Physical Chemistry Formulas

Physical Chemistry568 Qs

Structure of Atom

Planck's quantum theory, photoelectric effect, de Broglie relation, Heisenberg uncertainty principle, and quantum numbers.

Physical Chemistry703 Qs

Chemical Thermodynamics & Energetics

Enthalpy, entropy, Gibbs free energy, Hess law, Kirchhoff equation, and equilibrium constant temperature dependence.

Physical Chemistry486 Qs

Chemical & Ionic Equilibrium

K_p & K_c relations, pH calculations of weak acids/bases, buffer solutions, Henderson equation, and solubility product K_sp.

8 Formulas

Physical Chemistry685 Qs

Electrochemistry & Conductance

Nernst equation, Gibbs energy of cell, Kohlrausch law, Faraday laws of electrolysis, and molar conductivity.

Physical Chemistry671 Qs

Solutions & Colligative Properties

Raoult's law, relative lowering of vapor pressure, elevation in boiling point, depression in freezing point, and van 't Hoff factor.

8 Formulas

Physical Chemistry464 Qs

Chemical Kinetics

Rate laws, order and molecularity, integrated rate equations for zero and first order, half-life, and Arrhenius equation.

Physical Chemistry430 Qs

Solid State Chemistry

Unit cell geometry, density formula, packing efficiency of SC/BCC/FCC, radius ratios, and crystal defects.

7 Formulas Diagram

Formulas Reference (Physical Chemistry)

8 Chapters match

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Mole Concept & Stoichiometry

Mole concept, molarity, molality, normality, mole fraction, limiting reagent, and empirical formula.

Concentration Units (Molarity, Molality, Normality)

M = \frac{n_{\text{solute}}}{V_{\text{solution (L)}}}, \quad m = \frac{n_{\text{solute}}}{w_{\text{solvent (kg)}}}, \quad N = M \times n\text{-factor}

Fundamental quantitative measures of solution concentration.

Normality equals Molarity times valence n-factor.

Mole Fraction & Relation with Molality

X_A + X_B = 1, \quad m = \frac{X_B \times 1000}{(1 - X_B) M_A}

Converts mole fraction of solute X_B to molality m.

M_A is the molar mass of the solvent.

Average Molar Mass & Vapor Density

M_{\text{avg}} = \frac{\sum n_i M_i}{\sum n_i}, \quad \text{Molar Mass} = 2 \times \text{Vapor Density}

Molecular weight calculation for gas mixtures and unknown volatile compounds.

Vapor density is measured relative to Hydrogen gas.

Equivalent Weight & n-Factor Rules

E = \frac{\text{Molar Mass}}{n\text{-factor}}, \quad n\text{-factor}(\text{KMnO}_4): \begin{cases} 5 & \text{Acidic } (\text{Mn}^{2+}) \\ 3 & \text{Neutral } (\text{MnO}_2) \\ 1 & \text{Alkaline } (\text{MnO}_4^{2-}) \end{cases}

Calculates chemical equivalent weight. For acids: replaceable H⁺ count; bases: replaceable OH⁻ count; redox: change in oxidation state per molecule.

For K₂Cr₂O₇ in acidic medium: Cr goes from +6 to +3, giving n-factor = 2 × 3 = 6 (E = M / 6).

Law of Chemical Equivalence & Titration

N_1 V_1 = N_2 V_2, \quad \text{mEq} = N \times V_{\text{mL}} = \frac{\text{mass (g)}}{E} \times 1000

In any complete stoichiometric reaction, the number of gram-equivalents of all reactants and products are identical.

Does NOT require balancing the stoichiometric reaction equation beforehand.

Empirical Formula & Dulong-Petit Law

\text{MF} = (\text{EF})_n \quad \left(n = \frac{\text{Molar Mass}}{\text{EF Mass}}\right), \quad \text{Atomic Weight} \times \text{Specific Heat (cal/g)} \approx 6.4

Determination of true molecular formula from simplest atomic ratio and Dulong-Petit approximate atomic mass estimation for solid metals.

Valence of metal = (Approximate Atomic Weight) / (Equivalent Weight).

Limiting Reagent, Percentage Yield & Purity

\text{LR: Smallest } \frac{\text{Moles}}{\text{Coefficient}}, \quad \% \text{ Yield} = \frac{\text{Actual}}{\text{Theoretical}} \times 100\%, \quad \% \text{ Purity} = \frac{m_{\text{pure}}}{m_{\text{sample}}} \times 100\%

Limiting reagent is consumed first and dictates the maximum theoretical amount of product formed.

