Rounding Numbers, Significant Figures, and Unit Conversion in Chemistry

Rounding Numbers & Unit Conversion

Introduction

Accurate reporting of calculated values in chemistry requires understanding how to handle significant figures (SFs) and perform unit conversions. This ensures that results reflect the precision of the measurements and do not imply greater accuracy than the data supports.

Significant Figures (SFs)

Definition and Importance

• Significant figures are the digits in a measured value that are known with certainty plus one digit that is estimated.

• The number of SFs in a value indicates its precision. More SFs mean more information about the measurement.

• Calculations do not increase the number of SFs; they only propagate the uncertainty present in the original measurements.

Rules for Counting Significant Figures

• All nonzero digits are significant.

• Zeros between nonzero digits are significant.

• Leading zeros are not significant.

• Trailing zeros are significant only if there is a decimal point.

Significant Figures in Calculations

Multiplication and Division

When multiplying or dividing measured values, the result should have the same number of SFs as the value with the fewest SFs.

• Rule: The number of SFs in the result = the smallest number of SFs among the values used.

Example:

• $357 \times 12 = 4284$

• 357 has 3 SFs, 12 has 2 SFs. The result should have 2 SFs: $4.3 \times 10^3$

Another Example:

• $0.12 \times 4 = 0.48$ (0.12 has 2 SFs, 4 has 1 SF) → round to 1 SF: $0.5$

• $365 \div 12 = 30.41666...$ (365 has 3 SFs, 12 has 2 SFs) → round to 2 SFs: $30$ or $3.0 \times 10^1$

Addition and Subtraction

For addition and subtraction, the result should be reported to the same decimal place as the measurement with the least number of decimal places (i.e., the highest digit containing uncertainty).

• Rule: Report down to the highest digit that contains uncertainty among the values used.

Example:

0.378 mL

+ 42.3 mL

- 1.5833 mL

= 41.0947 mL

Rounded result: 41.1 mL (rounded to the tenths place, as 42.3 has the least decimal places)

Another Example:

170.4 lb

+ 0.07 lb

= 170.47 lb

Rounded result: 170.5 lb (rounded to the tenths place)

Combination of Multiplication/Division and Addition/Subtraction

When a calculation involves both types of operations, follow the order of operations and apply the appropriate SF rule at each step.

• Example: $5.00 \times 2.000 + 0.65$

• First, multiply: $5.00 \times 2.000 = 10.0$ (3 SFs)

• Then, add: $10.0 + 0.65 = 10.65$

• Final answer should be rounded to the tenths place (since 10.0 has the least decimal places): 10.7

Unit Conversion

Introduction

Unit conversion is essential in chemistry to express measurements in appropriate units. Conversion factors are considered to have infinite SFs, so they do not limit the SFs in the result.

Example: Calculating Mass from Density and Volume

• Given: Density of silver = 10.5 g/cm3; side of cube = 0.85 m

• Find: Mass in kg

Solution:

• Volume of cube: $V = (0.85\, \text{m})^3$

• Convert m3 to cm3: $1\, \text{m}^3 = (100\, \text{cm})^3 = 10^6\, \text{cm}^3$

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

• $\text{mass} = 10.5\, \frac{\text{g}}{\text{cm}^3} \times (0.85\, \text{m} \times 100\, \frac{\text{cm}}{\text{m}})^3$

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

• Final answer: $6.4 \times 10^3$ kg (rounded to 2 SFs, as 0.85 has 2 SFs)

Example: Fuel Efficiency Conversion

• Given: 16.0 L/100 km; convert to miles per gallon (mpg)

• Conversion factors: 1 mile = 1.609 km; 1 gallon = 3.785 L

• Set up the conversion:

Step	Calculation
Convert 100 km to miles	$100\, \text{km} \times \frac{1\, \text{mile}}{1.609\, \text{km}}$
Convert 16.0 L to gallons	$16.0\, \text{L} \times \frac{1\, \text{gal}}{3.785\, \text{L}}$
Calculate mpg	Result: 14.7 mpg (rounded to 3 SFs)

Interpretation: 14.7 mpg is not considered a good fuel efficiency for a car.

Summary Table: Significant Figures in Calculations

Operation	Rule for SFs	Example	Result (with correct SFs)
Multiplication/Division	Fewest SFs among inputs	$357 \times 12$	$4.3 \times 10^3$ (2 SFs)
Addition/Subtraction	Fewest decimal places among inputs	170.4 + 0.07	170.5 (rounded to tenths place)
Unit Conversion	Conversion factors have infinite SFs	0.85 m to cm	85 cm (2 SFs from original value)

Key Takeaways

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

• Apply the correct rules for SFs based on the type of mathematical operation.

• Unit conversions do not limit the number of SFs in the result.

• Proper reporting of results ensures scientific accuracy and integrity.

Additional info: The frog and jewelry images in the slides are decorative and do not pertain to the chemistry content.