Measurement, Significant Figures, and Unit Conversions in Chemistry

Measurement and Problem Solving

Scientific Notation

Scientific notation is a method used to express very large or very small numbers in a compact form. It is commonly used in chemistry to handle measurements that span many orders of magnitude.

• Format: A number is written as a × 10n, where a is a number between 1 and 10, and n is an integer.

• Example: The number 0.00056 can be written as $5.6 \times 10^{-4}$.

• Application: Scientific notation simplifies calculations and clearly indicates the precision of measurements.

Reading Measurements and Estimated Values

When recording measurements, always include all certain digits plus one estimated digit. The estimated digit reflects the uncertainty in the measurement.

• Certain digits: Determined directly from the instrument's scale.

• Estimated digit: The last digit, which is an approximation between the smallest scale divisions.

• Example: If a graduated cylinder shows 24.6 mL, the '6' is estimated.

Significant Figures

Significant figures (sig figs) indicate the precision of a measured or calculated quantity. They include all certain digits plus the first uncertain digit.

What Are Significant Figures?

• Definition: All the digits in a measurement that are known with certainty plus one final digit, which is somewhat uncertain or estimated.

How to Count Significant Figures

• All nonzero digits are significant.

• Zeros between nonzero digits are significant.

• Leading zeros (zeros before the first nonzero digit) are not significant.

• Trailing zeros in a number with a decimal point are significant.

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

• Example: 0.00450 has three significant figures (4, 5, and the trailing 0).

How Many Figures to Keep in a Calculation

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

• Addition/Subtraction: The result should have the same number of decimal places as the measurement with the fewest decimal places.

• Example: $2.34 \times 1.2 = 2.8$ (rounded to two significant figures).

Rules for Rounding

• If the digit to be dropped is less than 5, leave the last retained digit unchanged.

• If the digit to be dropped is 5 or greater, increase the last retained digit by one.

• Example: Rounding 2.347 to two decimal places gives 2.35.

SI Units for Various Quantities

The International System of Units (SI) is the standard system of measurement in science. Each physical quantity has a corresponding SI unit.

Quantity	SI 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

Using Prefixes with Units

SI prefixes are used to indicate multiples or fractions of units, making it easier to express very large or small quantities.

Prefix	Symbol	Factor
kilo-	k	$10^3$
centi-	c	$10^{-2}$
milli-	m	$10^{-3}$
micro-	μ	$10^{-6}$
nano-	n	$10^{-9}$

• Example: 1 kilometer (km) = $1 \times 10^3$ meters (m).

Dimensional Analysis (Unit Conversions)

Dimensional analysis is a systematic approach to problem solving that uses conversion factors to move from one unit to another.

One-Step Conversions

• Multiply the given value by a conversion factor that cancels the original unit and introduces the desired unit.

• Example: Convert 5.0 cm to meters: $5.0\ \mathrm{cm} \times \frac{1\ \mathrm{m}}{100\ \mathrm{cm}} = 0.050\ \mathrm{m}$

Multi-Step Conversions

• Use multiple conversion factors in sequence to reach the desired unit.

• Example: Convert 1200 mm to km:

  • $1200\ \mathrm{mm} \times \frac{1\ \mathrm{m}}{1000\ \mathrm{mm}} \times \frac{1\ \mathrm{km}}{1000\ \mathrm{m}} = 0.0012\ \mathrm{km}$

Conversions Raised to a Power

• When converting squared or cubed units, raise the conversion factor to the appropriate power.

• Example: Convert 5.0 cm2 to m2:

  • $5.0\ \mathrm{cm}^2 \times \left(\frac{1\ \mathrm{m}}{100\ \mathrm{cm}}\right)^2 = 5.0\ \mathrm{cm}^2 \times \frac{1}{10,000} = 0.00050\ \mathrm{m}^2$

Density

Density is a physical property that relates the mass of a substance to its volume. It is commonly used to identify substances and as a conversion factor in calculations.

How to Calculate Density

• Formula: $\mathrm{Density} = \frac{\mathrm{Mass}}{\mathrm{Volume}}$

• Units: Commonly expressed in g/cm3 or kg/m3.

• Example: If a sample has a mass of 10.0 g and a volume of 2.0 cm3, its density is $\frac{10.0\ \mathrm{g}}{2.0\ \mathrm{cm}^3} = 5.0\ \mathrm{g/cm}^3$.

Using Density as a Conversion Factor

• Density can be used to convert between mass and volume.

• Example: To find the mass of 3.0 cm3 of a substance with density 2.0 g/cm3:

  • Mass = Density × Volume = $2.0\ \mathrm{g/cm}^3 \times 3.0\ \mathrm{cm}^3 = 6.0\ \mathrm{g}$