Always balance the chemical reaction before dividing moles by stoichiometric coefficients.

Volume Strength of H₂O₂ & Parts Per Million (ppm)

\text{Volume Strength} = 11.2 \times M = 5.6 \times N, \quad \text{ppm} = \frac{w_{\text{solute}}}{w_{\text{solution}}} \times 10^6

Volume of O₂ liberated at STP from 1 volume of hydrogen peroxide solution upon decomposition, and ppm trace concentration unit.

20 volume H₂O₂ has molarity M = 20 / 11.2 ≈ 1.78 M and normality N = 20 / 5.6 ≈ 3.57 N.

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Structure of Atom

Planck's quantum theory, photoelectric effect, de Broglie relation, Heisenberg uncertainty principle, and quantum numbers.

Planck Quantum Energy & de Broglie Wavelength

E = h\nu = \frac{hc}{\lambda}, \quad \lambda = \frac{h}{mv} = \frac{h}{\sqrt{2m(KE)}}

Wave-particle duality relations.

Energy of one mole of photons (Einstein): E = N_A h ν.

Heisenberg's Uncertainty Principle

\Delta x \cdot \Delta p \ge \frac{h}{4\pi} = \frac{\hbar}{2}, \quad \Delta x \cdot \Delta v \ge \frac{h}{4\pi m}

Fundamental limit on simultaneous measurement of conjugate variables.

Also applies to energy and time: ΔE · Δt ≥ h / (4π).

Quantum Numbers & Nodal Surfaces

\text{Radial Nodes} = n - l - 1, \quad \text{Angular Nodes} = l, \quad \text{Total Nodes} = n - 1

Characterization of electronic orbitals (n, l, m, s).

Orbital angular momentum L = √(l(l+1)) ħ.

Bohr Model for Hydrogenic Atoms

r_n = 0.529 \frac{n^2}{Z} \text{ \AA}, \quad v_n = 2.18 \times 10^6 \frac{Z}{n} \text{ m/s}, \quad E_n = -13.6 \frac{Z^2}{n^2} \text{ eV/atom} = -\frac{2.18 \times 10^{-18} Z^2}{n^2} \text{ J}

Quantized orbital radius, electron orbital velocity, and total electronic energy levels for single-electron species (H, He⁺, Li²⁺, Be³⁺).

Kinetic Energy K = -E_n = +13.6 Z²/n² eV; Potential Energy U = 2E_n = -27.2 Z²/n² eV.

Einstein's Photoelectric Equation

h\nu = \Phi_0 + KE_{\text{max}} = h\nu_0 + e V_s, \quad KE_{\text{max}} = hc\left(\frac{1}{\lambda} - \frac{1}{\lambda_0}\right)

Conservation of energy in photoelectric emission where Φ₀ = hν₀ is work function and V_s is stopping potential.

Slope of stopping potential V_s versus incident frequency ν is universal constant h/e.

Orbital & Spin Angular Momentum

L_{\text{orbital}} = \sqrt{l(l+1)}\frac{h}{2\pi} = \sqrt{l(l+1)}\hbar, \quad \mu_s = \sqrt{n(n+2)} \text{ BM} \quad (n = \text{unpaired } e^-)

Spatial angular momentum of atomic orbitals and spin-only magnetic moment in Bohr Magnetons.

For s-orbital (l = 0), orbital angular momentum is strictly ZERO.

Rydberg Formula & Hydrogen Emission Spectral Series

\bar{\nu} = \frac{1}{\lambda} = R_H Z^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right), \quad N_{\text{lines}} = \frac{(n_2 - n_1)(n_2 - n_1 + 1)}{2}

Wavenumbers of emitted spectral lines. Lyman (n₁=1, UV), Balmer (n₁=2, visible), Paschen (n₁=3, IR), Brackett (n₁=4, IR), Pfund (n₁=5, far-IR).

Only the Balmer series falls in the human-visible region of electromagnetic spectrum.

Radial Probability Distribution Function

P(r)\,dr = 4\pi r^2 R_{n,l}^2(r)\,dr, \quad r_{\text{max probability}}(1s) = a_0 = 0.529\,\text{\AA}

Probability of finding an electron in a spherical shell of radius r and thickness dr surrounding the nucleus.

At r = 0, probability density ψ² is maximum for s-orbitals, but radial probability P(r) is zero due to the 4πr² geometric factor.

