Chapter 13: High-Yield Cheat Sheet

This is the MCAT general chemistry cheat sheet you wish you had the week before test day. Every formula, periodic trend, and rule from all 12 chapters, compressed into one page. Atomic structure, periodic trends, gas laws, equilibrium constants, Ksp, Ka/Kb, buffers, Henderson-Hasselbalch, rate laws, Gibbs free energy, electrochemical cell potentials, Nernst equation - all in one scrollable reference. If you can recite what is on this page, you have the gen chem toolkit for 100% of what the MCAT will hand you.

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1. Atomic Structure

Subatomic particles: proton (+1, in nucleus, ~1 amu), neutron (0, in nucleus, ~1 amu), electron (-1, in orbitals, ~0 amu, mass ≈ $\frac{1}{1836}$ of a proton).

• Atomic number (Z) = # protons (defines the element)
• Mass number (A) = # protons + # neutrons
• Isotopes = same Z, different A. Isoelectronic species = same electron count.

Quantum numbers:

Number	Symbol	Range	Meaning
Principal	$n$	1, 2, 3, …	Energy level / shell
Azimuthal	$\ell$	0 to $n-1$	Subshell (s=0, p=1, d=2, f=3)
Magnetic	$m_\ell$	$-\ell$ to $+\ell$	Orbital orientation
Spin	$m_s$	±½	Electron spin

Orbital capacities: s = 2, p = 6, d = 10, f = 14 electrons. Subshell energies: s < p < d < f (for a given n). Fill by Aufbau (lowest energy first), Hund’s rule (singly occupy degenerate orbitals with parallel spin before pairing), and the Pauli exclusion principle (no two electrons have all four quantum numbers equal).

Exceptions to Aufbau (half-filled/filled d shells are extra stable): Cr is [Ar] 4s¹ 3d⁵ (not 4s² 3d⁴); Cu is [Ar] 4s¹ 3d¹⁰.

Bohr model: $E_n = -13.6/n^2$ eV (for hydrogen). Photon energy $\Delta E = hf = hc/\lambda$ on transitions. Balmer series = visible (transitions to $n=2$); Lyman = UV (to $n=1$); Paschen = IR (to $n=3$).

Heisenberg uncertainty: $\Delta x \cdot \Delta p \ge h/(4\pi)$. You cannot know position and momentum exactly at the same time.

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2. The Periodic Table and Trends

Groups (columns): Group 1 = alkali metals (1 valence e⁻); Group 2 = alkaline earths (2); Group 17 = halogens (7); Group 18 = noble gases (full shell).

Periods (rows): period number = highest $n$ for that row’s valence electrons.

Trend	Direction of Increase	Why
Atomic radius	Down, left	More shells; less $Z_{\text{eff}}$
Ionic radius (cations)	Smaller than parent atom	Lost outer shell
Ionic radius (anions)	Larger than parent atom	Added e⁻ repulsion
Ionization energy (IE)	Up, right	Harder to remove e⁻ from small, high $Z_{\text{eff}}$ atom
Electron affinity (EA)	Up, right (mostly)	More favorable to add e⁻
Electronegativity (χ)	Up, right	Pauling scale: F = 4.0, O = 3.5, N = 3.0, C = 2.5, H = 2.1
Metallic character	Down, left	Opposite of EN
Acidity of binary acids (HX)	Down, right	Bond strength drops → easier H⁺ release

Successive IE: IE₁ < IE₂ < IE₃. A large jump signals a stable noble-gas configuration was reached.

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3. Bonding and Molecular Geometry

Bond types:

• Ionic (ΔEN > ~1.7): metal + nonmetal, full electron transfer
• Polar covalent (0.4 < ΔEN < 1.7): unequal sharing
• Nonpolar covalent (ΔEN < 0.4): equal sharing

Bond order vs bond length/energy: higher bond order = shorter, stronger bond. Triple > double > single.

Formal charge: $FC = V - L - B/2$, where V = valence e⁻, L = lone-pair e⁻, B = bonded e⁻. Preferred Lewis structure minimizes formal charges and places negative charge on the more electronegative atom.

VSEPR geometry:

Steric #	Electron Geometry	0 LP	1 LP	2 LP	3 LP
2	Linear	Linear (180°)
3	Trigonal planar	Trigonal planar (120°)	Bent
4	Tetrahedral	Tetrahedral (109.5°)	Trigonal pyramidal	Bent (~105°)
5	Trigonal bipyramidal	Trig. bipyramidal	Seesaw	T-shape	Linear
6	Octahedral	Octahedral (90°)	Square pyramidal	Square planar

Hybridization from steric #: 2 → sp, 3 → sp², 4 → sp³, 5 → sp³d, 6 → sp³d². Each σ bond = one sp-hybrid overlap; additional π bonds come from unhybridized p orbitals.

