Chemistry Lecture Notes: Scientific Notation, Dimensional Analysis, and Atomic Theory
Scientific Notation and Significant Figures
Scientific notation is the expression of a number as a product of a number between 1 and 10 and a power of 10: N = a \times 10^{n} , \quad 1 \le a < 10.- Examples discussed:
Large numbers: 1,200,000 = 1.2 \times 10^{6} (2 significant figures)
1,200,000,000 would be written as 1.2 \times 10^{9} (depending on significant figures kept; notes show the idea that the form is 1.x × 10^n).
Small numbers: 0.00000126 = 1.26 \times 10^{-6}.
Conversion between decimal and scientific notation:
Moving the decimal point to the right makes the exponent more negative by 1: i.e., in going from a to the right reduces the magnitude and adds a minus sign to the exponent (as explained informally by the speaker).
Moving the decimal point to the left increases the exponent by +1.
Mathematicians would formalize this as adjusting the exponent to keep the value the same; the instructor notes this may look like a minor dispute mathematicians would have, but the practical idea is to relate decimal movement to changes in the power of 10.
Notable constants and values in scientific notation:
{3.00 \times 10^{8}}\text{ m/s} for the speed of light (given as 2.996 \times 10^{8} \text{ m s}^{-1} in the notes).
6.02 \times 10^{23} (Avogadro’s number) is referenced as a standard number in chemistry contexts.
1.000 \times 10^{0} or any exact conversion factors are treated as exact with respect to significant figures.
Practical tip for calculations with scientific notation:
To add/subtract numbers in scientific notation, rewrite one or more terms to have the same exponent, then add/subtract the mantissas.
To multiply/divide, add/subtract exponents and multiply/divide mantissas:
Example: (2.0 \times 10^{2}) \times (3.0 \times 10^{1}) = (2.0 \times 3.0) \times 10^{2+1} = 6.0 \times 10^{3}.
Head’s up about significant figures during dimensional analysis:
Only the measured quantities determine the number of significant figures; conversion factors are exact and do not limit sig figs.
When averaging a set of measured values, the average has the same number of significant figures as the inputs if the inputs are considered exact multiples in the averaging process.
Dimensional analysis: carrying units through calculations
Example problem: determine density of an antifreeze sample from a four-quart volume and a mass in pounds, converting to grams and milliliters.
Given: a four-quart sample weighs 9.26 pounds; conversion factors:
1\ \text{liter} = 1.0567\ \text{quarts}
1\ \text{liter} = 1000\ \text{mL}
1\ \text{pound} = 453.59\ \text{g}
Step 1: convert volume from quarts to liters to milliliters
Start with 4.00 quarts and multiply by the appropriate conversion factor:
4.00\ \text{qt} \times \frac{1\ \text{L}}{1.0567\ \text{qt}} = 3.78\ \text{L} = 3.78\times 10^{3}\ \text{mL}.Note: 4.00 keeps 3 significant figures; 1.0567 is an exact conversion factor and does not limit sig figs.
Step 2: convert mass from pounds to grams
9.26\ \text{lb} \times \frac{453.59\ \text{g}}{1\ \text{lb}} = 4.20\times 10^{3}\ \text{g} (approximately; three significant figures retained).
Step 3: compute density (mass/volume)
Mass m = 4.20\times 10^{3}\ \text{g}; Volume V = 3.78\times 10^{3}\ \text{mL}.
Density: \rho = \frac{m}{V} = \frac{4.20\times 10^{3}}{3.78\times 10^{3}} \ \text{g/mL} \approx 1.11\ \text{g/mL}. (3 significant figures)
Summary rule for density units:
Density can be expressed as \rho = \dfrac{\text{mass}}{\text{volume}} with units such as \text{g/mL} or \text{g/cm}^3 (note: 1\ \text{cm}^3 = 1\ \text{mL}).
Dimensional analysis: a quick mental check while using calculators
When interpreting results for a calculation like 200 × 30 or 189 × 29, a reasonable outcome should be on the order of a few thousand (e.g., around 6\times 10^{3} for 200 × 30). If the calculator returns a vastly different magnitude (like ~400), re-check unit handling and magnitude estimation.
Quick recap: accuracy vs precision
Accuracy: how close a measurement (or average) is to the true value.
Precision: how close multiple measurements are to each other.
Two datasets can have similar accuracy but differ in precision (or vice versa).
Example data set discussion (illustrated with water density):
True value (literature): 0.9974\;\text{g/mL} at a given temperature.
Group A measurements: 0.9976, 0.9976, 0.9978 → average ~0.9970 g/mL; spread small (0.0002) suggesting precision.
Group B measurements: 0.9959, 0.9976, 0.9992 → average ~0.9976 g/mL; spread larger (0.0033).
Which dataset is more accurate? The one whose average is closer to the literature value (Group A in this example).