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Chemical Thermodynamics & Energetics

Enthalpy, entropy, Gibbs free energy, Hess law, Kirchhoff equation, and equilibrium constant temperature dependence.

First Law & Enthalpy Relation

\Delta U = q + w = q - P_{\text{ext}}\Delta V, \quad \Delta H = \Delta U + \Delta n_g R T

First law formulation with enthalpy-internal energy gaseous conversion.

Δn_g = (moles of gaseous products) - (moles of gaseous reactants).

Gibbs Free Energy & Spontaneity

\Delta G = \Delta H - T\Delta S, \quad \Delta G^\circ = -RT \ln K = -2.303 RT \log_{10} K

Universal thermodynamic criterion for spontaneity and chemical equilibrium.

Spontaneous if ΔG < 0; at equilibrium ΔG = 0; non-spontaneous if ΔG > 0.

Kirchhoff's Thermochemical Equation

\Delta H_{T_2} = \Delta H_{T_1} + \Delta C_p(T_2 - T_1), \quad \Delta U_{T_2} = \Delta U_{T_1} + \Delta C_v(T_2 - T_1)

Temperature dependence of standard reaction enthalpy and internal energy.

ΔC_p = ∑ C_p(products) - ∑ C_p(reactants).

Entropy Changes in Physical Processes

\Delta S = \frac{q_{\text{rev}}}{T}, \quad \Delta S_{\text{gas}} = n C_v \ln\frac{T_2}{T_1} + n R \ln\frac{V_2}{V_1} = n C_p \ln\frac{T_2}{T_1} - n R \ln\frac{P_2}{P_1}

Entropy change for state transitions of ideal gas.

In isothermal process, ΔS = n R ln(V₂/V₁) = -n R ln(P₂/P₁).

Van 't Hoff Isochore Equation

\ln\frac{K_2}{K_1} = \frac{\Delta H^\circ}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)

Temperature variation of equilibrium constant K.

For endothermic reactions (ΔH° > 0), K increases as T increases; for exothermic, K decreases.

Work Done: Reversible vs Irreversible (Isothermal & Adiabatic)

w_{\text{iso, rev}} = -2.303 n R T \log_{10}\left(\frac{V_2}{V_1}\right), \quad w_{\text{adiab, rev}} = \frac{n R(T_2 - T_1)}{\gamma - 1} = \frac{P_2 V_2 - P_1 V_1}{\gamma - 1}

Pressure-volume work for fundamental paths. In expansion, reversible work output is strictly greater than irreversible work.

Free expansion against vacuum (P_ext = 0) has w = 0, q = 0, ΔT = 0 for ideal gas (both isothermal and adiabatic).

Standard Enthalpy via Bond Energies & Hess's Law

\Delta_r H^\circ = \sum \text{B.E.}_{\text{reactants}} - \sum \text{B.E.}_{\text{products}} = \sum \Delta_f H^\circ_{\text{products}} - \sum \Delta_f H^\circ_{\text{reactants}}

Enthalpy calculation from chemical bond dissociation energies or standard enthalpies of formation.

Standard enthalpy of formation Δ_f H° of elements in their reference standard states (e.g. C_graphite, O₂ gas, Br₂ liquid) is zero by convention.

Molar Heat Capacities & Degrees of Freedom

C_p - C_v = R, \quad \gamma = \frac{C_p}{C_v} = 1 + \frac{2}{f} \implies \begin{cases} \text{Monoatomic } (f=3): \gamma = 5/3 \approx 1.67 \\ \text{Diatomic } (f=5): \gamma = 7/5 = 1.40 \end{cases}

Mayer's relation and adiabatic exponent γ connection to molecular translational and rotational degrees of freedom.

At high temperatures, vibrational modes become active, contributing an additional R to C_v.

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Chemical & Ionic Equilibrium

K_p & K_c relations, pH calculations of weak acids/bases, buffer solutions, Henderson equation, and solubility product K_sp.

Kp and Kc Relation

K_p = K_c (RT)^{\Delta n_g}, \quad \Delta n_g = \sum n_{\text{gas (prod)}} - \sum n_{\text{gas (react)}}

Connects partial pressure equilibrium constant K_p and molarity constant K_c.

If Δn_g = 0, K_p = K_c and equilibrium is independent of total pressure.

Ostwald's Dilution Law & pH of Weak Acid

\alpha = \sqrt{\frac{K_a}{C}}, \quad [H^+] = \sqrt{K_a C} = C\alpha, \quad \text{pH} = \frac{1}{2}[\text{p}K_a - \log_{10} C]

Degree of dissociation and hydronium concentration for weak monoprotic acid.

