Chapter 2 Measurement and Problem Solving: Scientific Notation, Significant Figures, and Precision

Measurement and Problem Solving

Scientific Notation: Writing Large and Small Numbers

Scientific notation is a method used to express very large or very small numbers in a compact form, making them easier to read and work with in chemistry.

• Unit: A unit is a standard agreed upon quantity by which other quantities are measured.

• Scientific Notation Format: Numbers are written as $a \times 10^n$, where a is the decimal part and n is the exponent.

• Example: $140,000,000,000 = 1.4 \times 10^{11}$

• Positive Exponent: Indicates multiplication by powers of ten (moves decimal to the right).

• Negative Exponent: Indicates division by powers of ten (moves decimal to the left).

Steps to Write Numbers in Scientific Notation:

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

2. Multiply this number by 10 raised to the number of places the decimal was moved.

3. If the decimal is moved to the left, the exponent is positive; if moved to the right, the exponent is negative.

Example: $5983 = 5.983 \times 10^3$

Application: The population of China in 2017 was approximately 1,387,000,000 people. In scientific notation: $1.387 \times 10^9$ people.

Precision and Reporting Measurements

Precision refers to the degree of reproducibility of a measurement. Scientists use significant figures to indicate the precision of a measurement.

• More digits: Higher precision.

• Fewer digits: Lower precision.

• Uncertainty: Usually indicated by the last reported digit.

Example: Reporting a temperature increase of 0.7°C means the temperature could be as much as 0.8°C or as little as 0.6°C.

Estimating Measurements

When reading instruments, estimate one digit beyond the smallest marking.

• Balance with 1 g markings: Estimate to the tenths place (e.g., 1.3 g).

• Balance with 0.1 g markings: Estimate to the hundredths place (e.g., 1.26 g).

• Graduated cylinder with 0.1 mL markings: Report to the tenths place (e.g., 2.9 mL).

Significant Figures

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

• All nonzero digits are significant.

• Interior zeros (zeros between nonzero digits) are significant.

• Trailing zeros after a decimal point are significant.

• Leading zeros (zeros to the left of the first nonzero digit) are not significant.

• Trailing zeros at the end of a number but before an implied decimal point are ambiguous and should be avoided by using scientific notation.

Examples:

• $1.05$ (three significant figures)

• $0.0110$ (three significant figures)

• $4.020$ (four significant figures)

• $50.00$ (four significant figures)

• $0.0005$ (one significant figure)

Exact Numbers: Numbers that are counted or defined quantities have an unlimited number of significant figures (e.g., 10 pencils, 100 cm in 1 m).

Counting Significant Figures in Ambiguous Numbers

Numbers like 2100 can be ambiguous unless written in scientific notation.

• $2100$ (no decimal): Ambiguous

• $2.1 \times 10^3$ (scientific notation): 2 significant figures

• $2.10 \times 10^3$: 3 significant figures

• $2.100 \times 10^3$: 4 significant figures

Rounding Significant Figures in Calculations

When rounding, only the last digit dropped is considered to decide rounding direction.

• If the digit dropped is less than 5, round down.

• If the digit dropped is 5 or more, round up.

Example: To round 2.349 to two significant figures, only the 4 in the hundredths place determines rounding: 2.349 rounds to 2.3.

Addition and Subtraction with Significant Figures

For addition and subtraction, the result has the same number of decimal places as the quantity with the fewest decimal places.

• Example: $5.74 + 0.823 + 2.651 = 9.214$ (answer rounded to two decimal places: 9.21)

Multiplication and Division with Significant Figures

For multiplication and division, the result has the same number of significant figures as the quantity with the fewest significant figures.

• Example: $4.56 \times 1.4 = 6.384$ (answer rounded to two significant figures: 6.4)

Tables

Significant Figures in Different Numbers

Number	Significant Figures
83.51	4
0.00350	3
2,100	Ambiguous
2.10 × 103	3
5,000	Ambiguous

Examples of Rounding

Original Value	Rounded to 2 Sig Figs
2.33	2.3
2.37	2.4
2.349	2.3

Key Formulas

• Scientific Notation: $a \times 10^n$

• Significant Figures in Addition/Subtraction: Result has the same number of decimal places as the least precise measurement.

• Significant Figures in Multiplication/Division: Result has the same number of significant figures as the least precise measurement.

Additional info: These notes cover essential concepts from "Measurement and Problem Solving" in introductory chemistry, including scientific notation, significant figures, precision, and rounding rules. They provide foundational skills for accurate measurement and calculation in chemical experiments.