General Chemistry I Notes

Fundamental Principles of Matter and Its States

• Definition of Matter, Mass, Volume, and Chemistry:

  • Matter is defined as anything that possesses mass ($m$) and occupies space (volume, $V$).

  • Chemistry is the scientific discipline dedicated to understanding matter, its composition, structure, transformation, and fundamental properties.

  • Matter is studied from a particulate perspective, meaning it is composed of microscopic and submicroscopic particles, including atoms, molecules, and subatomic particles (protons, neutrons, and electrons).

  • How these constituent particles assemble and interact directly dictates the macroscopic physical and chemical properties of matter.

• Particulate Building Blocks:

  • Atoms: The basic, submicroscopic particles that serve as the fundamental building blocks of ordinary matter.

  • Molecules: Chemical species formed when two or more atoms bind together in specific geometric arrangements via chemical bonds.

  • Particle-level interactions dictate macro-level behavioral characteristics across all physical states.

• The Three States of Matter:

  • Solid Matter:

  • Particle Arrangement: Atoms or molecules are packed closely together in fixed, rigid positions.

  • Particle Motion: Particles vibrate about fixed equilibrium positions but cannot slide past or move around one another.

  • Macroscopic Properties: Solids possess a fixed volume and a rigid, definite shape.

  • Compressibility: Incompressible because particles are already in direct physical contact.

  • Examples: Ice ($H_2O(s)$), aluminum ($Al$), and diamond ($C$).     

    

  • Liquid Matter:

  • Particle Arrangement: Atoms or molecules are packed about as closely as in solids, but inter-particle forces allow them spatial mobility.

  • Particle Motion: Particles are free to flow and move relative to one another.

  • Macroscopic Properties: Liquids possess a fixed, definite volume, but no fixed shape. Their fluidity allows them to assume the shape of their container.

  • Compressibility: Generally incompressible due to minimal intermolecular spacing.

  • Examples at Room Temperature: Water ($H_2O(l)$), ethyl alcohol ($C_2H_5OH$), and gasoline.     

    

  • Gaseous Matter:

  • Particle Arrangement: Large inter-particle distances exist between atoms or molecules relative to particle sizes.

  • Particle Motion: Particles move completely freely and independently in continuous random straight-line trajectories.

  • Macroscopic Properties: Gases have neither a fixed volume nor a fixed shape; they expand to fill the entirety of any container.

  • Compressibility: Highly compressible because the vast empty space between gas particles can be decreased significantly under applied pressure.

  • Example: Oxygen gas ($O_2(g)$) stored under high pressure in a compressed gas cylinder.     

    

Classification of Matter by Composition

• Pure Substances vs. Mixtures:

  • The classification of matter begins by evaluating particle uniformity and composition variability.

  • Pure Substance: Matter composed of only one single type of particle, exhibiting an invariant, fixed composition across all samples.

  • Mixture: Matter composed of two or more distinct types of particles combined in proportions that can vary from sample to sample.

• Categories of Pure Substances:

  • Elements:

  • Simplest pure substances that cannot be decomposed into simpler substances by chemical processes.

  • Consist of only a single type of atom (e.g., helium gas $He$, copper metal $Cu$, fluorine gas $F_2$).

  • Function as the primary chemical building blocks of all matter.     

    

  • Compounds:

  • Pure substances composed of two or more distinct elements chemically combined in fixed, definite stoichiometric ratios.

  • Can be broken down into constituent elements through chemical reactions.

  • All molecular or formula units of a pure compound are identical in composition and geometric structure (e.g., pure water $H_2O$, table salt $NaCl$, methyl alcohol $CH_3OH$).     

    

• Categories of Mixtures:

  • Heterogeneous Mixtures:

  • Mixtures in which components are not uniformly distributed throughout the sample volume.

  • Sample compositions vary depending on location, creating distinct physical regions or phases with differing properties.

  • Examples: Wet sand, iced coffee with floating ice cubes, oil and water salad dressing, freshly opened champagne (containing gas bubbles dispersed in liquid), and chocolate chip cookies.     

    

  • Homogeneous Mixtures (Solutions):

  • Mixtures that are thoroughly and uniformly mixed at the particulate level, exhibiting consistent composition throughout.

  • Any aliquot taken from any region of the mixture yields an identical composition.

  • Examples: Tea with dissolved sugar, air in a room, hot chocolate, flat soda, ale, and seawater.

