Essentials: Units, Measurements, and Problem Solving

Essentials: Units, Measurements, and Problem Solving

Measurement Types: Qualitative and Quantitative

In chemistry, observations and measurements are fundamental to understanding and describing matter. Measurements can be classified as either qualitative or quantitative.

• Qualitative observations: Descriptive in nature, such as changes in color or physical state. They do not involve numbers.

• Quantitative observations: Involve measurements and numerical values obtained from instruments, glassware, or other measuring devices. These can include counted values (e.g., number of cats per household).

• The type of measurement (qualitative vs. quantitative) determines the statistical methods used in data analysis.

What Are Measurements?

All measurements consist of two essential parts:

• Scalar or dimensional unit: The unit may be from the International System of Units (SI) or the English system. For example, 5.9 m means 5.9 meters, and 3.7 kg means 3.7 kilograms.

• Numerical value: Reflects the precision of the instrument or glassware used. For example, 25.0 cm or 1.00 ft.

Quantitative Measurement Errors

Errors in measurement can be classified as:

• Systematic (Determinate) Error: Consistent error in the same direction (either always higher or lower than the true value).

• Random (Indeterminate) Error: Error with equal probability of being higher or lower than the true value; difficult to correct or identify the source.

Standard Units of Measure (SI)

The SI system is the standard for scientific measurements:

• Length: meter (m)

• Mass: kilogram (kg)

• Time: second (s)

• Temperature: kelvin (K)

• Amount of substance: mole (mol), $6.02 \times 10^{23}$ units

• Electric current: ampere (A)

• Luminous intensity: candela (cd)

Metric System: Prefix Multipliers

Prefix multipliers are used to express units in powers of ten:

Prefix	Symbol	Decimal Equivalent	Power of 10
mega-	M	1,000,000	Base × 106
kilo-	k	1,000	Base × 103
deci-	d	0.1	Base × 10-1
centi-	c	0.01	Base × 10-2
milli-	m	0.001	Base × 10-3
micro-	μ or mc	0.000001	Base × 10-6
nano-	n	0.000000001	Base × 10-9
pico-	p	0.000000000001	Base × 10-12

Temperature Scales and Calculations

Temperature can be measured in Fahrenheit (°F), Celsius (°C), or Kelvin (K). The relationships between these scales are:

• Celsius to Kelvin: $T(K) = t(°C) + 273.15$

• Celsius to Fahrenheit: $°F = 1.8(°C) + 32$

Example: Convert 37.00°C to Kelvin: $37.00 + 273.15 = 310.15\ K$

Example: Convert 77 K to Celsius: $77 - 273.15 = -196\ ^\circ C$

Reliability of Measurements: Precision, Accuracy, and Uncertainty

• Precision: Closeness of a series of measurements to each other. High precision does not guarantee accuracy.

• Accuracy: Closeness of a measurement to the true or accepted value. High accuracy does not guarantee precision.

• Uncertainty: The estimated amount by which a measured value may differ from the true value, often reported as (± value).

Example: 23.45 ± 0.05 mL means the true value lies between 23.40 and 23.50 mL.

Significant Figures and Measurements

Significant figures reflect the precision of a measurement. Conversion factors are treated as exact values and have an infinite number of significant figures.

• Examples of exact values: 1 in = 2.54 cm, 100 pennies = $1, 1 mole = $6.02 \times 10^{23}$ units

Significant Figure Rules

• All nonzero digits are significant (e.g., 536 has three significant figures).

• Zeroes between nonzero digits are significant (e.g., 6703 has four significant figures).

• Leading zeroes are not significant (e.g., 0.0043 has two significant figures).

• Trailing zeroes after a nonzero digit and before a decimal point are not significant (e.g., 7000 has one significant figure).

• Trailing zeroes after a decimal point are significant (e.g., 50.0 has three significant figures).

Measurements and Significant Figures: Examples

• 50,003 km: five significant figures

• 400 L: one significant figure

• 0.04450 m: four significant figures

• 100 cm = 1 m: unlimited significant figures (exact value)

• 1.000 × 103: four significant figures

• 3.050 × 10-1 g: four significant figures

Precision of Laboratory Glassware: Precision of the Last Digit

• For instruments with a scale, the last digit is estimated between marks.

• Example: If the meniscus is between 4.5 and 4.6 mL, estimate to 4.56 mL.

Significant Figures and Scientific Notation

• 3010 in scientific notation: $3.01 \times 10^3$ (three significant figures)

• 0.0310 in scientific notation: $3.10 \times 10^{-2}$ (three significant figures)

Mathematical Operations and Significant Figures

• Multiplication and Division: The answer has the same number of significant figures as the measurement with the fewest significant figures.

