Measurement and Problem Solving in Chemistry

Measurement and Problem Solving

Measured vs. Exact Numbers

In chemistry, it is important to distinguish between measured and exact numbers. Measured numbers are obtained when a quantity is determined using a measuring instrument and always contain some degree of uncertainty. Exact numbers are obtained by counting or by definition and have no uncertainty.

• Measured numbers: Examples include 5.25 m, 500 L, 75 g. These are subject to measurement error.

• Exact numbers: Examples include 75 m&m’s in a bowl, 24 students in a class. These are counted and have no uncertainty.

Additional info: Exact numbers are also found in defined relationships, such as 1 dozen = 12 items.

Uncertainty in Measurements

All measurements have a degree of uncertainty due to limitations in the measuring instrument and the observer. The last digit in a measurement is always uncertain.

• Never write a number without units if units are available (e.g., 3.1 miles, not just 3.1).

• In the measurement 19.257 mL, the digit 7 is uncertain, while all other digits are certain.

Measurement and Units

Scientific measurements use standardized units. There are two main systems: the English system and the metric system. The metric system is preferred in science due to its use of powers of ten, making conversions straightforward.

• English system: Uses units like feet, miles, pounds, etc., with irregular conversion factors.

• Metric system: Uses units like meters, liters, grams, etc., with conversions based on powers of ten.

Accuracy and Precision

Accuracy refers to how close a measurement is to the true value, while precision refers to how close repeated measurements are to each other. Both are important in scientific measurements.

• Accuracy: Measurements close to the actual value (e.g., 4.20 cm, 4.25 cm, 4.30 cm for a nail that is actually 4.25 cm).

• Precision: Measurements that are close to each other, even if not accurate (e.g., three measurements of 4.10 cm).

• Both: Three measurements of 4.25 cm are both accurate and precise.

Length Measurements

Length is measured using rulers or meter sticks, typically in meters (m) or centimeters (cm). The degree of uncertainty depends on the precision of the measuring instrument.

• Rulers with more hash marks (smaller divisions) provide more precise measurements.

Mass Measurements

Mass is the amount of matter in a substance and is measured in grams (g) using a balance. The uncertainty depends on the sensitivity of the balance.

• On Earth, mass and weight are often used interchangeably, but technically, weight is the force of gravity on an object.

• Balances measure mass by comparing the object to a standard mass, canceling out gravity's effect.

Volume Measurements

Volume is the amount of space occupied by a substance and is measured in liters (L), milliliters (mL), or cubic centimeters (cm3). Various laboratory equipment is used for measuring volume, such as graduated cylinders, pipets, and burets.

• 1 mL = 1 cm3 (interchangeable)

Graduated Cylinder

Graduated cylinders are used to measure liquid volume. Always read the volume at the bottom of the meniscus (the curve formed by the liquid).

Pipets

Pipets (or pipettes) are used to deliver a fixed volume of liquid. Volumetric pipets are highly accurate, while Pasteur pipets are used for transferring small amounts.

Buret

Burets are long, calibrated glass tubes with a stopcock for precise delivery of liquids, commonly used in titrations.

Exponential Numbers and Scientific Notation

Scientific notation is a convenient way to express very large or very small numbers using powers of ten. The format is N x 10n, where N is a number between 1 and 10, and n is an integer.

• Positive exponent: Move decimal to the left (e.g., 5,983 = 5.983 x 103).

• Negative exponent: Move decimal to the right (e.g., 0.00034 = 3.4 x 10-4).

Significant Digits (Significant Figures)

Significant figures (sig figs) reflect the precision of a measurement. The last digit is always uncertain. Rules for determining significant figures:

• All non-zero digits are significant.

• Zeroes between non-zero digits are significant.

• Zeroes to the right of a decimal point and after a non-zero digit are significant.

• Leading zeroes are not significant; they only locate the decimal point.

• Trailing zeroes before a decimal point are ambiguous; use scientific notation to clarify.

Rounding and Significant Figures

When rounding, look at the right-most non-significant digit:

• If 5 or greater, round up the last significant digit.

• If 4 or less, leave the last significant digit unchanged.

Calculations with Significant Figures

Addition and Subtraction

The answer should have the same number of decimal places as the measurement with the least decimal places.

Multiplication and Division

The answer should have the same number of significant figures as the measurement with the least significant figures.

Unit Equations and Unit Factors

Unit conversion factors (unit factors) are ratios of two equivalent quantities. They are used to convert between units in calculations.

• Example: 1 dollar = 10 dimes gives two unit factors: (1 dollar/10 dimes) or (10 dimes/1 dollar).

Exactly Equivalent Relationships

Some unit relationships are defined exactly and have an infinite number of significant figures (e.g., 1 foot = 12 inches). English-metric conversions are not exact and must consider significant figures.

Unit Analysis Problem Solving

Unit analysis (dimensional analysis) is a method for solving problems by carrying units through the calculation. This ensures correct units and helps avoid mistakes.

• Write the units for the answer.

• Arrange unit factors so that all units cancel except the desired unit.

• Check for correct units and significant figures in the final answer.

Metric Prefixes and Conversions

Metric prefixes indicate powers of ten and are paired with base units to express different magnitudes. Common prefixes include kilo- (103), centi- (10-2), milli- (10-3), micro- (10-6), and nano- (10-9).

• 1 km = 103 m

• 1 mm = 10-3 m

• 1 µs = 10-6 s

Volume by Calculation

The volume of a regular solid can be calculated using the formula:

$V = l \times w \times h$

where l is length, w is width, and h is height. All units must be the same before multiplying.

Volume by Displacement

The volume of an irregular solid can be determined by measuring the amount of liquid it displaces in a graduated cylinder:

$\text{Volume of solid} = \text{Volume}_{\text{final}} - \text{Volume}_{\text{initial}}$

Density

Density is the amount of mass per unit volume and is calculated as:

$d = \frac{\text{mass}}{\text{volume}}$

• Units: g/cm3 or g/mL for liquids and solids; g/L for gases.

• The density of water is 1.00 g/mL.

• If an object sinks in water, its density is greater than 1.00 g/mL; if it floats, its density is less.

The Percent Concept

Percent expresses the amount of a single quantity in relation to the whole sample:

$\text{Percent} = \frac{\text{Part}}{\text{Whole}} \times 100$

• Example: If a ring contains 20.0 g of gold and 14.3 g of silver, the percent gold is $\frac{20.0}{20.0 + 14.3} \times 100$.