Comprehensive Introduction to Chemistry Definitions, Units, and Mathematical Foundations

Importance and Definition of Chemistry

• Chemistry is critical to understanding the observable world. The properties of matter are fundamentally determined by the properties of atoms and molecules.

• Chemists study atoms and molecules to understand:

  • The composition of matter.

  • The structure of atoms and molecules.

  • How structure and composition result in specific properties of matter.

• Chemistry is often referred to as the Central Science because it connects physical sciences with life sciences (such as biology and medicine) and applied sciences (such as engineering).

• Chemistry is defined as the scientific study of matter and its properties, the changes that matter undergoes, and the energy associated with those changes.

Atoms, Molecules, and the Properties of Substances

• Atoms are submicroscopic particles that serve as the building blocks of matter. Although they are the fundamental units, free atoms are uncommon in nature.

• Atoms bond together in specific geometric arrangements to form molecules.

• The properties of all substances are dependent on the specific atoms that compose them.

• Properties of a substance depend on the properties of the molecules that compose it. For example, specific molecular structures determine if a substance is:

  • Drinkable and necessary for life (Water).

  • Capable of burning skin on contact and used in rocket fuel.

• Chemists seek to understand the relationship between the types of atoms in a molecule, how they bond, and resulting substance properties.

Matter and the States of Matter

• Matter is defined as anything that has mass and volume. It consists entirely of atoms and molecules.

• Composition of matter refers to the types and amounts of simpler substances that make up matter.

• Matter commonly occurs in three physical forms known as states:

  • Solid Matter: Atoms or molecules pack closely together in fixed locations. While they vibrate, they do not move past each other. This results in a fixed volume and a rigid shape.

  • Liquid Matter: Atoms or molecules pack nearly as closely as in solids, but they are free to move past one another. Consequently, liquids have a fixed volume but no fixed shape, taking the shape of their container.

  • Gaseous Matter: Atoms or molecules have a significant amount of space between them and are free to move relative to one another. Gases assume both the shape and volume of their container.

• Compressibility: Gases are compressible because of the space between particles. Liquids are not compressible.

Physical and Chemical Changes and Properties

• Matter undergoes two primary types of changes:

  • Physical Change: This alters only the state or appearance of the substance, not its composition. Atoms or molecules do not change into a different type. An example is water changing from a liquid state to a gaseous state: $H_2O_{(l)} \rightarrow H_2O_{(g)}$.

  • Chemical Change (Chemical Reaction): This involves the rearrangement of atoms, transforming original atoms or molecules into entirely different substances. It involves making a new compound.

• Physical Property: A property observed without changing the composition of a substance via chemical reaction. Examples include smell, melting point, boiling point, color, hardness, and density.

• Chemical Property: A property observed only by changing the composition of a substance through a chemical reaction. Examples include flammability, corrosiveness, and reactivity.

Energy, Work, and Systems

• Physical and chemical changes are typically accompanied by energy changes. For example, evaporating water molecules absorb energy, while burning wood releases energy.

• Energy is the capacity to do work. Work is defined as the action of a force.

• Total energy is the sum of kinetic energy and potential energy: $\text{Total Energy} = \text{Kinetic Energy} + \text{Potential Energy}$.

  • Kinetic Energy: Energy associated with an object's motion.

  • Potential Energy: Energy associated with an object's position or composition.

• Energy Conversion: A weight at the top of a building has high potential energy and no kinetic energy. As it falls, it accelerates, converting potential energy into kinetic energy. Upon hitting the ground, this is converted into thermal energy.

• Thermal Energy: A type of kinetic energy arising from the motion of the individual atoms and molecules that compose an object.

• Law of the Conservation of Energy: Energy changes form, but it is neither created nor destroyed.

• Stability: Systems with high potential energy tend to be unstable and change in ways that lower their potential energy. Chemical potential energy arises primarily from the forces between the electrically charged particles that compose atoms and molecules.

The Scientific Method

• The scientific method is a problem-solving process based on observation and experiment. Information is used to formulate a hypothesis.

• Hypothesis: A proposed explanation for a phenomenon that must be testable and falsifiable.

• Experiment: A procedure carried out to generate observations that either support or refute a hypothesis.

• Variables: Parameters in an experiment that can have more than one value. Experiments are controlled to measure the effect of one variable on another while keeping all others constant.

• Scientific Law: A brief statement that summarizes past observations and predicts future ones based on a series of similar observations.

• Scientific Theory: Based on well-established hypotheses, theories serve as models for how nature behaves. Unlike laws, theories attempt to explain why nature behaves the way it does. They represent the pinnacle of scientific knowledge and are the most accurate models of the physical world.

