Physical Chemistry for Engineers 2: Comprehensive Study Notes

Definition and Classification of Chemical Equilibrium

Definition of Chemical Equilibrium: It is a well-established fact that many reactions do not go to completion. They proceed to a certain point and then apparently stop, leaving considerable amounts of unaffected reactants. Under a given set of conditions (temperature, pressure, concentration), the point at which a reaction stops is always the same; there exists a definitely fixed relationship among the concentrations. This state is called equilibrium.
Rate Balance: At equilibrium, the rate at which reactants disappear to form products is exactly equal to the rate at which products interact to reform the reacting substances.
Homogeneous Equilibria: Established in a system where only one phase occurs (only gases, a single liquid, or a single solid phase). * Examples: * Ammonia Equilibrium: $3/2H_2(g) + 1/2N_2(g) \rightleftharpoons NH_3(g)$ * Phosgene Equilibrium: $CO(g) + Cl_2(g) \rightleftharpoons COCl_2(g)$ * Dissociation of Hydrogen Sulfide: $2H_2S(g) \rightleftharpoons 2H_2(g) + S_2(g)$
Heterogeneous Equilibria: Established in a system in which more than a single phase appears (solid and gas, liquid and gas, solid and liquid, or solid and solid). * Examples: * Dissociation of Cupric Oxide: $4CuO(s) \rightleftharpoons 2Cu_2O(s) + O_2(g)$ * Carbon Disulfide Equilibrium: $C(s) + S_2(g) \rightleftharpoons CS_2(g)$ * Dissociation of Ammonium Carbamate: $NH_2COONH_4(s) \rightleftharpoons 2NH_3(g) + CO_2(g)$

The Thermodynamic Equilibrium Constant and Gaseous Reactions

Reaction Gibbs Energy Equation: For a general reaction $aA + bB \rightleftharpoons cC + dD$ , the Gibbs energy change is given by: $\Delta G = \Delta G^0 + RT \ln(K_a)$ * $\Delta G^0$ : Gibbs free energy change in the standard state. * $K_a$ : Thermodynamic equilibrium constant.
Constants and Temperature: $K_a$ at equilibrium must be constant and independent of all other factors EXCEPT temperature.
Spontaneity Criteria: * If \Delta G < 0: Forward reaction is spontaneous. * If \Delta G > 0: Reverse reaction is spontaneous. * If $\Delta G = 0$ : Reaction is at equilibrium; the equation becomes $\Delta G^0 = -RT \ln(K_a)$ .
Lewis Fugacity Rule: The activity of any gas in a mixture is equal to the partial pressure of the gas multiplied by the activity coefficient of the pure gas at the total pressure of the mixture ( $a_i = \gamma_i P_i = \gamma_i N_i P_T$ ).
Relationship of $K_p$ to $K_a$ : $K_a = K_p K_{\gamma}$ * $K_p$ : Equilibrium constant expressed in pressures (must use atmospheres for this substitution). * For ideal gases or real gases at zero pressure, $\gamma = 1$ , therefore $K_{\gamma} = 1$ and $K_a = K_p$ . * For nonideal gases at pressures above zero, $\gamma$ deviates from unity; $K_{\gamma}$ is determined by the total pressure of the system.
Relationship of $K_p$ to $K_c$ : $K_p = K_c (RT)^{\Delta n_g}$

