs)

  • If before an implied decimal point (e.g., 420), they are ambiguous and should be avoided by using scientific notation.

• Leading zeros (zeros to the left of the first nonzero digit) are not significant. Example: 0.038 (2 sig figs)

• Exact numbers (from counting or defined quantities) have an unlimited number of significant figures. Examples: 5 pencils, 2 molecules, 1 dozen = 12, 1 m = 100 cm

Examples of Significant Figures

• 45872 (5 sig figs; all digits are certain except the last, which is estimated)

• 0.0060 (2 sig figs; leading zeros are not significant, trailing zero after decimal is significant)

• 420 (ambiguous; could be 2 or 3 sig figs, use scientific notation to clarify)

• 420. (3 sig figs; decimal point indicates significance)

Measuring and Estimating with Significant Figures

• When measuring, always record all certain digits plus one estimated digit.

• Different observers may estimate the last digit slightly differently, but the number of significant figures remains the same.

Table: Types of Zeros and Their Significance

Type of Zero	Example	Significant?
Leading zeros	0.0025	No
Captive zeros	3005	Yes
Trailing zeros (after decimal)	16.00	Yes
Trailing zeros (before decimal, ambiguous)	420	Ambiguous

Exact Numbers

• Numbers obtained by counting or defined relationships (e.g., 1 dozen = 12) are considered to have infinite significant figures.

• Conversion factors in equations are often exact.

Practice Problems

• Express the following in scientific notation:

  • 0.00082 = $8.2 \times 10^{-4}$

  • 25,000 = $2.5 \times 10^{4}$

• Express the following in decimal notation:

  • $6.32 \times 10^{-7}$ m = 0.000000632 m

  • $4.5 \times 10^{2}$ mm = 450 mm

• How many significant figures are in each measured quantity?

  • 5050 (3 sig figs)

  • 500 (1 sig fig, unless written as 5.00 x 102 for 3 sig figs)

  • 0.001 (1 sig fig)

Additional info: These notes are based on textbook content and lecture slides for a General Chemistry course, focusing on the essential skills of expressing numbers in scientific notation and determining significant figures for accurate scientific communication.