Valid only when degree of dissociation α ≤ 0.05 (5%).

Henderson-Hasselbalch Equation (Buffer Solutions)

\text{pH} = \text{p}K_a + \log_{10}\left(\frac{[\text{Salt}]}{[\text{Acid}]}\right), \quad \text{pOH} = \text{p}K_b + \log_{10}\left(\frac{[\text{Salt}]}{[\text{Base}]}\right)

Calculates pH of acidic and basic buffer solutions resistent to pH change.

Maximum buffer capacity occurs when [Salt] = [Acid], giving pH = pKa.

Salt Hydrolysis pH Formulas

\text{Weak Acid + Strong Base: } \text{pH} = 7 + \frac{1}{2}(\text{p}K_a + \log_{10} C), \quad \text{Weak Acid + Weak Base: } \text{pH} = 7 + \frac{1}{2}(\text{p}K_a - \text{p}K_b)

Hydrolysis of salts producing alkaline or acidic aqueous solutions.

Salt of weak acid and weak base pH is independent of concentration C.

Solubility Product (K_sp) & Common Ion Effect

A_x B_y (s) \rightleftharpoons x A^{y+} + y B^{x-} \implies K_{\text{sp}} = x^x y^y S^{x+y}, \quad S' = \frac{K_{sp}}{[\text{Common Ion}]^y}

Equilibrium constant for sparingly soluble salts, and suppression of solubility in presence of shared ions.

Common ion effect drastically decreases solubility S without altering K_sp.

Strong Acid + Weak Base Salt Hydrolysis

\text{pH} = 7 - \frac{1}{2}(\text{p}K_b + \log_{10} C), \quad K_h = \frac{K_w}{K_b}, \quad h = \sqrt{\frac{K_w}{K_b C}}

Cationic hydrolysis producing an acidic solution (e.g. NH₄Cl).

Degree of hydrolysis h increases upon dilution.

Buffer Capacity & Effective Buffer Range

\beta = \frac{d B}{d\text{pH}} = 2.303 \frac{C_{\text{acid}} C_{\text{salt}}}{C_{\text{acid}} + C_{\text{salt}}}, \quad \text{Effective Range: } \text{pH} = \text{p}K_a \pm 1

Quantitative measure of buffer resistance to pH change upon addition of strong acid/base.

Maximum buffer capacity occurs at pH = pKa (equimolar ratio of conjugate acid-base pair).

Van 't Hoff Isochore & Temperature Dependence

\ln\left(\frac{K_2}{K_1}\right) = \frac{\Delta H^\circ}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right), \quad \text{Inert gas at const. } V: \Delta n_g \text{ no shift}

Variation of equilibrium constant K with absolute temperature T governed by standard enthalpy of reaction ΔH°.

Endothermic reactions (ΔH° > 0) have higher K at higher temperatures; exothermic reactions have lower K.

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Electrochemistry & Conductance

Nernst equation, Gibbs energy of cell, Kohlrausch law, Faraday laws of electrolysis, and molar conductivity.

Nernst Equation (at 298 K)

E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{0.0591}{n} \log_{10} Q

Computes non-standard cell potential under varying ion concentrations.

At equilibrium, E_cell = 0 and Q = K_eq.

Standard Cell Potential & Free Energy

\Delta G^\circ = -n F E^\circ_{\text{cell}}, \quad \log_{10} K = \frac{n E^\circ_{\text{cell}}}{0.0591}

Connects electrochemical cell voltage to standard Gibbs energy and equilibrium constant.

F = Faraday constant ≈ 96500 C/mol.

Molar Conductivity & Kohlrausch's Law

\Lambda_m = \frac{\kappa \times 1000}{M} \; (\text{S cm}^2\text{mol}^{-1}), \quad \Lambda_m^\circ = \nu_+ \lambda_+^\circ + \nu_- \lambda_-^\circ

Independent migration of ions at infinite dilution.

Degree of dissociation of weak electrolyte: α = Λ_m / Λ_m°.

Faraday's Laws of Electrolysis

w = Z I t = \left(\frac{E}{96500}\right) I t, \quad \frac{w_1}{E_1} = \frac{w_2}{E_2}

Mass w of substance liberated at electrode by current I in time t.

Equivalent weight E = Molar mass / n-factor.