Properties and Data of Gaseous Equilibria

Equilibrium Generalizations: 1. Adding excess reactants tends to drive a reaction further to completion regarding reactants not in excess. 2. Initial presence of products tends to decrease the extent of conversion of reactants to products.
The Ammonia Equilibrium: Reaction: $3/2H_2(g) + 1/2N_2(g) \rightleftharpoons NH_3(g)$ . * Investigators: Haber and co-workers, Nernst and Jellinek, Larson and Dodge. * Data at 10 atm: At $T = 350^\circ C$ , $\text{\%}NH_3 = 7.35$ , $K_p = 0.0266$ . At $T = 450^\circ C$ , $\text{\%}NH_3 = 2.04$ , $K_p = 0.00659$ . * Data at 50 atm: At $T = 350^\circ C$ , $\text{\%}NH_3 = 25.11$ , $K_p = 0.0278$ .
The Phosgene Equilibrium: Reaction: $CO(g) + Cl_2(g) \rightleftharpoons COCl_2(g)$ . * Investigators: Max Bodenstein and Heinrich Plaut. * Data at $394.8^\circ C$ : Initial pressures $P_{Cl_2,i} = 351.4\,mmHg$ , $P_{CO,i} = 342.0\,mmHg$ , total equilibrium pressure $P_T = 439.5\,mmHg$ . If $x$ is the pressure drop of $Cl_2$ , $x$ is also the partial pressure of phosgene.
Dissociation of Hydrogen Sulfide: Reaction: $2H_2S(g) \rightleftharpoons 2H_2(g) + S_2(g)$ . * Investigators: Preuner and Schupp. * Data at 1 atm: * At $750^\circ C$ : $\alpha = 0.055, K_p = 0.000091$ * At $1132^\circ C$ : $\alpha = 0.307, K_p = 0.0260$
Effect of Inert Gases: Inert gases do not affect the thermodynamic constant $K_a$ , but they modify activity coefficients ( $\gamma$ s) and $K_{\gamma}$ , thus changing $K_p$ . They affect partial pressures at a given total pressure, inducing a shift in the reaction extent to meet the demands of the equilibrium constant.

Heterogeneous and Hydrate Equilibria

Condensed Phases: The activity of a pure solid or liquid is taken as unity up to high pressures. They are disregarded in writing $K_p$ expressions (referred to as condensed equilibrium constants).
Dissociation of Cupric Oxide ( $CuO$ ): $4CuO(s) \rightleftharpoons 2Cu_2O(s) + O_2(g)$ . * Investigators: F. Hastings Smith and H.R. Robert. * Data: At $900^\circ C, K_p = P_{O_2} = 12.5\,mmHg$ . At $1080^\circ C, K_p = 388.0\,mmHg$ .
Carbon Disulfide Equilibrium: $C(s) + S_2(g) \rightleftharpoons CS_2(g)$ . * Investigator: F. Koref. * Data: $K_p$ values range from $5.45$ to $5.73$ across various volumes of $CS_2$ and $S_2$ .
Dissociation of Ammonium Carbamate: $NH_2COONH_4(s) \rightleftharpoons 2NH_3(g) + CO_2(g)$ . * Investigators: T.R. Briggs and V. Migidichian. * Data at $30^\circ C$ : Initial ammonia pressure $e_1 = 0, P_T = 125.0\,mmHg, K_p = 2.89 \times 10^{-5}$ . At $e_1 = 168.6, P_T = 194.2\,mmHg, K_p = 2.93 \times 10^{-5}$ .
Effect of Pressure: Influence is predicted by Le Chatelier's principle. Volumes of condensed phases are disregarded as they are negligible compared to gases.
Equilibria in Hydrates: The constant is $K_p = P^x$ , where $x$ is the moles of vapor and $P$ is the vapor pressure above the solid phases. The vapor pressure of a hydrate-anhydride pair is constant. * Aqueous Vapor Pressure pair examples at $25^\circ C$ : * $MgSO_4 \cdot 7H_2O - MgSO_4 \cdot 6H_2O: P = 11.5\,mmHg$ * $CuSO_4 \cdot 5H_2O - CuSO_4 \cdot 3H_2O: P = 7.80\,mmHg$ * $Na_2HPO_4 \cdot 12H_2O - Na_2HPO_4 \cdot 7H_2O: P = 19.13\,mmHg$

Temperature Variation of Equilibrium Constants

General Form for $K_a$ : $\frac{d \ln K_a}{dT} = \frac{\Delta G^0}{RT^2}$
Integrated Form for $K_p$ : $\ln\left(\frac{K_{p2}}{K_{p1}}\right) = \frac{\Delta H^0}{R} \left(\frac{T_2 - T_1}{T_1 T_2}\right)$
Linear Form (Calculator Y = A + BX): $\ln K_p = -\frac{\Delta H^0}{R} \left(\frac{1}{T}\right) + C$ * $Y = \ln K_p$ * $X = 1/T$ * $B = -\Delta H^0 / R$
Variation of $K_c$ : $\frac{d \ln K_c}{dT} = \frac{\Delta E^0}{RT^2}$ , where $\Delta E^0$ is the heat of reaction at constant volume.

