Measurement and Problem Solving in Chemistry: Scientific Notation, Significant Figures, and Unit Conversions

Measurement and Problem Solving

Uncertainty in Measurements

All scientific measurements contain some degree of uncertainty, which is indicated by the last reported digit. Understanding and reporting uncertainty is crucial in scientific communication and decision-making.

• Uncertainty: The uncertainty in a measurement is reflected in the last digit reported. For example, a temperature increase of 0.6 °C is reported as 0.6 ± 0.1 °C, meaning the actual value could be between 0.5 °C and 0.7 °C.

• Significance: The degree of certainty can influence important decisions, such as those in environmental policy.

• Estimated Digits: In any measurement, all digits except the last are certain; the last digit is estimated.

Scientific Notation

Structure of Scientific Notation

Scientific notation is a method for expressing very large or very small numbers in a compact form. It consists of two parts:

• Decimal Part: A number between 1 and 10.

• Exponential Part: 10 raised to an integer exponent, n.

For example: $1.2 \times 10^{-10}$

Interpreting Exponents

The exponent in scientific notation indicates how many times the decimal part is multiplied or divided by 10.

• Positive Exponent: Multiply by 10 n times.

• Negative Exponent: Divide by 10 n times.

Examples:

• $1 \times 10^1 = 10$

• $1 \times 10^2 = 100$

• $1 \times 10^3 = 1000$

• $10^{-1} = \frac{1}{10} = 0.1$

• $10^{-2} = \frac{1}{10 \times 10} = 0.01$

• $10^{-3} = \frac{1}{10 \times 10 \times 10} = 0.001$

Converting Numbers to Scientific Notation

To convert a number to scientific notation:

1. Find the decimal part (a number between 1 and 10).

2. Determine the exponent by counting how many places the decimal point moves.

3. Multiply the decimal part by $10^n$, where n is the number of places moved.

Examples:

• $5983 = 5.983 \times 10^3$ (decimal moved 3 places to the left; exponent is positive)

• $0.00034 = 3.4 \times 10^{-4}$ (decimal moved 4 places to the right; exponent is negative)

Significant Figures

Identifying Significant Figures

Significant figures (sig figs) reflect the precision of a measurement. The rules for determining which digits are significant are:

• Nonzero digits: Always significant.

• Interior zeros: Zeros between nonzero digits are significant.

• Trailing zeros: Zeros to the right of a nonzero digit and after a decimal point are significant.

• Leading zeros: Zeros to the left of the first nonzero digit are not significant; they only locate the decimal point.

• Ambiguous zeros: Trailing zeros before an implied decimal point are ambiguous and should be avoided.

Exact numbers (from counting or defined quantities) have an unlimited number of significant figures.

Examples of Significant Figures

• 35 (two significant figures)

• 40.00 (four significant figures)

• 0.0400 (three significant figures)

• $5.83 \times 10^3$ (three significant figures)

• Dozen = 12 (unlimited significant figures)

Significant Figures in Calculations

When performing calculations, the result should reflect the precision of the input data.

• Multiplication/Division: The result has the same number of significant figures as the factor with the fewest significant figures.

• Addition/Subtraction: The result has the same number of decimal places as the quantity with the fewest decimal places.

• Rounding: Round only the final answer, not intermediate steps, to avoid cumulative errors. If the last digit dropped is 4 or less, round down; if 5 or more, round up.

Example (Multiplication):

• $2.02 \times 89.665 \times 0.10 = 18.1$ (rounded to two significant figures)

Example (Addition):

• $5.74 + 0.823 + 2.651 = 9.21$ (rounded to two decimal places)

Units of Measurement

SI Base Units

The International System of Units (SI) is the standard for scientific measurements. The main SI base units are:

Quantity	Unit	Symbol
Length	meter	m
Mass	kilogram	kg
Time	second	s
Temperature	kelvin	K
Amount of substance	mole	mol
Electric current	ampere	A
Luminous intensity	candela	cd

Length, Mass, and Time Standards

• Meter: Defined as the distance light travels in vacuum in $1/299,792,458$ seconds.