• Revision: Scientific knowledge is subject to continual experimentation. If observations disprove a law or theory, it is discarded or modified. Theories can never be conclusively proven because future experiments may reveal flaws.

Mass and the Law of Conservation of Mass

• Antoine Lavoisier formulated the Law of Conservation of Mass in the 18th century by observing the mass of objects before and after burning them in sealed containers.

• Law of Conservation of Mass: In a chemical reaction, matter is neither created nor destroyed.

• The total mass of the reactants must equal the total mass of the products: $\text{Total mass of reactants} = \text{Total mass of the products}$.

Units of Measurement and the Metric System

• Units are standard quantities used to specify measurements. Major systems include the U.S. customary units and the Metric system.

• Scientists use the International System of Units (SI), which is based on the metric system. The failure to use consistent units can be costly, as seen in the 1999 Mars Climate Orbiter loss, which cost $125$ million dollars.

• Standard SI Base Units:

  • Mass: kilogram ($kg$)

  • Length: meter ($m$)

  • Time: second ($s$)

  • Temperature: kelvin ($K$)

  • Amount of substance: mole ($mol$)

  • Electric current: ampere ($A$)

  • Luminous intensity: candela ($cd$)

• Metric Prefix Multipliers:

  • mega- (M): $\text{Base} \times 10^6$ ($1,000,000$)

  • kilo- (k): $\text{Base} \times 10^3$ ($1,000$)

  • deci- (d): $\text{Base} \times 10^{-1}$ ($0.1$)

  • centi- (c): $\text{Base} \times 10^{-2}$ ($0.01$)

  • milli- (m): $\text{Base} \times 10^{-3}$ ($0.001$)

  • micro- ($\mu$ or mc): $\text{Base} \times 10^{-6}$ ($0.000001$)

  • nano- (n): $\text{Base} \times 10^{-9}$ ($0.000000001$)

  • pico- (p): $\text{Base} \times 10^{-12}$ ($0.000000000001$)

Mass, Weight, and Intensive vs. Extensive Properties

• Mass: A measure of the quantity of matter. It is an extensive property.

• Weight: A measure of the gravitational pull exerted on an object.

• Extensive Property: A property that depends on the amount of a substance (e.g., mass, volume).

• Intensive Property: A property that is independent of the amount of the substance (e.g., density, melting point, boiling point).

Dimensional Analysis and Problem Solving

• Conversion factors are ratios used to express the same quantity in different units (e.g., $1\,\text{mi} = 5280\,\text{ft}$).

• Dimensional analysis uses units as a guide. Units are treated like algebraic quantities that can be multiplied and divided.

• General Problem Solving Strategy:

  1. Identify the starting point (given information).

  2. Identify the endpoint (requested information).

  3. Determine the conversion factors that relate the two.

  4. Devise and follow a plan.

• Examples of Conversions:

  • Lhasa elevation conversion: $11,995\,\text{ft} \times \frac{1\,\text{mile}}{5280\,\text{ft}} = 2.27\,\text{miles}$.

  • Feet into centimeters: $11,995\,\text{ft} \times \frac{12\,\text{in}}{1\,\text{ft}} \times \frac{2.54\,\text{cm}}{1\,\text{in}} = 365,607.6\,\text{cm}$.

  • Miles to Kilometers ($1\,\text{km} \approx 0.62\,\text{mile}$): $56\,\text{miles} \times \frac{1\,\text{km}}{0.62\,\text{mile}} = 90\,\text{km}$.

  • Pounds to Milligrams ($2.2\,\text{lb} = 1\,\text{kg}$): $7.00\,\text{lb} \times \frac{1\,\text{kg}}{2.2\,\text{lb}} \times \frac{1000\,\text{g}}{1\,\text{kg}} \times \frac{1000\,\text{mg}}{1\,\text{g}} = 3,181,818\,\text{mg}$.

Derived Units: Volume and Density

• Derived units are combinations of other units (e.g., $\text{miles per hour}$).

• Volume: A measure of space, defined as any unit of length cubed ($\text{length}^3$).

  • $1\,\text{m}^3 = 1000\,\text{dm}^3$

  • $1\,\text{dm}^3 = 1\,\text{L} = 1000\,\text{cm}^3 = 1000\,\text{mL}$

  • $1\,\text{cm}^3 = 1\,\text{mL}$

  • To convert units raised to a power, the conversion factor must also be raised to that power: $(2.54\,\text{cm})^3 = (1\,\text{in})^3 \rightarrow 16.39\,\text{cm}^3 = 1\,\text{in}^3$.