Solutions of Nonelectrolytes: Definitions and Classification

Types of Mixtures: 1. Coarse mixture: Individual particles are discernable and mechanically separable (e.g., salt and sugar). 2. Colloidal dispersion: Finer particles, heterogeneity not readily apparent but exists (e.g., clay shaken with water). 3. True solution: Constituents cannot be mechanically separated; every part is like every other part; constitutes a homogeneous phase (e.g., sugar in water).
Terms: * Solute: Substance that dissolves. * Solvent: Substance in which solution takes place. * Unsaturated: Contains less than the maximum dissolvable amount. * Saturated: Contains the maximum solute for a given temperature. * Supersaturated: Contains more than the solvent can normally dissolve. * Non-electrolytic: Solute persists in molecular, uncharged form. * Electrolytic: Solute dissociates into electrically charged ions.
Solubility Factors: * Nature: "Like dissolves like." Completely miscible (e.g., ethyl alcohol and water), completely immiscible (e.g., water and mercury), or partially miscible (e.g., ether and water). * Temperature: If heat is evolved at saturation, solubility decreases with rising temperature. If heat is absorbed, solubility increases. * Pressure: Small effect unless gases are involved.
Concentration Expressions: * Weight basis (temperature-independent): Percent by weight, weight per weight of constituent, molality (moles solute per kg solvent), mole fraction. * Volume basis (temperature-dependent): Percent by volume, weight per volume, molarity (moles per liter), normality (equivalents per liter).

Thermodynamic Properties of Solutions and the Solution Process

Total Free Energy ( $G$ ): $G = n_1 G_1 + n_2 G_2 + \dots$ , where $G_i$ are partial molal free energies.
Total Entropy ( $S$ ) and Enthalpy ( $H$ ): $S = n_1 S_1 + n_2 S_2$ ; $H = n_1 H_1 + n_2 H_2$ .
Relation: $G = H - TS$ and $G_i = H_i - TS_i$ .
Partial Molal Free Energy: $G_i = G_i^0 + RT \ln(a_i)$ . * Standard state for miscible substances is often the pure substance where $a_i = 1$ and $G_i = G_i^0$ . * Activity Coefficient ( $\gamma'_i$ ): Converts mole fraction ( $N_i$ ) to activity ( $a_i$ ): $a_i = \gamma'_i N_i$ . * Infinite Dilution Reference: As $N_1 \rightarrow 1$ , $\gamma'_1 \rightarrow 1$ . For solute $a_2 = \gamma'_2 N_2$ ; as $N_2 \rightarrow 0$ , $\gamma'_2 \rightarrow 1$ . * Concentration Basis: * Molar (moles/L): $a_2 = f_2 C_2$ ; as $C_2 \rightarrow 0$ , $f_2 \rightarrow 1$ . * Molal: $a_2 = \gamma_2 m_2$ ; as $m_2 \rightarrow 0$ , $\gamma_2 \rightarrow 1$ .
Mixing Thermodynamics (Binary): * $\Delta G_m = n_1(G_1 - G_1^0) + n_2(G_2 - G_2^0) = n_1 RT \ln(a_1) + n_2 RT \ln(a_2)$ * $\Delta V_m = n_1(V_1 - V_1^0) + n_2(V_2 - V_2^0)$ * $\Delta S_m = n_1(S_1 - S_1^0) + n_2(S_2 - S_2^0)$ * $\Delta H_m = n_1(H_1 - H_1^0) + n_2(H_2 - H_2^0)$ * $\Delta G_m = \Delta H_m - T\Delta S_m$
Phase Equilibrium: For equilibrium in multicomponent systems across phases, the partial molal free energy of each component must be the same in all phases ( $G'_i = G''_i = G'''_i$ ).

Vapor Pressure and Ideal Solutions

Equilibrium Between Solution and Vapor: For a volatile constituent $i$ , the activity is given by the ratio of fugacities: $a_i = \frac{f_i(g)}{f_i^0(g)}$ .
Ideal Vapor Behavior: $a_i = \frac{P_i}{P_i^0}$ , where $P_i$ is the partial pressure above the solution and $P_i^0$ is the vapor pressure of the pure constituent.
Free Energy of Mixing (Ideal Gases): $\Delta G_m = n_1 RT \ln\left(\frac{P_1}{P_1^0}\right) + n_2 RT \ln\left(\frac{P_2}{P_2^0}\right)$ .
Ideal Solution Definition: One where activity equals mole fraction ( $a_i = N_i$ ) for all conditions. * $\Delta G_m = n_1 RT \ln(N_1) + n_2 RT \ln(N_2)$ . * $\Delta V_m = 0$ : No volume change on mixing. * $\Delta H_m = 0$ : No evolution or absorption of heat. * $\Delta S_m = -R[n_1 \ln(N_1) + n_2 \ln(N_2)]$ .
Raoult's Law: $P_1 = N_1 P_1^0$ and $P_2 = N_2 P_2^0$ .
Vapor Mole Fraction ( $Y_1$ ): $Y_1 = \frac{N_1 P_1^0}{N_1 P_1^0 + (1 - N_1)P_2^0}$ .