Intermolecular forces (weakest → strongest):

• London dispersion (all molecules; stronger with more electrons/surface area)
• Dipole-dipole (polar molecules)
• Hydrogen bonding (H bound to N, O, or F)
• Ion-dipole (ion + polar molecule)
• Ionic > any IMF

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4. Stoichiometry and Compounds

Mole: $6.022 \times 10^{23}$ particles. Molar mass = g/mol from periodic table.

$$n = \frac{m}{M}, \qquad N = n \cdot N_A$$

Percent composition of element X: $\%X = \dfrac{(\text{\# X}) \cdot M_X}{M_{\text{compound}}} \times 100$.

Empirical formula = smallest whole-number ratio of atoms. Molecular formula = actual count (integer multiple of empirical formula; divide molecular mass / empirical mass).

Limiting reagent: compute moles of product each reagent could make, pick the smallest. That reagent is limiting. Theoretical yield = product from limiting reagent; percent yield = actual/theoretical × 100.

Balancing: conserve atoms of each element; conserve charge in redox half-reactions. Balance metals first, then nonmetals, then H, then O (for combustion-like reactions).

Reaction types (memorize the pattern):

• Combination: A + B → AB
• Decomposition: AB → A + B
• Single displacement: A + BC → AC + B (more active metal displaces)
• Double displacement (metathesis): AB + CD → AD + CB (swap ions, often precipitation)
• Combustion: CₓHᵧ + O₂ → CO₂ + H₂O
• Neutralization: acid + base → salt + H₂O
• Redox: electron transfer (see §11)

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5. Gases

Ideal gas law:
$PV = nRT$

with $R = 0.0821~\mathrm{L\cdot atm/(mol\cdot K)} = 8.314~\mathrm{J/(mol\cdot K)}$. T must be in kelvin.

At STP (0°C, 1 atm): 1 mol = 22.4 L.

Combined gas law (fixed n): $\dfrac{P_1 V_1}{T_1} = \dfrac{P_2 V_2}{T_2}$. Holding any variable constant gives you the classic named laws:

Law	Constant	Relationship
Boyle’s	n, T	$PV$ = const (P ∝ 1/V)
Charles’	n, P	$V/T$ = const
Gay-Lussac’s	n, V	$P/T$ = const
Avogadro’s	P, T	$V/n$ = const

Dalton’s law (partial pressures): $P_{total} = \sum P_i$. Mole fraction $X_i = n_i/n_{total}$, so $P_i = X_i P_{total}$.

Graham’s law of effusion: $\dfrac{r_1}{r_2} = \sqrt{\dfrac{M_2}{M_1}}$. Lighter molecules effuse faster.

Kinetic molecular theory (ideal): gas particles have negligible volume and no IMFs; collisions are elastic; $KE_{avg} \propto T$. Real gases deviate most at high P / low T (molecular volume + IMFs matter). Van der Waals equation corrects for both.

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6. Solutions

Concentration units:

• Molarity (M): mol solute / L solution
• Molality (m): mol solute / kg solvent
• Mole fraction (X): $n_{A}$ / $n_{\text{total}}$
• Normality (N): equivalents / L (H⁺ or e⁻ depending on context)
• Percent (% w/w, % w/v, % v/v)

Dilution: $M_1 V_1 = M_2 V_2$ (moles conserved).

Solubility rules (memorize):

• All group 1 and ammonium salts are soluble.
• All nitrates, acetates, perchlorates are soluble.
• Halides are soluble except with Ag⁺, Pb²⁺, Hg₂²⁺.
• Sulfates are soluble except with Ba²⁺, Pb²⁺, Ca²⁺ (slightly), Sr²⁺.
• Carbonates, phosphates, sulfides, hydroxides mostly insoluble (except with group 1 / ammonium).

Solubility product (Ksp): for $M_aX_b(s) \rightleftharpoons aM + bX$, $K_{sp} = [M]^a [X]^b$. Common-ion effect: adding a product ion shifts equilibrium left, lowering solubility.