• Classification Classification Master Reference Table:

  | Substance | Element / Compound / Mixture | Homogeneous / Heterogeneous |   | :--- | :--- | :--- |   | Helium Gas ($He$) | Element | N/A (Pure Substance) |   | Water ($H_2O$) | Compound | N/A (Pure Substance) |   | Lemonade | Mixture | Homogeneous / Heterogeneous |   | Ale | Mixture | Homogeneous |   | Seawater | Mixture | Homogeneous |   | Air in a Room | Mixture | Homogeneous |   | Table Salt ($NaCl$) | Compound | N/A (Pure Substance) |   | Methyl Alcohol ($CH_3OH$) | Compound | N/A (Pure Substance) |   | Fluorine Gas ($F_2$) | Element | N/A (Pure Substance) |   | Copper ($Cu$) | Element | N/A (Pure Substance) |   | Iced Coffee | Mixture | Heterogeneous |   | Oil and Water Salad Dressing | Mixture | Heterogeneous |   | Freshly Opened Champagne | Mixture | Heterogeneous |   | Chocolate Chip Cookie | Mixture | Heterogeneous |   | Hot Chocolate | Mixture | Homogeneous |   | Flat Soda | Mixture | Homogeneous |

Physical vs. Chemical Changes and Signs of Chemical Reaction

• Physical Changes:

  • Process altering only state, shape, phase, or physical appearance without modifying underlying chemical composition.

  • The chemical identity of the constituent atoms or molecules remains completely invariant.

  • Examples: Sublimation of solid dry ice ($CO_2(s) \rightarrow CO_2(g)$), dissolution of milk powder in hot tea, mixing fruit punch powder with liquid water, melting of solid candle wax.

• Chemical Changes:

  • Process altering the fundamental chemical composition of matter.

  • Atoms undergo bond breaking and bond formation to rearrange into chemically distinct product substances.

  • Examples: Combustion of gasoline inside an internal combustion engine, burning of plant leaves, fermentation of grape sugars into ethanol wine, rust formation on iron nails ($Fe \rightarrow Fe_2O_3$   

    

• Comprehensive Physical vs. Chemical Practice Table:

  | Phenomenon / Process | Physical Change | Chemical Change |   | :--- | :---: | :---: |   | Dry ice subliming | $✓$ | |   | Gasoline combusting in an engine | | $✓$ |   | Making fruit punch from powder and water | $✓$ | |   | Leaves burning | | $✓$ |   | Lighting a candle (tricky: melting wax is physical, wax combustion is chemical) | $✓$ | $✓$ |   | Grapes fermenting into wine | | $✓$ |   | Milk dissolving in hot tea | $✓$ | |

• Seven Universal Signs / Indicators of Chemical Change:

  1. Gas Evolution (Bubbling / Effervescence): Release of gas bubbles in a liquid medium without boiling.

  2. Precipitation: Formation of an insoluble solid phase from a liquid solution reaction.

  3. Color Change: Distinct, unexpected shift in light absorption profile.

  4. Temperature Change: Spontaneous release (exothermic) or absorption (endothermic) of thermal energy.

  5. Emission of Light: Bioluminescence or incandescence generated during chemical structural reorganization.

  6. Volume Change: Non-additive volume shifts due to modified density and bonding structure.

  7. Change in Smell or Taste: Generation of new volatile olfactory or gustatory chemical molecules.

The Scientific Method

• Empirical Foundation:

  • Scientific knowledge is inherently empirical, meaning it is derived directly from objective observation and physical experimentation.

  • The scientific method provides a systematic process for understanding natural phenomena by observing nature, proposing testable mechanisms, and performing rigorous experimental verification.

• Core Components of the Scientific Method:

  • Observations (Data):

  • Initial records and measurements detailing characteristics or behavior of nature.

  • Qualitative Data: Observational descriptions that are subjective (e.g., color, physical appearance, texture).

  • Quantitative Data: Objective empirical measurements captured using calibrated instrumentation paired with standardized units.

  • Hypothesis Formulation:

  • A tentative, testable interpretation or explanation of observed natural phenomena.

  • Historical Metaphor/Example: Antoine Lavoisier explained combustion observations by hypothesizing that burning substances chemically combine with a specific gaseous component of air (oxygen).

  • A valid hypothesis must be falsifiable, meaning it must make predictions capable of being proven wrong through experimental testing.