• Addition and Subtraction: The answer has the same number of decimal places as the measurement with the fewest decimal places.

Multiplication and Division Operation Examples

• 3.56 cm × 4 cm = 14.24 cm2 → report as 10 cm2

• 2.050 mL × 5.0 mL = 10.25 mL2 → report as $1.0 \times 10^1$ mL2

• 45.0 cm / 9 cm = 5.0 cm → report as 5 cm

Addition and Subtraction Operation Examples

• 23.467 in + 313.21 in = 336.68 in (report to two decimal places)

• 456.32 cm - 0.68 cm = 456 cm (report to the nearest whole number)

Density: An Intensive Physical Property of Matter

Density is a key property in chemistry, defined as mass per unit volume:

Formula: $\text{Density} = \frac{\text{mass}}{\text{volume}}$

• Intensive property: Independent of the amount of substance.

• Extensive properties: Mass and volume are extensive, dependent on the amount.

• Density of liquids and gases is affected by temperature.

Practice Problem: Density

• Problem: Mercury (Hg) has a density of 13.6 g/cm3. What is the mass in grams for 95 mL of mercury?

• Strategy:

  1. Convert mL to cm3 (1 mL = 1 cm3).

  2. Rearrange the equation to solve for mass: $m = d \times V$

  3. Substitute values: $m = (13.6\ \text{g/cm}^3) \times (95\ \text{cm}^3) = 1292\ \text{g} = 1.3 \times 10^3\ \text{g}$

Introduction to Energy and Its Units

Energy is the capacity to do work, and is involved in all physical and chemical changes.

• Work: The action of a force applied across a distance.

• Electrostatic force: The push or pull on objects with an electrical charge.

Energy Overview

• Conservation of energy: The First Law of Thermodynamics states that energy is conserved in the universe.

• Kinetic energy: Energy associated with movement. $KE = \frac{1}{2}mv^2$

• Potential energy: Energy associated with position or composition.

• Energy can be converted from one form to another (e.g., chemical to mechanical).

Energy Terminology

• System: The part of the universe under study (e.g., a chemical reaction).

• Surroundings: Everything outside the system.

• Universe: System + surroundings.

Exothermic vs. Endothermic: Directionality of Heat (Energy) Flow

• Endothermic: Heat transfers from surroundings to the system. The system's energy increases, surroundings' energy decreases, and the temperature of the surroundings decreases.

• Exothermic: Heat transfers from the system to the surroundings. The system's energy decreases, surroundings' energy increases, and the temperature of the surroundings increases.

Energy Units and Energy Conversion Factors

• Calorie (cal): The amount of heat needed to raise the temperature of 1 g of water by 1°C. $4.184\ \text{J} = 1\ \text{cal}$

• Kilocalorie (kcal): $1\ \text{kcal} = 1000\ \text{cal} = 4184\ \text{J}$

• Joule (J): SI unit of energy. $4.184\ \text{J} = 1\ \text{cal}$

• Kilojoule (kJ): $1\ \text{kJ} = 1000\ \text{J}$

• Diet Calorie (Cal or C): $1\ \text{Cal} = 1\ \text{kcal} = 1000\ \text{cal}$

• Kilowatt-hour (kWh): $1\ \text{kWh} = 3.60 \times 10^6\ \text{J}$

Dimensional Analysis: Strategy for Solving Problems

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

• Conversion factors are relationships between two units, often derived from equivalence statements (e.g., 1 inch = 2.54 cm).

• Arrange conversion factors so that units cancel appropriately.

• Multiple conversion factors can be strung together as needed.

Example: $2.00\ \text{ft} \times \frac{12\ \text{in}}{1\ \text{ft}} = 24.0\ \text{inches}$

Strategy for Solving Problems

1. Sort out the information: Identify given quantities and units, and what needs to be calculated. Identify necessary mathematical relationships or definitions.

2. Devise a strategy (plan): Determine if each step involves a conversion factor or equation. Ensure units cancel properly.

3. Solve the problem: Apply mathematical operation rules for significant figures and cancel units as you proceed.

4. Check the answer: Ensure the final unit is correct and the value is reasonable.

How to Use Dimensional Analysis

• Arrange conversion factors so the starting unit cancels.

• String conversion factors as needed to reach the desired unit.

• General formula: Given unit × (find unit / given unit) = find unit

• Example: $200 \times \frac{12\ \text{in}}{1\ \text{ft}} = 2400\ \text{inches}$