Concentration Cell EMF

E_{\text{cell}} = \frac{0.0591}{n} \log_{10}\frac{C_2}{C_1} \quad (C_2 > C_1)

EMF generated purely by concentration difference of identical electrodes.

E°_cell is identically ZERO for concentration cells.

Debye-Hückel-Onsager Equation (Strong Electrolytes)

\Lambda_m = \Lambda_m^\circ - A\sqrt{C}

Linear variation of molar conductance with square root of concentration for completely dissociated strong electrolytes.

Extrapolation to C → 0 gives limiting molar conductance Λ_m° for strong electrolytes (cannot be extrapolated for weak electrolytes).

Solubility of Sparingly Soluble Salts via Conductance

S = \frac{\kappa \times 1000}{\Lambda_m^\circ} \; (\text{mol/L}), \quad K_{sp} = [M^{z+}]^x [X^{z-}]^y

Determining microscopic solubility S and solubility product K_sp from electrolytic specific conductance κ.

Specific conductance of salt alone is κ_salt = κ_solution - κ_water.

Cell Thermodynamics & Fuel Cell Efficiency

\Delta S = n F \left(\frac{\partial E}{\partial T}\right)_P, \quad \Delta H = -nFE + nFT\left(\frac{\partial E}{\partial T}\right)_P, \quad \eta_{\text{fuel cell}} = \frac{\Delta G^\circ}{\Delta H^\circ}

Entropy, enthalpy of cell reaction derived from temperature coefficient (∂E/∂T)_P, and theoretical efficiency of H₂-O₂ fuel cells.

If (∂E/∂T)_P > 0, cell absorbs heat from surroundings during operation and ΔS > 0.

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Solutions & Colligative Properties

Raoult's law, relative lowering of vapor pressure, elevation in boiling point, depression in freezing point, and van 't Hoff factor.

Raoult's Law (Liquid-Liquid Solutions)

P_{\text{total}} = P_A^\circ X_A + P_B^\circ X_B, \quad Y_A = \frac{P_A}{P_{\text{total}}} = \frac{P_A^\circ X_A}{P_{\text{total}}}

Partial vapor pressure of volatile components and mole fraction Y_A in vapor phase.

Positive deviation has P_total > Raoult's prediction (ΔH_mix > 0, ΔV_mix > 0).

Relative Lowering of Vapor Pressure (RLVP)

\frac{P_A^\circ - P_A}{P_A^\circ} = i X_B = i \frac{n_B}{n_A + n_B} \approx i \frac{w_B M_A}{M_B w_A}

Vapor pressure reduction caused by dissolving non-volatile solute B.

For dilute solutions: (P° - P)/P = i (n_B / n_A) is exact.

Boiling Point Elevation & Freezing Depression

\Delta T_b = i K_b m, \quad \Delta T_f = i K_f m \quad \left(m = \text{molality} = \frac{1000 w_B}{M_B w_A}\right)

Thermal colligative shifts dependent strictly on particle concentration.

For water: K_b = 0.52 K kg/mol, K_f = 1.86 K kg/mol.

Osmotic Pressure

\pi = i C R T = i \left(\frac{n_B}{V}\right) R T

Hydrostatic pressure required to stop inward osmosis through semipermeable membrane.

Best colligative property for determining molar mass of polymers and biomolecules.

Van 't Hoff Factor (Dissociation & Association)

i = 1 + (n - 1)\alpha \; (\text{Dissociation}), \quad i = 1 - \left(1 - \frac{1}{n}\right)\alpha \; (\text{Association}), \quad i = \frac{M_{\text{theoretical}}}{M_{\text{observed}}}

Ratio of observed colligative property to theoretical calculated colligative property.

For complete dimerization (n = 2, α = 1), i = 0.5.

Henry's Law of Gas Solubility

P = K_H \cdot \chi, \quad K_H \propto T \implies \text{Solubility decreases as } T \text{ increases}

Partial pressure of gas above liquid is proportional to mole fraction χ of gas dissolved in the liquid.

Aquatic species are more comfortable in cold water because higher dissolved O₂ is present at lower temperatures (smaller K_H).

Thermodynamic Expressions for K_b and K_f

K_b = \frac{R T_b^2 M_{\text{solvent}}}{1000 \Delta H_{\text{vap}}} = \frac{R T_b^2}{1000 l_v}, \quad K_f = \frac{R T_f^2 M_{\text{solvent}}}{1000 \Delta H_{\text{fus}}} = \frac{R T_f^2}{1000 l_f}

Ebullioscopic and cryoscopic constants determined solely by properties of the pure solvent.