  • Scientific Law:

  • A brief, concise statement summarizing vast bodies of past empirical observations and predicting future outcomes under specific conditions.

  • Example: The Law of Conservation of Mass, which states that matter is neither created nor destroyed during a chemical reaction.

  • Unlike legislative laws, scientific laws cannot be violated by choice; however, they operate within specific defined boundary conditions.

  • A law describes what happens in nature.

  • Scientific Theory:

  • A comprehensive, well-substantiated framework of hypotheses providing a deep explanation of how and why natural phenomena occur.

  • Examples: Charles Darwin's Theory of Evolution, Albert Einstein's General Theory of Relativity, John Dalton's Atomic Theory.

  • Theories are continuously validated by experimental testing and used to predict future experimental outcomes.

  • A theory can never be absolute or 100% conclusively proven, as novel experimental techniques or future observations retain the potential to reveal flaws or boundary limitations.

Measurement Systems and SI Base Units

• Importance of Standardized Measurement:

  • Numerical data lacking units are scientifically meaningless because units state the relative scale and standardized framework of the quantity measured.

• Major Unit Systems:

  • Metric System: Utilized globally in scientific and general applications; built systematically upon base units related by powers of $10$.

  • English System: Utilized in commercial applications within the United States; composed of historically arbitrary, functionally unrelated conversion ratios (e.g., $1 \text{foot} = 12 \text{inches} = 0.333 \text{yards} = \frac{1}{5280} \text{miles}$).

  • International System of Units (SI): Global scientific standard derived from the French Système International d'Unités.

• SI Base Units Table:

  

  | Physical Quantity | Name of SI Unit | Symbol |   | :--- | :--- | :---: |   | Length | Meter | $\text{m}$ |   | Mass | Kilogram | $\text{kg}$ |   | Time | Second | $\text{s}$ |   | Temperature | Kelvin | $\text{K}$ |   | Amount of Substance | Mole | $\text{mol}$ |   | Electric Current | Ampere | $\text{A}$ |   | Luminous Intensity | Candela | $\text{cd}$ |

• Metric System Prefixes (Mandatory Memorization Set):

  

  | Prefix | Symbol | Exponential Meaning | Standard Decimal Value | Unit Equation Representative Ratio ($\text{x} = \text{g, m, L}$) |   | :--- | :---: | :---: | :--- | :--- |   | Mega | $\text{M}$ | $10^6$ | $1,000,000$ | $1\u2009\text{Mx} = 10^6\u2009\text{x}$ |   | Kilo | $\text{k}$ | $10^3$ | $1,000$ | $1\u2009\text{kx} = 10^3\u2009\text{x}$ |   | Deka | $\text{da}$ | $10^1$ | $10$ | $1\u2009\text{dax} = 10^1\u2009\text{x}$ |   | Deci | $\text{d}$ | $10^{-1}$ | $0.1$ | $1\u2009\text{dx} = 10^{-1}\u2009\text{x}$ |   | Centi | $\text{c}$ | $10^{-2}$ | $0.01$ | $1\u2009\text{cx} = 10^{-2}\u2009\text{x}$ |   | Milli | $\text{m}$ | $10^{-3}$ | $0.001$ | $1\u2009\text{mx} = 10^{-3}\u2009\text{x}$ |   | Micro | $\mu$ | $10^{-6}$ | $0.000001$ | $1\u2009\mu\text{x} = 10^{-6}\u2009\text{x}$ |   | Nano | $\text{n}$ | $10^{-9}$ | $0.000000001$ | $1\u2009\text{nx} = 10^{-9}\u2009\text{x}$ |

Measurement Reliability: Precision, Accuracy, and Error Analysis

• Core Definitions:

  • Accuracy: The closeness of a measured value to the actual, true, or accepted reference value ($\text{Accuracy} = \text{Correctness}$).

  • Precision: The closeness of a series of repeated measurements to one another, reflecting measurement reproducibility ($\text{Precision} = \text{Consistency}$).

• Error Classifications:

  • Random Error: Unpredictable, equal-probability error causing measurements to fluctuate high or low relative to the true value. Generally stems from human estimation or minor environmental fluctuations.

  • Systematic Error: Predictable error shifting measurements consistently in one direction (always too high or always too low). Stems from miscalibrated balances or faulty equipment.