Deviation from Raoult's Law and Azeotropes

Classification of Binary Miscible Liquid Pairs: * Type 1: Total vapor pressure is intermediate between pure components (e.g., Benzene - Toluene, Carbon tetrachloride - Benzene). * Type 2: Exhibit a maximum in total vapor pressure curve; positive deviation from Raoult's Law (e.g., Benzene - Ethyl alcohol, Acetone - Carbon disulfide). * Type 3: Exhibit a minimum in total vapor pressure curve; negative deviation (e.g., Chloroform - Acetone, Pyridine - Acetic acid, Water - Nitric acid).
Azeotropes: Mixtures of Type II and III where liquid and vapor compositions are identical at a specific point. They are not definite compounds; composition changes with total pressure.
Fractional Distillation and Fractionating Columns: * Equilibrium distillation: vapor is in equilibrium with the total mass of boiling liquid. * Fractional distillation: separation via a fractionating column (Still A, Column D, Condenser F). * Operation: The vapor bubbles through liquid layers on "bubble caps"; each plate acts as a miniature still, redistributing components so the vapor becomes richer in the more volatile constituent while the liquid becomes richer in the less volatile one.
Boiling Point Diagrams: * Type I: Intermediate boiling points. * Type II: Minimum boiling point. * Type III: Maximum boiling point. * Lever Rule for Weight Ratio: $\frac{W_1}{W_2} = \frac{bc}{ab} = \frac{x_v - x}{x - x_l}$ .

Solubility of Partially Miscible Liquid Pairs

Type I: Maximum Solution Temperature: (e.g., Water and Aniline). At $100^\circ C$ , mixtures between compositions A and $A_1$ yield two layers. The temperature where solubility becomes complete is the Critical Solution Temperature or Consolute Temperature.
Type II: Minimum Solution Temperature: (e.g., Triethylamine in Water). Completely miscible below $18.5^\circ C$ . Curve exhibits a minimum critical solution temperature.
Type III: Maximum and Minimum Solution Temperature: (e.g., Nicotine in Water). Upper critical temperature at $60.8^\circ C$ (34% nicotine); also has a lower critical temperature.
Type IV: Without Critical Solution Temperature: (e.g., Ethyl ether and Water). Only partially soluble at all temperatures where the solution exists.
Relative Weights: Weight calculation of layers via $\frac{\text{Weight of Water Layer}}{\text{Weight of Amine Layer}} = \frac{\text{distance ab}}{\text{distance ca}}$ .

Colligative Properties of Nonelectrolyte Solutions

Colligative Property Definition: Depends only on the number of particles in solution, not their nature.
Vapor Pressure Lowering: $\Delta P = P^0 - P = N_2 P^0$ . * Molecular Weight Determination: $\frac{P^0 - P}{P^0} = \frac{W_2 M_1}{W_1 M_2}$ .
Boiling Point Elevation (Ebullioscopy): $\Delta T_b = T - T_0 = K_b m$ . * $K_b$ (Molal BP Elevation Constant): $\Delta T_b$ for a 1-molal solution. * Calculation: $K_b = \frac{R T_0^2}{1000 \Delta H_v}$ . * Molecular Weight: $M_2 = \frac{1000 W_2 K_b}{W_1 \Delta T_b}$ .
Freezing Point Lowering (Cryoscopy): $\Delta T_f = T_0 - T = K_f m$ . * $K_f$ (Cryoscopic Constant): $\Delta T_f$ for a 1-molal solution. * Calculation: $K_f = \frac{R T_0^2}{1000 \Delta H_f}$ . * Common $K_f$ values: Water ( $1.86$ ), Benzene ( $5.12$ ), Camphor ( $37.7$ ).
Osmotic Pressure ( $P$ or $\Pi$ ): Observed when a semipermeable membrane (e.g., copper ferrocyanide $Cu_2Fe(CN)_6$ ) separates solution from solvent. * Relation to Vapor Pressure: $P = \frac{RT}{V_1^0} \ln\left(\frac{P_1^0}{P_1}\right)$ . * Van't Hoff Equation: $P = CRT$ (Analogous to ideal gas law; applies to dilute solutions < 0.2\,M).