• Density: The ratio of mass to volume: $\text{Density} = \frac{\text{Mass}}{\text{Volume}}$.

  • It is a characteristic intensive physical property that varies with temperature.

  • SI Units: $kg/m^3$. More common units: $g/cm^3$ or $g/mL$.

  • Example Calculation: A lead bar with mass $9832\,\text{g}$ and volume $867\,\text{mL}$ has a density of $\frac{9832\,\text{g}}{867\,\text{mL}} = 11.34\,\text{g/mL}$.

  • Density as a conversion factor: To find the mass of $95\,\text{mL}$ of Mercury ($\text{density} = 13.6\,\text{g/cm}^3$): $95\,\text{mL} \times \frac{1\,\text{cm}^3}{1\,\text{mL}} \times \frac{13.6\,\text{g}}{1\,\text{cm}^3} = 1292\,\text{g}$.

Temperature and Heat

• Temperature ($T$): A measure of how hot or cold an object is relative to another.

• Heat: Energy that flows from an object with higher temperature to one with lower temperature.

• Major Scales:

  • Fahrenheit ($^{\circ}\text{F}$): Water freezes at $32\,^{\circ}\text{F}$ and boils at $212\,^{\circ}\text{F}$.

  • Celsius ($^{\circ}\text{C}$): Water freezes at $0.00\,^{\circ}\text{C}$ and boils at $100\,^{\circ}\text{C}$.

  • Kelvin ($K$): The SI unit. Water freezes at $273\,K$ and boils at $373\,K$. Absolute zero is $0\,K$.

• Conversion Formulas:

  • Celsius to Fahrenheit: $\text{}^{\circ}\text{C} = \frac{(\text{}^{\circ}\text{F} - 32)}{1.8}$. Example: $21\,^{\circ}\text{C}$ is $69.8\,^{\circ}\text{F}$.

  • Celsius to Kelvin: $K = \text{}^{\circ}\text{C} + 273.15$. Example: $21\,^{\circ}\text{C}$ is $294.15\,K$.

Reliability in Measurement and Significant Figures

• Scientific measurements are reported to reflect the certainty of the measurement. All digits are certain except for the last digit, which is estimated.

• Example: $1.450\,\text{ppm}$ implies $1.450 \pm 0.001\,\text{ppm}$, whereas $1.45\,\text{ppm}$ implies $1.45 \pm 0.01\,\text{ppm}$.

• Significant Figure Rules:

  • All non-zero values are significant (e.g., $536$ has $3$ sig figs).

  • Interior zeros (between non-zero digits) are significant (e.g., $6703$ has $4$ sig figs).

  • Leading zeros (left of the first non-zero digit) are NOT significant (e.g., $0.0043$ has $2$ sig figs).

  • Trailing zeros:

    • After a decimal point are significant (e.g., $50.0$ has $3$ sig figs).

    • Before an implied decimal point are generally not significant (e.g., $7000$ has $1$ sig fig).

• Exact Numbers: These have no uncertainty. Sources include accurate counting of objects, defined quantities (cma in a meter), or integral numbers in formulas ($\text{Radius} = \frac{d}{2}$).

• Scientific Notation: Used to express large/small numbers easily (e.g., $3,010,000 = 3.01 \times 10^6$; $0.00004052 = 4.052 \times 10^{-5}$).

Significant Figures in Mathematical Operations

• Multiplication and Division: The result is determined by the value with the least number of significant figures.

  • Example: $4010\,(3\,\text{sig figs}) \times 0.0031\,(2\,\text{sig figs}) \times 121.0\,(4\,\text{sig figs}) = 1500\,(2\,\text{sig figs})$.

• Addition and Subtraction: The result is determined by the value with the fewest decimal places.

  • Example: $0.03 + 1.425 + 0.3279 = 1.78$ (limited by two decimal places in $0.03$).

• Rounding:

  • Round down if the last digit is $4$ or less; round up if $5$ or more.

  • Do not round until the final answer is reached.

• Combined Operations: Follow the order of operations and track sig figs at each step.

  • Example: $\frac{(1.03 - 0.9137)}{0.699} = 0.17$ (The subtraction result $0.1163$ has $2$ sig figs, which limits the final quotient).

Accuracy, Precision, and Error

• Accuracy: How close a measured value is to the actual or true value.

• Precision: How close a series of measurements are to one another (reproducibility).

• Systematic Error: Causes measurements to be consistently too high or too low, affecting accuracy but potentially maintaining precision.

• Random Error: Has an equal probability of being too high or too low. Most measurements contain random error, but it averages out over many trials.