• Weighing Case Study Analysis (True Mass of Lead Block = $10.00\u2009\text{g}$):

  

  | Trial | Student A Data | Student B Data | Student C Data |   | :--- | :---: | :---: | :---: |   | Trial 1 | $10.49\u2009\text{g}$ | $9.78\u2009\text{g}$ | $10.03\u2009\text{g}$ |   | Trial 2 | $9.79\u2009\text{g}$ | $9.82\u2009\text{g}$ | $9.99\u2009\text{g}$ |   | Trial 3 | $9.92\u2009\text{g}$ | $9.75\u2009\text{g}$ | $10.03\u2009\text{g}$ |   | Trial 4 | $10.31\u2009\text{g}$ | $9.80\u2009\text{g}$ | $9.98\u2009\text{g}$ |   | Average | $10.13\u2009\text{g}$ | $9.79\u2009\text{g}$ | $10.01\u2009\text{g}$ |

• Student A Evaluation: Data are both inaccurate (average $10.13\u2009\text{g}$ deviates from $10.00\u2009\text{g}$) and imprecise (large trial spread from $9.79\u2009\text{g}$ to $10.49\u2009\text{g}$ due to high random error).

• Student B Evaluation: Data are precise (trial spread is tight, ranging from $9.75\u2009\text{g}$ to $9.82\u2009\text{g}$) but inaccurate (average $9.79\u2009\text{g}$ is consistently lower than true mass due to systematic error).

• Student C Evaluation: Data display minimal systematic or random error; data are both highly accurate (average $10.01\u2009\text{g}$ matches true value) and highly precise (trial spread ranges from $9.98\u2009\text{g}$ to $10.03\u2009\text{g}$).

  • Isopropyl Alcohol Density Problem:

• Experimental density determinations: Trial 1 = $0.775\u2009\text{g/mL}$, Trial 2 = $0.675\u2009\text{g/mL}$, Trial 3 = $0.833\u2009\text{g/mL}$, Trial 4 = $0.861\u2009\text{g/mL}$.

• Precision Evaluation: Results are not precise. The values range widely from $0.675$ to $0.861\u2009\text{g/mL}$, showing poor consistency.

• Accuracy Evaluation (True Density = $0.786\u2009\text{g/mL}$):

  • Experimental Average Calculation:       $\text{Average} = \frac{0.775 + 0.675 + 0.833 + 0.861}{4} = \frac{3.144}{4} = 0.786\u2009\text{g/mL}$

  • The experimental average ($0.786\u2009\text{g/mL}$) is accurate because it matches the true density, even though individual trial precision was poor.

• Uncertainty Bounds: Experimental equipment uncertainties are reported with tolerance ranges (e.g., $\pm 0.1\u2009\text{g}$, $\pm 0.01\u2009\text{g}$, or $\pm 0.0001\u2009\text{g}$).

Significant Figures, Measurement Rules, and Scientific Notation

• Origin and Purpose of Significant Figures:

  • Scientific measurements preserve instrument precision by recording all digits known with absolute certainty plus one single estimated (uncertain) digit.

  • Example Value: In a reported distance of $5.213\u2009\text{cm}$, the digits $5.21$ are known with absolute certainty, whereas the final digit $3$ is estimated.

• Instrument Reading Examples:

  • Unsubdivided Scale ($1\u2009\text{cm}$ markings): Reading aligned between $5$ and $6\u2009\text{cm}$ is reported as $5.4\u2009\text{cm}$ ($5$ is certain, $0.4$ is estimated; $2$ significant figures).

  • Subdivided Scale ($0.1\u2009\text{cm}$ markings): Reading aligned between $5.3$ and $5.4\u2009\text{cm}$ is reported as $5.36\u2009\text{cm}$ ($5.3$ is certain, $0.06$ is estimated; $3$ significant figures).

  • Graduated Cylinder Reading: Volume is read at the bottom of the curved liquid meniscus. With markings spaced every $0.1\u2009\text{mL}$, the bottom of a meniscus resting between $4.5$ and $4.6\u2009\text{mL}$ is recorded to the hundredths place as $4.56\u2009\text{mL}$.   

    

  • Digital Balance Precision: Mass shown as $34.70\u2009\text{g}$ indicates certainty down to the tenths place with an estimated trailing zero at the hundredths place ($4$ significant figures).

• Universal Rules for Determining Significant Figures:

  1. Non-Zero Digits: All non-zero integers are significant ($7.314$ contains $4$ sig figs; $73.14$ contains $4$ sig figs).

