Chapter 2: Chemistry and Measurements – Study Notes

Chemistry and Measurements

Measured Numbers

In chemistry, measurements are fundamental for quantifying physical properties such as length, mass, and time. These measurements are obtained using specific tools and are reported as measured numbers.

• Measured numbers are values determined using a measuring instrument (e.g., ruler, balance).

• They always include some degree of uncertainty, reflected in the last digit, which is estimated.

• Examples: Measuring the length of an object with a ruler, weighing a sample on a balance.

Reporting Length

When reporting the length of an object, it is important to record all certain digits plus one estimated digit. This process ensures the measurement reflects the precision of the instrument used.

• Observe the numerical values at the ends of the object on the measuring scale.

• Estimate the last digit by visually dividing the space between the smallest marked lines.

• The estimated digit is the final digit reported for a measured number.

Example: If the end of an object is between the 4-cm and 5-cm marks, and appears halfway, report the length as 4.5 cm.

Precision in Measurement

The precision of a measurement depends on the smallest division on the measuring tool.

• If a ruler is marked every 1 cm, the measurement might be reported as 3.0 cm (the estimated digit is in the tenths place).

• If a ruler is marked every 0.1 cm, the measurement might be reported as 4.55 cm (the estimated digit is in the hundredths place).

Example: Reporting a length as 4.55 cm when the object is halfway between 4.5 cm and 4.6 cm on a ruler marked every 0.1 cm.

Significant Figures

Definition and Importance

Significant figures (SFs) are the digits in a measured number that are known with certainty plus one estimated digit. They reflect the precision of the measurement and are crucial in scientific calculations.

• All nonzero digits are significant.

• Zeros may or may not be significant, depending on their position.

Rules for Identifying Significant Figures

• Nonzero numbers are always significant.

• Zeros between nonzero digits are significant (e.g., 50.08 km has 4 SFs).

• Zeros at the end of decimal numbers are significant (e.g., 200.0 m has 4 SFs).

• Zeros at the beginning of decimal numbers are not significant (e.g., 0.005 km has 1 SF).

• Zeros at the end of non-decimal numbers (without a decimal point) are not significant (e.g., 1300 g has 2 SFs).

• Scientific notation: Only the digits in the coefficient are significant (e.g., $2.650 \times 10^4$ m has 4 SFs).

Examples of Significant Figures

Measured Number	Significant Figures	Explanation
50.08 km	4	Zeros between nonzero digits are significant.
200.0 m	4	Zeros at the end of a decimal number are significant.
0.005 km	1	Zeros at the beginning of a decimal number are not significant.
1300 g	2	Zeros at the end of a non-decimal number are not significant.
$2.650 \times 10^4$ m	4	Scientific notation: only digits in the coefficient are significant.

Exact Numbers

Definition and Examples

Exact numbers are values obtained by counting or by definition, not by measurement. They have an infinite number of significant figures and do not affect the number of significant figures in calculations.

• Obtained by counting (e.g., 8 cookies, 3 coins).

• Defined relationships (e.g., 1 ft = 12 in, 1 kg = 1000 g).

Example: The number of coins in a collection (e.g., 3 coins) is exact; the conversion 1 min = 60 s is exact by definition.

Practice and Application

Identifying Significant Figures

• 30 m: 2 SFs

• 50 L: 2 SFs

• 0.008 g: 1 SF

• 30.0 m: 3 SFs

Writing Numbers in Scientific Notation

• 02650 m: $2.650 \times 10^3$ m (4 SFs; zeros preceding 2 are not significant)

• 0.026 g: $2.6 \times 10^{-2}$ g (2 SFs; zeros before 2 are not significant)

• 44000 L: $4.4 \times 10^4$ L (2 SFs; zeros at end without decimal are not significant)

Summary Table: Significant Figures in Various Numbers

Number	Significant Figures	Type
0.00476	3	Measured
25.300	5	Measured
2.050 × 103	3	Measured (Scientific Notation)
5.80 × 102	3	Measured (Scientific Notation)

Key Equations and Concepts

• Scientific Notation: $a \times 10^n$ where a is the coefficient containing all significant figures.

• Significant Figures in Calculations: The result of a calculation should be reported with the same number of significant figures as the measurement with the least number of significant figures.

Additional info:

• Significant figures are essential for reporting the precision of measurements and for proper scientific communication.

• Exact numbers do not limit the number of significant figures in calculations.