Colligative properties (depend on particle count, not identity; use van’t Hoff factor $i$):

• Vapor pressure lowering (Raoult’s law): $P_A = X_A P^\circ_A$
• Boiling point elevation: $\Delta T_b = i K_b m$
• Freezing point depression: $\Delta T_f = i K_f m$
• Osmotic pressure: $\Pi = i M R T$

Ionic compounds have $i$ ≈ # ions (NaCl → 2, CaCl₂ → 3); non-electrolytes have $i$ = 1.

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7. Kinetics

Rate law (for $aA + bB \rightarrow \text{products}$):
$\text{rate} = k[A]^x [B]^y$

Exponents $x, y$ are the orders - determined experimentally, not from stoichiometry.

Integrated rate laws:

Order	Rate Law	Integrated	Half-life
0	$-d[A]/dt = k$	$[A] = [A]_0 - kt$	$t_{1/2} = [A]_0 / (2k)$
1	$-d[A]/dt = k[A]$	$\ln[A] = \ln[A]_0 - kt$	$t_{1/2} = \ln 2 / k \approx 0.693/k$
2	$-d[A]/dt = k[A]^2$	$1/[A] = 1/[A]_0 + kt$	$t_{1/2} = 1/(k[A]_0)$

Only first order has a half-life independent of concentration.

Arrhenius equation:
$k = A e^{-E_a / RT}$

Higher T → higher k. Higher $E_a$ → more T-sensitive.

Catalysts lower $E_a$ (both forward and reverse), increase both rate constants, but do not change $K_{eq}$ or $\Delta G$. They are not consumed.

Reaction coordinate diagram: transition state at the peak; intermediate in a trough between two peaks (multi-step). Rate-determining step = highest $E_a$ transition in the mechanism.

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8. Equilibrium

Law of mass action for $aA + bB \rightleftharpoons cC + dD$:
$K = \frac{[C]^c [D]^d}{[A]^a [B]^b}$

Pure solids and liquids do not appear in K (activity = 1). Gas-phase: use $K_p$ with partial pressures; $K_p = K_c (RT)^{\Delta n}$.

Reaction quotient Q: same expression, current (non-equilibrium) concentrations.

• Q < K → reaction proceeds forward
• Q > K → reaction proceeds reverse
• Q = K → at equilibrium

Le Châtelier’s principle: system responds to stress by shifting to counteract it.

Stress	Shift
Add reactant	Forward (→ products)
Remove product	Forward
↑ P (↓ V), gases	Toward fewer moles of gas
↑ T, exothermic forward	Reverse (treat heat as product)
↑ T, endothermic forward	Forward
Catalyst	No shift (only speeds equilibration)
Inert gas at constant V	No shift

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9. Acids and Bases

Definitions:

• Arrhenius: acid → H⁺ in water; base → OH⁻
• Brønsted-Lowry: acid = H⁺ donor; base = H⁺ acceptor (conjugate pairs differ by one H)
• Lewis: acid = electron-pair acceptor; base = electron-pair donor

Autoionization of water: $K_w = [\mathrm{H^+}][\mathrm{OH^-}] = 10^{-14}$ at 25°C.

$\mathrm{pH} = -\log[\mathrm{H^+}], \quad \mathrm{pOH} = -\log[\mathrm{OH^-}], \quad \mathrm{pH} + \mathrm{pOH} = 14$

Strong acids (memorize): HCl, HBr, HI, HNO₃, H₂SO₄ (first H only), HClO₄, HClO₃. Strong bases: group 1 hydroxides (LiOH, NaOH, KOH, RbOH, CsOH), heavy group 2 (Ca(OH)₂, Sr(OH)₂, Ba(OH)₂).

Weak acid equilibrium: $K_a = [\mathrm{H^+}][\mathrm{A^-}]/[\mathrm{HA}]$. Larger $K_a$ = stronger acid = smaller $pK_a$.

Conjugate pairs: $K_a \cdot K_b = K_w$, so $pK_a + pK_b = 14$.

Titration curves:

Titration	Equivalence pH	Indicator
Strong acid + strong base	7	Bromthymol blue
Weak acid + strong base	> 7 (basic; conj. base dominates)	Phenolphthalein
Strong acid + weak base	< 7 (acidic)	Methyl red/orange
Polyprotic	Multiple equivalence points	Multiple pKa values

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10. Thermodynamics and Thermochemistry

System vs surroundings. Signs: Q > 0 = heat absorbed by system (endothermic); W > 0 = work done by system (expansion).