  2. Interior Zeros: Zeros located between non-zero digits are always significant ($60.052$ contains $5$ sig figs; $6033$ contains $4$ sig figs).

  3. Trailing Zeros:

  • Trailing zeros following non-zero digits are significant if a decimal point is explicitly written ($4.70$ contains $3$ sig figs; $3500.$ contains $4$ sig figs; $30.2000$ contains $6$ sig figs; $0.0923700\u2009\%$ contains $6$ sig figs).

  • Trailing zeros are ambiguous or insignificant if no decimal point is written ($100$ contains $1$ sig fig; $81000$ contains $2$ sig figs; $25000$ contains $2$ sig figs).

  1. Leading Zeros: Zeros located to the left of the first non-zero digit serve solely as decimal place keepers and are never significant ($0.0032$ contains $2$ sig figs; $0.0000328$ contains $3$ sig figs; $0.000001070$ contains $4$ sig figs).

  2. Exact Numbers:

  • Derived from direct counting of discrete objects (e.g., $12\u2009\text{eggs}$) or defined equivalences (e.g., $1\u2009\text{inch} = 2.54\u2009\text{cm}$).

  • Possess an infinite number of significant figures ($\infty$) and never limit the precision of calculated results.

• Significant Figures and Scientific Notation Practice Matrix:

  | Measurement / Quantity | Number of Significant Figures | Standard Scientific Notation |   | :--- | :---: | :---: |   | $9605.6\u2009\text{lb}$ | $5$ | $9.6056 \times 10^3\u2009\text{lb}$ |   | $0.00180\u2009\text{oz}$ | $3$ | $1.80 \times 10^{-3}\u2009\text{oz}$ |   | $81000\u2009\text{in}$ | $2$ | $8.1 \times 10^4\u2009\text{in}$ |   | $201.90\u2009\text{m}$ | $5$ | $2.0190 \times 10^2\u2009\text{m}$ |   | $0.000001070\u2009\text{cm}$ | $4$ | $1.070 \times 10^{-6}\u2009\text{cm}$ |   | $3500.\u2009\text{g}$ | $4$ | $3.500 \times 10^3\u2009\text{g}$ |   | $0.000000050\u2009\text{s}$ | $2$ | $5.0 \times 10^{-8}\u2009\text{s}$ |   | $12\u2009\text{eggs}$ | Infinite ($\infty$) | Infinite ($\infty$) |   | $45.6\u2009\text{ft}$ | $3$ | $4.56 \times 10^1\u2009\text{ft}$ |   | $6033\u2009\text{sec}$ | $4$ | $6.033 \times 10^3\u2009\text{sec}$ |   | $0.0000328\u2009\text{cm}^2$ | $3$ | $3.28 \times 10^{-5}\u2009\text{cm}^2$ |   | $25000\u2009\text{m/s}$ | $2$ | $2.5 \times 10^4\u2009\text{m/s}$ |   | $30.2000\u2009\text{nautical miles}$ | $6$ | $3.02000 \times 10^1\u2009\text{nautical miles}$ |   | $0.0923700\u2009\%$ | $6$ | $9.23700 \times 10^{-2} \%$ |

• Standard Rounding Rules:

  • If the leftmost digit to be dropped is less than $5$, retain the preceding digit without change.

  • If the leftmost digit to be dropped is $5$ or greater, increase the preceding digit by $1$ unit.

  • Demonstration: Rounding $3.74966 \times 10^4$ to $3$ sig figs gives $3.75 \times 10^4$.

  • Rounding Practice Problems:

  • $0.00735139\u2009\text{cm}$ rounded to $4$ sig figs $\rightarrow 0.007351\u2009\text{cm}$ ($7.351 \times 10^{-3}\u2009\text{cm}$).

  • $2293\u2009\text{ft}$ rounded to $3$ sig figs $\rightarrow 2290\u2009\text{ft}$ ($2.29 \times 10^3\u2009\text{ft}$).

  • $0.004859\u2009\text{g}$ rounded to $2$ sig figs $\rightarrow 0.0049\u2009\text{g}$ ($4.9 \times 10^{-3}\u2009\text{g}$).

  • $27536.710\u2009\text{cm}$ rounded to $4$ sig figs $\rightarrow 27540\u2009\text{cm}$ ($2.754 \times 10^4\u2009\text{cm}$).