How to compare spread: max – min for each group (Group A ~0.0021; Group B ~0.0086).
Transition to Section 2: Atoms, Molecules, Ions (Chapter 2 focus)
The Periodic Table and Periodic Behavior
The atom: what it looks like; particles inside the atom; history of atomic theory; how to symbolize atoms and ions; chemical formulas; naming compounds.
Periodic table: elements arranged in groups/columns; properties repeat (periodic behavior) as you move across the table.
Mendeleev’s early table and modern periodic table:
Elements in a column tend to have similar properties.
Visual cues on the periodic table:
Boxes show element symbol, atomic number, usually state at room temperature (solids in black, liquids in blue, gases in red), metals (yellow), nonmetals (green), metalloids (purple).
Key groups to remember:
Alkali metals (Group 1): very reactive metals.
Alkaline earth metals (Group 2): reactive metals but less than Group 1.
Halogens (Group 17): very reactive nonmetals.
Noble gases (Group 18): very nonreactive (inert) gases.
Other groups/types:
Oxygen group (Calcogens, often called Chalcogens) (Group 16).
Nitrogen group (pnictogens, Group 15).
Transition metals: metals in the center block; many different oxidation states.
Main-group elements: Groups 1, 2, and 13–18 (excluding transition metals).
Top elements in each group (examples):
Noble gases: Helium (top of Group 18).
Alkaline earth metals: Beryllium (top of Group 2).
Alkali metals: Hydrogen sits atop Group 1 but is not a metal; beneath it is Lithium (top metal in Group 1).
Calcogens/Chalcogens: Oxygen is at the top of Group 16.
Halogens: Fluorine at the top of Group 17.
Group-to-compound ideas:
Many alkali metals form ionic compounds with halogens in a 1:1 ratio in simple salts.
The discussion hints at ionic bonding as the electrostatic attraction between a metal donating an electron and a nonmetal accepting it.
Element symbols and sample identifications:
Chlorine: symbol Cl; Group 17 (halogen).
Calcium: symbol Ca; Group 2 (alkaline earth metal).
Sodium: symbol Na; Group 1 (alkali metal); named from Natrium (Latin) used in some languages; symbol Na reflects historical naming.
Fluorine: symbol F; Group 17 (halogen).
Lightest members of groups (top elements):
Noble gases: Helium (top).
Alkaline earth metals: Beryllium (top).
Alkali metals: Hydrogen sits above Li; Li is the first metal in Group 1.
Calcogens: Oxygen is at the top of Group 16.
Atomic theory: historical development and key ideas
Greek roots of the atomic concept: cutting matter into smaller pieces until you reach indivisible units (atomos).
Dalton’s postulates (five):
All matter is composed of atoms; atoms are the smallest units that participate in chemical change.
An element consists of only one type of atom, with characteristic mass; all atoms of that element have same mass.
A compound consists of two or more elements in a small whole-number ratio of atoms, constant across samples.
Atoms are rearranged but not created or destroyed during chemical reactions (conservation of atoms).
Atoms of different elements have different properties and masses.
Implications of Dalton’s postulates:
Existence of a fixed ratio of atoms in compounds (e.g., water = H:O in a fixed ratio).
Conservation of mass in chemical reactions because atoms are merely rearranged.
Limitations and refinements after Dalton:
Atoms are not indivisible: subatomic particles (electrons, protons, neutrons) exist.
Subatomic particles and the development of the atomic model
Thomson’s cathode-ray tube experiments:
Discovered electrons; cathode rays are deflected by electric and magnetic fields; electrons carry negative charge.
Conclusion: atoms contain negatively charged particles (electrons).
Early model: plum pudding model (electrons scattered through a positively charged matrix).
Millikan’s oil-drop experiment:
Measured elementary charge e = 1.6 imes 10^{-19}\ \text{C} per electron (observations show charges are multiples of this value).
Rutherford’s gold foil experiment:
Most alpha particles pass through a thin gold foil; occasionally, some are deflected at large angles or back-scattered.
Conclusion: atoms contain a very small, dense, positively charged nucleus; most of the atom is empty space.
Resulting nuclear model: nucleus contains protons (and later, neutrons); electrons occupy the surrounding space.
Evolution of atomic model (post-Rutherford): electrons in defined orbitals around a tiny, dense nucleus; nucleus contains protons and neutrons; electrons are far lighter and occupy most of the atom’s volume.
Nucleus: protons, neutrons, and electrons
Nuclear composition:
Protons: positively charged; located in the nucleus; mass ~1 amu.
Neutrons: electrically neutral; located in the nucleus; mass ~1 amu.
Electrons: negatively charged; located outside the nucleus; mass negligible compared to protons/neutrons; mass ~0 amu (in common approximations).
Units and mass concepts:
Atomic mass unit (amu): a standard unit used to express atomic and molecular masses; used to compare masses of nuclei.