First law: $\Delta U = Q - W$ (physics convention) or $\Delta U = Q + W$ (chem convention, where W is work done on system). MCAT gen-chem uses the chemistry convention.

Enthalpy: $\Delta H = Q_p$ (heat at constant P). $\Delta H < 0$ exothermic, $\Delta H > 0$ endothermic.

Hess’s law: enthalpy is a state function. $\Delta H_{rxn} = \sum \Delta H_f^\circ (\text{products}) - \sum \Delta H_f^\circ (\text{reactants})$. $\Delta H_f^\circ$ of elements in their standard state = 0.

Bond enthalpies: $\Delta H_{rxn} \approx \sum (\text{bonds broken}) - \sum (\text{bonds formed})$.

Entropy (S): disorder. $\Delta S > 0$ for: solid → liquid → gas, dissolving, ↑ T, ↑ moles of gas.

Gibbs free energy:
$\Delta G = \Delta H - T\Delta S$

$\Delta H$	$\Delta S$	Result
–	+	Spontaneous at all T
+	–	Non-spontaneous at all T
–	–	Spontaneous only at low T
+	+	Spontaneous only at high T

Equilibrium connection:
$\Delta G^\circ = -RT \ln K, \qquad \Delta G = \Delta G^\circ + RT \ln Q$

At equilibrium, $\Delta G = 0$ and $Q = K$.

Heat capacity: $Q = mc\Delta T$ (specific heat) or $Q = nC\Delta T$ (molar heat capacity). Calorimetry: $Q_{released} = -Q_{absorbed}$.

Phase changes (constant T): $Q = n\Delta H_{fus}$ (melting) or $n\Delta H_{vap}$ (vaporization). Phase diagram: triple point (solid, liquid, gas coexist); critical point (liquid/gas distinction disappears). Water is unusual: solid less dense than liquid, so the solid/liquid line has a negative slope.

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11. Oxidation-Reduction

OIL RIG: Oxidation Is Loss of electrons, Reduction Is Gain.

• Oxidizing agent = gets reduced (takes electrons).
• Reducing agent = gets oxidized (gives electrons).

Assigning oxidation numbers:

• Element alone: 0
• Monatomic ion: charge
• O: usually -2 (peroxide -1, OF₂ +2)
• H: +1 with nonmetal, -1 with metal
• F: always -1
• Sum = total charge of species

Balancing redox (half-reaction method):

1. Write two half-reactions (oxidation, reduction).
2. Balance atoms other than O, H.
3. Balance O with H₂O; balance H with H⁺ (acidic) or add OH⁻ to both sides (basic).
4. Balance charge with e⁻.
5. Multiply half-reactions so e⁻ cancel; add them.

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12. Electrochemistry

Galvanic (voltaic) cell: spontaneous; chemical → electrical; $E_{cell}^\circ > 0$; $\Delta G < 0$. Electrolytic cell: external energy forces non-spontaneous reaction; $E_{cell}^\circ < 0$; $\Delta G > 0$.

Anode / cathode:

• Anode = oxidation (AN OX). In galvanic, anode is negative; in electrolytic, anode is positive.
• Cathode = reduction (RED CAT). In galvanic, cathode is positive; in electrolytic, cathode is negative.

Standard cell potential:
$E_{cell}^\circ = E_{cathode}^\circ - E_{anode}^\circ$

Both tabulated as reduction potentials - do not flip the sign of the anode half when using this formula.

Key relations:
$\Delta G^\circ = -nFE^\circ_{cell}, \qquad \Delta G^\circ = -RT \ln K$

with $F = 96{,}485~\mathrm{C/mol~e^-}$. So $E^\circ > 0 \Leftrightarrow \Delta G^\circ < 0 \Leftrightarrow K > 1$ (spontaneous forward).

Electrolysis / Faraday’s law: moles of electrons = $It/F$. Grams deposited = $(M \cdot I \cdot t) / (n \cdot F)$. More current or more time = more mass plated.

Concentration cells: same species on both sides at different concentrations. $E^\circ = 0$, but $E \ne 0$ because of Q term. Electrons flow from dilute anode to concentrated cathode.

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13. The Master Formula Sheet

Single-source reference - what to put on flashcards per section.

Atomic: $E_n = -13.6/n^2$ eV; $\Delta E = hc/\lambda$; $\Delta x \Delta p \ge h/(4\pi)$.

Stoichiometry: $n = m/M$; $N = n N_A$; moles before = moles after (for conservation).