• Significant Figures in Mathematical Operations:

  • Addition and Subtraction Rule:

  • The final calculated result is rounded to match the same number of decimal places as the quantity carrying the fewest decimal places (limiting place value uncertainty).

  • Worked Example 1:       $\begin{aligned} 37.68\u2009\text{L} \\ 6.71862\u2009\text{L} \\ + 108.428\u2009\text{L} \\ \hline 152.82664\u2009\text{L} \rightarrow 152.83\u2009\text{L} \quad (\text{rounded to } 2 \text{ decimal places due to } 37.68) \end{aligned}$

  • Worked Example 2a: $504.2 + 12.8 + 760 = 1277 \rightarrow 1280$ (or $1.28 \times 10^3$, limited by $760$ which has uncertainty in the tens place).

  • Worked Example 2b: $504.2 + 12.8 + 760. = 1277.0 \rightarrow 1277$ (limited by $760.$, units place).

  • Worked Example 2c: $504.2 + 12.8 + 760.0 = 1277.0$ (limited to the tenths place).

  • Multiplication and Division Rule:

  • The final calculated result is rounded to contain the same number of significant figures as the factor with the fewest significant figures.

  • Worked Example 1:       $\frac{(4.2 \times 10^3)(15.94)}{2.255 \times 10^{-4}} = 2.9688692 \times 10^8 \rightarrow 3.0 \times 10^8 \quad (\text{limited to } 2 \text{ sig figs by } 4.2 \times 10^3)$

  • Worked Example 2a: $31.66 \times 2.83 = 89.5978 \rightarrow 89.6$ (limited to $3$ sig figs by $2.83$).

  • Worked Example 2b:       $\frac{(3.722 \times 10^7)(7.1061 \times 10^6)}{(5.2 \times 10^{-3})(2.38 \times 10^2)} = 2.137... \times 10^{14} \rightarrow 2.1 \times 10^{14} \quad (\text{limited to } 2 \text{ sig figs by } 5.2 \times 10^{-3})$

  • Worked Example 2c:       $\frac{(1.6 \times 10^7)(3.269 \times 10^5)}{(8.449 \times 10^{-11})(2.3508 \times 10^3)} = 2.6334... \times 10^{19} \rightarrow 2.6 \times 10^{19} \quad (\text{limited to } 2 \text{ sig figs by } 1.6 \times 10^7)$

Density, Temperature Scales, and Conversion Formulas

• Density Concepts and Formulas:

  • Density ($d$ or $\rho$): Defined as the ratio of mass ($m$) per unit volume ($V$):     $d = \frac{m}{V}$

  • Density is an intensive physical property, meaning its value is independent of the quantity of material present.

  • Mass ($m$) and volume ($V$) are extensive physical properties, meaning their values depend directly upon sample size.

  • Units of Density: Typically expressed as $\text{g/cm}^3$, $\text{g/mL}$, or $\text{g/cc}$, where $1\u2009\text{mL} = 1\u2009\text{cm}^3 = 1\u2009\text{cc}$. The official SI derived unit is $\text{kg/m}^3$

• Density Practice Calculations:

  • Problem 1: An unknown liquid sample has a mass of $131.85\u2009\text{g}$ and occupies a volume of $150.\u2009\text{mL}$. Calculate liquid density.     $d = \frac{m}{V} = \frac{131.85\u2009\text{g}}{150.\u2009\text{mL}} = 0.87900...\u2009\text{g/mL} \rightarrow 0.879\u2009\text{g/mL} \quad (3\u2009\text{sig figs constrained by } 150.\u2009\text{mL})$

  • Problem 2: If the density of butter is $0.86\u2009\text{g/mL}$, calculate the volume occupied by $427\u2009\text{g}$ of butter.     $V = \frac{m}{d} = \frac{427\u2009\text{g}}{0.86\u2009\text{g/mL}} = 496.51...\u2009\text{mL} \rightarrow 5.0 \times 10^2\u2009\text{mL} \text{ (or } 500\u2009\text{mL showing } 2\u2009\text{sig figs)}$

• Temperature Scales and Thermal Relationships:

  • Temperature: A quantitative measure of the average kinetic energy ("hotness") of particles in a system.

  • As molecular speed increases, Kelvin temperature increases proportionally.   

