Rounding Numbers, Significant Figures, and Unit Conversion in Chemistry

Rounding Numbers & Significant Figures (SFs)

Introduction to Significant Figures

Significant figures (SFs) are the digits in a measured value that carry meaning regarding its precision. Correctly reporting calculated values using measured data is essential in chemistry to reflect the true uncertainty and reliability of results.

• Measured value with more SFs: Indicates greater precision and more information.

• Calculation does not increase information: The number of SFs in a result cannot exceed the precision of the least precise measurement used.

Rules for Rounding in Calculations

• Multiplication and Division: The result should have the same number of SFs as the measurement with the fewest SFs used in the calculation.

• Addition and Subtraction: The result should be rounded to the same decimal place as the measurement with the least number of decimal places (i.e., the highest digit containing uncertainty).

Examples of Rounding

• Multiplication Example: $12 \times 357 = 4284$ (12 has 2 SFs, 357 has 3 SFs; result rounded to 2 SFs: $4.3 \times 10^3$)

• Division Example: $365 \div 12 = 30.41666...$ (rounded to 2 SFs: $30$ or $3.0 \times 10^1$)

• Addition Example: $0.378\,\text{mL} + 42.3\,\text{mL} - 1.5833\,\text{mL} = 41.0947\,\text{mL}$ (rounded to 41.1 mL, since the least precise value is to the tenths place)

Unit Conversion in Chemistry

Introduction to Unit Conversion

Unit conversion is a fundamental skill in chemistry, allowing for the translation of measurements between different systems (e.g., metric to imperial). Conversion factors are considered to have infinite SFs, as they are defined values.

• Conversion factors: Treated as exact numbers with infinite SFs (e.g., $1\,\text{m} = 100\,\text{cm}$).

• Volume conversion: $1\,\text{m}^3 = (1\,\text{m})^3 = (100\,\text{cm})^3 = 10^6\,\text{cm}^3$ (not $100\,\text{cm}^3$).

Worked Example: Density and Mass Calculation

• Problem: The density of silver is $10.5\,\text{g/cm}^3$. What is the mass in kilograms of a cube of silver that measures $0.85\,\text{m}$ on each side?

• Solution:

  • Calculate volume: $V = (0.85\,\text{m})^3$

  • Convert $\text{m}^3$ to $\text{cm}^3$: $1\,\text{m}^3 = 10^6\,\text{cm}^3$

  • Calculate mass: $\text{mass} = \text{density} \times \text{volume}$

  • Convert grams to kilograms: $1\,\text{kg} = 1000\,\text{g}$

  Calculation: $\text{mass} = (10.5\,\text{g/cm}^3) \times (0.85\,\text{m} \times 100\,\text{cm}/1\,\text{m})^3 \times (1\,\text{kg}/1000\,\text{g})$ $= 6448.312\,\text{kg}$ Rounded to 2 SFs: $6.4 \times 10^3\,\text{kg}$

Significant Figures in Mixed Operations

Combining Multiplication/Division with Addition/Subtraction

When calculations involve both multiplication/division and addition/subtraction, apply the SF rules for each operation in sequence.

• Example: $5.00 \times 2.000 + 0.65$

  • First, multiply: $5.00 \times 2.000 = 10.00$ (3 SFs)

  • Then, add: $10.00 + 0.65 = 10.65$

  • Final answer should be rounded according to the decimal place of the least precise value in the addition (hundredths place): $10.65$

Unit Conversion Example: Fuel Efficiency

Converting Units for Practical Applications

Unit conversion is often required to compare measurements in different systems, such as converting fuel efficiency from liters per 100 kilometers to miles per gallon.

• Conversion factors: $1\,\text{mi} = 1.609\,\text{km}$, $1\,\text{gal} = 3.785\,\text{L}$

• Example: A car uses gasoline at a rate of $16.0\,\text{L}/100\,\text{km}$. Convert to miles per gallon (mpg):

  • Set up conversion: $16.0\,\text{L}/100\,\text{km} \times (1\,\text{gal}/3.785\,\text{L}) \times (100\,\text{km}/1.609\,\text{mi})$

  • Calculate: $4.7$ gallons per 100 miles, or $14.7$ mpg

  • Interpretation: Not a good fuel efficiency for a car.

Summary Table: Significant Figures Rules

Operation	Rule for SFs	Example
Multiplication/Division	Result has same SFs as the value with the fewest SFs	$12 \times 357 = 4.3 \times 10^3$ (2 SFs)
Addition/Subtraction	Result rounded to the highest digit containing uncertainty	$0.378 + 42.3 - 1.5833 = 41.1$ (tenths place)
Unit Conversion	Conversion factors have infinite SFs	$1\,\text{m} = 100\,\text{cm}$
Mixed Operations	Apply SF rules for each step sequentially	$5.00 \times 2.000 + 0.65 = 10.65$

Key Takeaways

• Always identify the number of significant figures in measured values before performing calculations.

• Report results with the correct number of significant figures to reflect the true uncertainty of the measurements.

• Unit conversions use exact values and do not limit the number of significant figures in the result.

• Proper handling of significant figures is essential for scientific accuracy and communication.

Additional info: The lecture also emphasizes the importance of understanding significant figures for reporting results in chemistry, and provides practical examples for exam preparation.