The nucleus is extremely small relative to the overall size of the atom: nucleus ~10^{-15} m; the atom ~10^{-10} m, giving a size difference of about 5 orders of magnitude.
Isotopes and atomic structure refinements
Isotopes: atoms of the same element (same number of protons Z) with different numbers of neutrons, hence different masses, but typically similar chemical properties.
Isotopic notation and mass number:
Mass number A = Z + N, where N is the number of neutrons.
Atomic number Z equals the number of protons.
In isotope notation, X is the element symbol; A is the mass number; Z is the atomic number; charge q indicates the ionic state.
Example working through iodine:
Iodine (I) has Z = 53 (53 protons).
If an isotope has A = 127, then N = A − Z = 127 − 53 = 74 neutrons.
If the ion carries a −1 charge, the number of electrons is Z + 1 (54 electrons for I−).
So this iodine ion would be represented as ^{127}_{53}I^{−} in standard isotope notation, indicating A = 127, Z = 53, charge −1.
The mass number vs. mass of protons/neutrons:
The mass number is an integer; it is the sum of protons and neutrons and is used for isotopic identity, not the precise mass in grams.
Notation for ions and atoms
General ion notation idea (as described in the transcript): a square/rectangular schematic with four boxes labeled to show:
Element symbol (e.g., I for iodine) in the interior area.
Mass number A (e.g., 127) and proton number Z (e.g., 53) associated with the nucleus.
The charge of the ion (e.g., −1) to indicate the electron count relative to protons.
Standard chemical isotope notation (commonly used in chemistry):
An atom with mass number A and atomic number Z is written as ^{A}{Z}X, and for an ion, the charge is shown to the top-right as ^{A}{Z}X^{q} where q is the ionic charge (e.g., ^{127}_{53}I^{−} for iodide-127).
Key definitions and practical concepts to remember
Atomic number (Z): number of protons in the nucleus.
Mass number (A): A = Z + N (protons + neutrons).
Isotopes: same Z (same element) but different A (different N).
Ions: atoms with a net charge due to loss or gain of electrons.
Electron charge: -e = -1.602 \times 10^{-19} \text{ C} (unit negative charge per electron).
Electron mass is negligible compared to proton/neutron masses; proton and neutron masses are about 1 amu each.
Quick connections to real-world relevance
Dimensional analysis is essential in lab work for converting measurements to consistent units (e.g., grams to kilograms, liters to milliliters).
Understanding density is critical in material characterization, antifreeze formulation, and quality control in manufacturing.
Atomic theory underpins chemistry naming, reaction stoichiometry, and material science.
Summary of historical models and experimental milestones (timeline-style gist)
Dalton’s atomic theory: all matter is made of atoms; atoms of same element identical; compounds formed by fixed ratios; atoms rearranged but not created/destroyed.
Thomson’s plum pudding model: electrons embedded in a positively charged matrix (mass mostly not concentrated in the electrons).
Millikan’s oil-drop experiment: determined the elementary charge e; established charge quantization at the level of individual electrons.
Rutherford’s gold foil experiment: discovered a tiny, dense nucleus; led to the nuclear model with electrons around a central nucleus.
Real-world practice problems to reinforce concepts
Convert a large number to scientific notation and back, ensuring the mantissa is between 1 and 10 and tracking significant figures.
Perform a dimensional analysis problem similar to the antifreeze density example and report the final density with correct units and significant figures.
Identify the group of a given element and predict basic bonding tendencies (e.g., alkali metals form ionic compounds with halogens in 1:1 ratios).
Given an isotope notation, determine Z, A, N, and the number of electrons for a neutral atom vs. an ion.
Compare accuracy and precision using sample data sets and determine which set is more accurate and/or more precise based on proximity to a true value and the spread of measurements.
Note on a few transcript-specific points to be aware of:
Some examples in the transcript have minor inconsistencies (e.g., a stated 1,000,200 example). The correct standard representation for 1,000,200 in scientific notation would be 1.0002 \times 10^{6}. Always align to the standard form with the mantissa between 1 and 10 unless teaching a specific illustrative exception.
When converting with conversion factors, remember that exact numbers (conversion factors) do not limit significant figures; only measured values do.
The transcript emphasizes showing units explicitly during dimensional analysis to avoid mistakes (e.g., making sure quarts cancel, converting quarts → liters → milliliters correctly).
Key Formulas and Notation (summary)
Scientific notation: N = a \times 10^{n}, \quad 1 \le a < 10.
Density: \rho = \dfrac{m}{V},\quad \text{with units such as } \dfrac{\text{g}}{\text{mL}} \text{ or } \dfrac{\text{g}}{\text{cm}^3}.
Mass number: A = Z + N.
Isotopic notation (conventional): ^{A}_{Z}X^{q} where q is the ionic charge.
Atomic number: Z = \text{number of protons}.
Mass number: A = Z + N,\quad N = A - Z.