    

  | Reference Phenomenon | Kelvin Scale ($\text{K}$) | Celsius Scale ($^\circ\text{C}$) | Fahrenheit Scale ($^\circ\text{F}$) |   | :--- | :---: | :---: | :---: |   | Water Boiling Point | $373.15\u2009\text{K}$ | $100.\u2009^\circ\text{C}$ | $212.\u2009^\circ\text{F}$ |   | Human Body Temperature | $310.15\u2009\text{K}$ | $37.0\u2009^\circ\text{C}$ | $98.6\u2009^\circ\text{F}$ |   | Standard Room Temperature | $298.15\u2009\text{K}$ | $25.0\u2009^\circ\text{C}$ | $77.0\u2009^\circ\text{F}$ |   | Water Freezing Point | $273.15\u2009\text{K}$ | $0.0\u2009^\circ\text{C}$ | $32.0\u2009^\circ\text{F}$ |

• Temperature Scale Conversion Equations:   $T_{^\circ\text{C}} = \frac{T_{^\circ\text{F}} - 32}{1.8}$   $T_{^\circ\text{F}} = 1.8 \times T_{^\circ\text{C}} + 32$   $T_{\text{K}} = T_{^\circ\text{C}} + 273.15$

• Temperature Conversion Practice Problems:

  • Convert $25.0\u2009^\circ\text{C}$ to $^\circ\text{F}$:     $T_{^\circ\text{F}} = 1.8 \times 25.0 + 32 = 45.0 + 32 = 77.0\u2009^\circ\text{F}$

  • Convert $-40.\u2009^\circ\text{C}$ to $^\circ\text{F}$:     $T_{^\circ\text{F}} = 1.8 \times (-40.) + 32 = -72 + 32 = -40.\u2009^\circ\text{F} \quad (\text{Intersection point of both scales!})$

  • Convert $-193.5\u2009^\circ\text{C}$ to Kelvin:     $T_{\text{K}} = -193.5 + 273.15 = 79.65 \rightarrow 79.7\u2009\text{K} \quad (\text{rounded to tenths place})$

  • Convert $1130.50\u2009\text{K}$ to $^\circ\text{F}$:     $\text{Step 1: } T_{^\circ\text{C}} = 1130.50 - 273.15 = 857.35\u2009^\circ\text{C}$     $\text{Step 2: } T_{^\circ\text{F}} = 1.8 \times 857.35 + 32 = 1543.23 + 32 = 1575.23\u2009^\circ\text{F}$

Dimensional Analysis and Multi-Step Unit Conversions

• Methodology of Dimensional Analysis:

  • Dimensional analysis treats units as algebraic quantities that are multiplied, divided, and canceled.

  • Unit Equation: Statement of equivalence between two quantities (e.g., $2.54\u2009\text{cm} = 1\u2009\text{in}$).

  • Conversion Factor: Fractional ratio equal to $1$ constructed from a unit equation, arranged so given units cancel out:     $\text{given quantity} \times \left(\frac{\text{desired unit}}{\text{given unit}}\right) = \text{sought quantity}$

• Reference Conversion Factors & Equivalence Table:

  • $1\u2009\text{pound} = 16\u2009\text{oz}$

  • $1\u2009\text{ton} = 2000\u2009\text{pounds}$

  • $1\u2009\text{foot} = 12\u2009\text{inches}$

  • $1\u2009\text{yard} = 3\u2009\text{feet}$

  • $1\u2009\text{mile} = 5280\u2009\text{ft}$

  • $1\u2009\text{gallon} = 4\u2009\text{quarts}$

  • $1\u2009\text{quart} = 2\u2009\text{pints}$

  • $1\u2009\text{quart} = 32\u2009\text{fluid oz}$

  • $2.2\u2009\text{pounds} = 1\u2009\text{kg}$

  • $2.54\u2009\text{cm} = 1\u2009\text{inch}$

  • $1\u2009\text{gallon} = 3.785\u2009\text{L}$

  • $1609\u2009\text{m} = 1\u2009\text{mi}$

  • $640\u2009\text{acre} = 1\u2009\text{mi}^2$

  • $1\u2009\text{mL} = 1\u2009\text{cm}^3$

• Exhaustive Practice Conversion Problems (Full Step-by-Step Derivations):

  • Problem 1: Convert $843\u2009\text{cL}$ to liters     $843\u2009\text{cL} \times \frac{10^{-2}\u2009\text{L}}{1\u2009\text{cL}} = 8.43\u2009\text{L}$