Gases: $PV = nRT$; $P_1 V_1 / T_1 = P_2 V_2 / T_2$; $P_i = X_i P_{total}$; $r_1/r_2 = \sqrt{M_2/M_1}$.

Solutions: $M_1 V_1 = M_2 V_2$; $\Delta T_b = iK_b m$; $\Delta T_f = iK_f m$; $\Pi = iMRT$.

Kinetics: rate = $k[A]^x[B]^y$; $t_{1/2}$(1st) = $\ln 2 / k$; $k = Ae^{-E_a/RT}$.

Equilibrium: $K = \prod [\text{prod}]^{\nu} / \prod [\text{react}]^{\nu}$; $K_p = K_c (RT)^{\Delta n}$; Q ≶ K decides shift direction.

Acid-base: $K_w = 10^{-14}$; pH + pOH = 14; $K_a K_b = K_w$; $\mathrm{pH} = \mathrm{p}K_a + \log\frac{[\mathrm{A^-}]}{[\mathrm{HA}]}$.

Thermo: $\Delta G = \Delta H - T\Delta S$; $\Delta G^\circ = -RT \ln K$; $Q = mc\Delta T$; Hess’s law.

Redox/electro: $E_{cell}^\circ = E_{cathode}^\circ - E_{anode}^\circ$; $\Delta G^\circ = -nFE^\circ$; Nernst; mass = $MIt/(nF)$.

Constants: $R = 8.314$ J/(mol·K) = 0.0821 L·atm/(mol·K); $N_A = 6.022 \times 10^{23}$; $F = 96{,}485$ C/mol; $K_w = 10^{-14}$ at 25°C; 1 atm = 760 torr = 101.3 kPa.

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14. Top 25 Facts to Memorize

1. Electrons fill lowest-energy orbitals first (Aufbau), one per orbital before pairing (Hund), no duplicates (Pauli).
2. Cr and Cu exceptions: 4s¹ 3d⁵ and 4s¹ 3d¹⁰ for extra stability of half-filled/filled d.
3. Periodic trends up and right: IE, EA, EN, $Z_{\text{eff}}$ all increase; atomic radius decreases.
4. Electronegativity order: F > O > N ≈ Cl > Br > I > S > C > H.
5. VSEPR geometries from steric number - 2 linear, 3 trigonal planar, 4 tetrahedral, 5 trig bipyramidal, 6 octahedral; lone pairs shape the molecular geometry.
6. H-bonding requires H bonded to N, O, or F.
7. Ideal gas law: $PV = nRT$; at STP, 1 mol = 22.4 L.
8. Real gases deviate at high P and low T.
9. Graham’s law: lighter gases effuse faster; rate ratio = √(M₂/M₁).
10. Colligative properties depend on particle count: use van’t Hoff factor $i$.
11. Strong acids/bases dissociate completely. Know the 7 strong acids and group 1/heavy group 2 hydroxides.
12. Weak acid pH at half-equivalence: pH = pKa.
13. Buffer range: ±1 pH unit of pKa.
14. K only depends on T. Changing concentrations shifts the system toward the new equilibrium; only T changes K itself.
15. Le Châtelier: system shifts to undo the stress.
16. First-order half-life is independent of [A]₀: $t_{1/2} = 0.693/k$.
17. Catalysts lower $E_a$; do not change $K$ or $\Delta G$.
18. Gibbs sign rules: negative ΔG = spontaneous; positive = non-spontaneous.
19. ΔG° vs ΔG: ΔG° is at standard conditions; ΔG uses current Q. At equilibrium, ΔG = 0 but ΔG° generally ≠ 0.
20. Hess’s law: enthalpy is a state function - you can add reactions.
21. Redox mnemonic: OIL RIG (Oxidation Is Loss, Reduction Is Gain).
22. Galvanic cell: spontaneous, anode = negative. Electrolytic: forced, anode = positive.
23. $\Delta G^\circ = -nFE^\circ$: positive E° = spontaneous.
24. Nernst equation explains why batteries “die”: as Q → K, E → 0.
25. Ksp common-ion effect: adding a product ion lowers solubility.

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15. Test-Day Mnemonics

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Next Steps

If something on this page feels unfamiliar, open the corresponding chapter and re-derive the relationship from first principles. Gen chem is the section where every correct answer starts with the right equation - once you pick it, the algebra is usually one or two steps. This cheat sheet gets you 80% recall on test day; the full chapters get you the reasoning to solve unfamiliar passages under the clock. Bookmark this page and return to it the night before every practice test and on the morning of the real thing.