• Kilogram: Defined by the mass of a platinum-iridium block kept at the International Bureau of Weights and Measures.

• Second: Defined by 9,192,631,770 periods of radiation from a cesium-133 atom.

Mass vs. Weight

• Mass: The quantity of matter in an object; does not depend on gravity.

• Weight: The force of gravity acting on mass; depends on gravitational field.

Prefix Multipliers

Prefix multipliers are used to express units that are much larger or smaller than the base unit.

Symbol	Meaning	Multiplier
T	trillion	1,000,000,000,000
G	billion	1,000,000,000
M	million	1,000,000
k	thousand	1,000
c	hundredth	0.01
m	thousandth	0.001
μ	millionth	0.000001
n	billionth	0.000000001
p	trillionth	0.000000000001

Derived Units

Derived units are formed from combinations of base units. For example, volume is a derived unit:

• Volume: $1 \text{ m}^3$, $1 \text{ cm}^3$, $1 \text{ mm}^3$

Unit Conversions and Dimensional Analysis

Dimensional Analysis

Dimensional analysis is a method for converting between units using conversion factors. Units are treated as algebraic quantities that can be multiplied, divided, and canceled.

• Always write numbers with their units.

• Include units in all calculations and ensure they flow logically from start to finish.

Conversion Factors

Conversion factors are ratios of equivalent quantities used to convert from one unit to another.

• Example: $1 \text{ in.} = 2.54 \text{ cm}$ (exact)

• Conversion factor: $\frac{2.54 \text{ cm}}{1 \text{ in.}}$ or $\frac{1 \text{ in.}}{2.54 \text{ cm}}$

Always check that the final units are correct and the magnitude makes sense.

Solution Maps

A solution map is a visual outline of the steps required to solve a unit conversion problem.

• Example: To convert inches to centimeters: $\text{inches} \rightarrow \text{centimeters}$ using $1 \text{ in.} = 2.54 \text{ cm}$

General Problem-Solving Strategy

1. Sort: Organize the given information.

2. Strategize: Create a solution map.

3. Solve: Perform calculations, paying attention to significant figures and units.

4. Check: Ensure the answer makes physical sense and units are correct.

Multistep Unit Conversions

For multistep conversions, each step should use a conversion factor that cancels the previous unit and introduces the next.

• Example: $\text{cm} \rightarrow \text{in.} \rightarrow \text{ft}$

Units Raised to a Power

When converting units raised to a power, the conversion factor must also be raised to that power.

• Example: $1 \text{ in.} = 2.54 \text{ cm}$, so $1 \text{ in.}^3 = (2.54 \text{ cm})^3 = 16.387 \text{ cm}^3$

Density

Definition and Calculation

Density is a physical property defined as the mass of a substance divided by its volume.

• Formula: $d = \frac{m}{V}$

• Units: Commonly expressed in $\text{g/cm}^3$ or $\text{g/mL}$

Example: A liquid sample with mass 27.2 g and volume 22.5 mL has density $d = \frac{27.2 \text{ g}}{22.5 \text{ mL}} = 1.21 \text{ g/mL}$

Density as a Conversion Factor

Density can be used to convert between mass and volume.

• Example: If density is $1.32 \text{ g/cm}^3$, to find the volume for 68.4 g: $\text{Volume} = \frac{68.4 \text{ g}}{1.32 \text{ g/cm}^3} = 51.8 \text{ cm}^3$

Densities of Common Substances

Substance	Density (g/cm3)
Charcoal, Oak	0.57
Ethanol	0.789
Ice	0.92
Aluminum	2.7
Titanium	4.50
Iron	7.86
Copper	8.96
Lead	11.4
Gold	19.3
Platinum	21.4

Summary and Learning Objectives

• Express very large and very small numbers using scientific notation.

• Report measured quantities with the correct number of digits.

• Determine significant figures in measurements and calculations.

• Round results appropriately.

• Convert between units, including those raised to a power.

• Calculate density and use it as a conversion factor.

Example Application: The importance of correct unit conversions is illustrated by the loss of the Mars orbiter due to a unit conversion error between kilometers and feet.