  • Problem 2: Convert $0.00350\u2009\text{kg}$ to $\mu\text{g}$     $0.00350\u2009\text{kg} \times \frac{10^3\u2009\text{g}}{1\u2009\text{kg}} \times \frac{1\u2009\mu\text{g}}{10^{-6}\u2009\text{g}} = 3.50 \times 10^6\u2009\mu\text{g}$

  • Problem 3: Convert $86.0\u2009\text{ton}$ to ounces     $86.0\u2009\text{ton} \times \frac{2000\u2009\text{lb}}{1\u2009\text{ton}} \times \frac{16\u2009\text{oz}}{1\u2009\text{lb}} = 2,752,000\u2009\text{oz} \rightarrow 2.75 \times 10^6\u2009\text{oz}$

  • Problem 4: Convert $1.00 \times 10^{-2}\u2009\text{gallons}$ to fluid ounces     $1.00 \times 10^{-2}\u2009\text{gal} \times \frac{4\u2009\text{qt}}{1\u2009\text{gal}} \times \frac{32\u2009\text{fl oz}}{1\u2009\text{qt}} = 1.28\u2009\text{fl oz}$

  • Problem 5: Convert $4.5 \times 10^{-2}\u2009\text{qt}$ to centiliters     $4.5 \times 10^{-2}\u2009\text{qt} \times \frac{1\u2009\text{gal}}{4\u2009\text{qt}} \times \frac{3.785\u2009\text{L}}{1\u2009\text{gal}} \times \frac{1\u2009\text{cL}}{10^{-2}\u2009\text{L}} = 4.258...\u2009\text{cL} \rightarrow 4.3\u2009\text{cL}$

  • Problem 6: Convert $100\u2009\text{mph}$ ($\text{miles/hour}$) to meters per second ($\text{m/s}$)     $\frac{100\u2009\text{mi}}{1\u2009\text{h}} \times \frac{1609\u2009\text{m}}{1\u2009\text{mi}} \times \frac{1\u2009\text{h}}{3600\u2009\text{s}} = 44.694...\u2009\text{m/s} \rightarrow 45\u2009\text{m/s}$

  • Problem 7: Units Raised to a Power Conversion Mechanics

  • When converting units raised to an exponent (e.g., area or volume units), both the numerical coefficient and the unit unit within the conversion factor must be raised to that power.     

    

        $\text{Derivation Example: } 2.54\u2009\text{cm} = 1\u2009\text{in} \implies (2.54\u2009\text{cm})^2 = (1\u2009\text{in})^2 \implies (2.54)^2\u2009\text{cm}^2 = 1^2\u2009\text{in}^2 \implies 6.4516\u2009\text{cm}^2 = 1\u2009\text{in}^2$

  • Problem 8: Convert $1.30 \times 10^{30}\u2009\text{nm}^2$ to acres     $1.30 \times 10^{30}\u2009\text{nm}^2 \times \left(\frac{10^{-9}\u2009\text{m}}{1\u2009\text{nm}}\right)^2 \times \left(\frac{1\u2009\text{mi}}{1609\u2009\text{m}}\right)^2 \times \frac{640\u2009\text{acre}}{1\u2009\text{mi}^2} = 3.21 \times 10^8\u2009\text{acre}$

  • Problem 9: Convert $30.\u2009\text{L}$ to $\mu\text{m}^3$     $30.\u2009\text{L} \times \frac{1000\u2009\text{mL}}{1\u2009\text{L}} \times \frac{1\u2009\text{cm}^3}{1\u2009\text{mL}} \times \left(\frac{10^{-2}\u2009\text{m}}{1\u2009\text{cm}}\right)^3 \times \left(\frac{1\u2009\mu\text{m}}{10^{-6}\u2009\text{m}}\right)^3 = 3.0 \times 10^{16}\u2009\mu\text{m}^3$

  • Problem 10: Convert $9.12 \times 10^{15}\u2009\text{nm}$ to miles     $9.12 \times 10^{15}\u2009\text{nm} \times \frac{10^{-9}\u2009\text{m}}{1\u2009\text{nm}} \times \frac{1\u2009\text{mi}}{1609\u2009\text{m}} = 5.668... \times 10^3\u2009\text{mi} \rightarrow 5.67 \times 10^3\u2009\text{mi}$