Numbers, Units, and Significant Figures in General Chemistry

Numbers, Units, and Significant Figures

Introduction

Accurate measurement and reporting of numerical data are fundamental skills in general chemistry. This topic covers the identification of significant figures, performing calculations with correct significant figures, unit conversions, and proper measurement techniques using laboratory instruments.

Counting Significant Figures

Definition of Significant Figures

Significant figures (SFs) are the digits in a number that carry meaningful information about its precision. They include all certain digits plus one estimated digit.

• Nonzero digits are always significant.

• Zeros between nonzero digits are significant.

• Leading zeros (zeros before nonzero digits) are not significant.

• Trailing zeros in a decimal number are significant.

• Trailing zeros in a whole number without a decimal point are ambiguous.

Examples:

• 6.022 has 4 significant figures.

• 0.00708 has 3 significant figures.

• 890 (with no decimal) has 2 or 3 significant figures (ambiguous).

• 0.00239 L has 3 significant figures.

• 93.10 cm has 4 significant figures.

• 4.0000 cm has 5 significant figures.

• 1.00 x 103 has 3 significant figures.

• 1.70 x 10-6 g/cm3 has 3 significant figures.

Significant Figures in Calculations

Rules for Calculations

• Multiplication and Division: The result should have the same number of significant figures as the measurement with the fewest significant figures.

• Addition and Subtraction: The result should be reported to the same decimal place as the least precise measurement.

Examples:

• $(4.68 imes 0.0319) = 0.149$ (rounded to 3 significant figures)

• $(7.50)^2 = 56.3$ (rounded to 3 significant figures)

• $(500.0 / 0.020) = 25000$ (rounded to 2 significant figures)

• $(0.177 + 0.4555 - 0.4) = 0.23$ (rounded to 2 decimal places)

• $(68.78 - 19.11) / 9.59811 = 5.18$ (rounded to 3 significant figures)

Unit Conversions

Introduction to Unit Conversions

Unit conversions are essential for expressing measurements in appropriate units for calculations and reporting. The factor-label method (dimensional analysis) is commonly used.

• Identify the starting unit and the desired unit.

• Use conversion factors to cancel units until the desired unit is reached.

Examples:

• Convert 100 mg to g: $100 ext{ mg} imes rac{1 ext{ g}}{1000 ext{ mg}} = 0.10 ext{ g}$

• Convert 450 m to km: $450 ext{ m} imes rac{1 ext{ km}}{1000 ext{ m}} = 0.45 ext{ km}$

• Convert 2330 °F to K: $K = rac{5}{9}(2330 - 32) + 273 = 1506 ext{ K}$

Measurement and Reporting

Using Laboratory Instruments

When measuring with rulers or graduated cylinders, always record:

• The smallest marked place-value (e.g., tenths, hundredths).

• The estimated place-value (one digit beyond the smallest mark).

• Report the measurement with the correct number of significant figures.

Examples:

• Object length measured with a ruler marked in centimeters: 13.3 cm

• Object length measured with a ruler marked in millimeters: 13.35 cm

• Object length measured with a ruler marked in tenths of a centimeter: 13.345 cm

Volume Measurement in Graduated Cylinders

Reading Volumes

To report the volume of liquid in a graduated cylinder:

• Read the bottom of the meniscus at eye level.

• Estimate one digit beyond the smallest graduation.

• Report the volume with the correct number of significant figures.

Examples:

• Cylinder A: 18.0 mL

• Cylinder B: 21.5 mL

Summary Table: Significant Figures in Common Measurements

Measurement	Number of Significant Figures	Notes
6.022	4	All digits are significant
0.00708	3	Leading zeros not significant
890	2 or 3	Ambiguous without decimal
93.10 cm	4	Trailing zero after decimal is significant
4.0000 cm	5	All zeros after decimal are significant
1.00 x 103	3	Scientific notation clarifies significant figures

Additional info:

• Significant figures are crucial for reporting the precision of measurements and calculations in chemistry.

• Always use the correct number of significant figures in laboratory reports and calculations to reflect the accuracy of the data.