K_b and K_f are independent of the nature of the solute dissolved.

Azeotropes & Deviations from Raoult's Law

\text{Positive Dev: } \Delta H_{\text{mix}} > 0, \; \Delta V > 0 \to \text{Min-Boiling Az.}; \quad \text{Negative Dev: } \Delta H < 0, \; \Delta V < 0 \to \text{Max-Boiling Az.}

Constant-boiling binary liquid mixtures that distill without changing composition in liquid and vapor phases.

Acetone + Chloroform shows negative deviation due to intermolecular C-H···O hydrogen bonding.

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Chemical Kinetics

Rate laws, order and molecularity, integrated rate equations for zero and first order, half-life, and Arrhenius equation.

Rate of Reaction & Rate Law

\text{Rate} = -\frac{1}{a}\frac{d[A]}{dt} = k [A]^x [B]^y, \quad \text{Order} = x + y

Stoichiometric rate definition and experimental power dependence.

Order can be zero, fractional, or negative; molecularity is always integer.

Integrated Rate Laws (Zero & First Order)

[A]_t = [A]_0 - kt \; (\text{Zero Order}), \quad \ln\frac{[A]_0}{[A]_t} = kt \iff [A]_t = [A]_0 e^{-kt} \; (\text{First Order})

Concentration-time profiles for fundamental reaction mechanisms.

First-order equation in log10: k = (2.303/t) * log10([A]₀ / [A]_t).

Half-Life Formulas

t_{1/2} = \frac{[A]_0}{2k} \; (\text{Zero Order}), \quad t_{1/2} = \frac{\ln 2}{k} = \frac{0.693}{k} \; (\text{First Order}), \quad t_{1/2} \propto \frac{1}{[A]_0^{n-1}}

Time required for reactant concentration to reduce to half its initial value.

For first-order: t_75% = 2 * t_50%, and t_99.9% ≈ 10 * t_50%.

Arrhenius Equation & Activation Energy

k = A e^{-E_a / RT} \iff \log_{10}\frac{k_2}{k_1} = \frac{E_a}{2.303 R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)

Temperature dependence of rate constant governed by activation energy barrier E_a.

Plot of ln k vs 1/T has slope -E_a / R.

Catalyst Mechanism

\Delta E_a = E_a - E_a', \quad \frac{k_{\text{cat}}}{k_{\text{uncat}}} = e^{\Delta E_a / RT}

A catalyst provides an alternate pathway with lower activation energy, accelerating both forward and backward rates equally.

A catalyst does NOT change equilibrium constant K or ΔG.

Second-Order & General n-th Order Integrated Laws

\text{2nd Order: } \frac{1}{[A]_t} - \frac{1}{[A]_0} = kt \implies t_{1/2} = \frac{1}{k [A]_0}, \quad n\text{-th Order: } \frac{1}{n-1}\left(\frac{1}{[A]_t^{n-1}} - \frac{1}{[A]_0^{n-1}}\right) = kt

Integrated rate law and half-life for second-order and general n-th order reactions (n ≠ 1).

Plot of 1/[A]_t versus t for a second-order reaction yields a straight line with slope +k.

Parallel (Branched) First-Order Reactions

k_{\text{eff}} = k_1 + k_2, \quad \frac{[B]_t}{[C]_t} = \frac{k_1}{k_2}, \quad \% \text{ Yield of } B = \frac{k_1}{k_1 + k_2} \times 100\%, \quad E_{a,\text{eff}} = \frac{k_1 E_{a1} + k_2 E_{a2}}{k_1 + k_2}

Kinetics where reactant A decomposes simultaneously into two distinct products B and C.

Product ratio [B]/[C] is constant and equal to k₁/k₂ at all times throughout the reaction.

Consecutive Reactions & Maximum Intermediate Time

A \xrightarrow{k_1} B \xrightarrow{k_2} C \implies t_{\text{max}} = \frac{\ln(k_2 / k_1)}{k_2 - k_1}, \quad [B]_{\text{max}} = [A]_0 \left(\frac{k_1}{k_2}\right)^{\frac{k_2}{k_2 - k_1}}

Sequential first-order decay. Intermediate concentration [B] rises to a peak and then decays to zero.

If k₂ >> k₁, intermediate B reacts immediately upon formation, validating the Steady State Approximation: d[B]/dt ≈ 0.

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Solid State Chemistry

Unit cell geometry, density formula, packing efficiency of SC/BCC/FCC, radius ratios, and crystal defects.