Nernst Distribution Law and Chemical Equilibrium in Solution

Distribution Law: A substance distributes between two solvents such that $\frac{a_A}{a_B} = K$ or $\frac{C_A}{C_B} = K$ at equilibrium.
Series of Extractions: * Amount unextracted: $W_n = W \left(\frac{K V_1}{K V_1 + V_2}\right)^n$ , where $V_1$ is volume of solvent, $V_2$ is extraction solvent, and $n$ is number of extractions. * Amount extracted: $W - W_n = W \left[1 - \left(\frac{K V_1}{K V_1 + V_2}\right)^n\right]$ .
Equilibrium Constant in Solution ( $K_a$ ): $K_a = \frac{a_C^c a_D^d}{a_A^a a_B^b}$ . * Relationship: $K_a = K_c K_f$ .

Solutions of Electrolytes and Arrhenius Theory

Colligative Properties of Electrolytes: Higher than nonelectrolytes of the same concentration.
Van't Hoff Factor ( $i$ ): $i = \frac{\text{Colligative effect for electrolyte}}{\text{Colligative effect for nonelectrolyte at same conc.}}$ . * $i = \frac{\Delta P}{\Delta P_0} = \frac{\Delta T_b}{\Delta T_{b0}} = \frac{\Delta T_f}{\Delta T_{f0}} = \frac{P}{P_0}$ .
Arrhenius Theory (1887): Electrolytes dissociate into ions such that total positive charge equals total negative charge.
Degree of Ionization ( $\alpha$ ): $i = 1 + \alpha(n - 1)$ where $n$ is the total number of ions per molecule. * Calculation: $\alpha = \frac{i - 1}{n - 1}$ . * Reaction: $A_x B_y \rightleftharpoons xA + yB$ , total molality $m_t = m[1 + \alpha(x + y - 1)]$ .
Debye-Huckel Theory (Interionic Attraction): For strong electrolytes (assumed completely ionized).
Ionic Strength ( $\mu$ ): $\mu = 1/2 \sum C_i z_i^2$ , where $C_2$ is concentration and $z_i$ is valence.
Debye-Huckel Factor $i$ (at $0^\circ C$ ): $i = v(1 - 0.375 z_+ z_- \sqrt{\mu})$ .

Electrochemistry: Units and Electrolytic Conduction

Ohm's Law: $I = E / R$ .
Units: * Ampere: International ampere deposits $0.00111800\,g$ of silver in 1 second. * Ohm: Resistance of a mercury column $106.300\,cm$ long, $14.4521\,g$ at $0^\circ C$ . * Coulomb ( $Q$ ): $Q = It$ . * Faraday ( $F$ ): $96,490\,abs.\,coulombs$ (or $96,487$ ). * Joule ( $w$ ): $w = EIt = EQ$ . ( $1\,joule = 1 \times 10^7\,ergs = 0.2390\,cal$ ). * Watt ( $p$ ): Unit of power; $p = EI = w/t$ .
Types of Conductors: 1. Electronic: Conduction by electron migration; stationary ions; resistance increases with temperature (e.g., metals). 2. Electrolytic: Conduction by ion migration; transport of matter; resistance decreases with temperature; chemical changes at electrodes.
Electrode Reactions (Example HCl): * Cathode: $2H^+ + 2e^- \rightarrow H_2(g)$ . * Anode: $2Cl^- \rightarrow Cl_2(g) + 2e^-$ .
Faraday's Laws of Electrolysis: * 1st Law: Mass of substance is proportional to quantity of electricity passed. * 2nd Law: Masses are proportional to equivalent weights ( $96,487\,C$ yields 1 equivalent weight).

Transference and Conductance

Transference Numbers ( $t$ ): Fraction of total current carried by an ion. * $t_+ = \frac{v_+}{v_+ + v_-}$ and $t_- = \frac{v_-}{v_+ + v_-}$ . * $t_+ + t_- = 1$ .
Hittorf's Rule: $\frac{\text{loss at cathode}}{\text{loss at anode}} = \frac{v_+}{v_-} = \frac{t_+}{t_-}$ .
Determination Methods: 1. Hittorf Method: Observing concentration changes at electrodes. 2. Moving Boundary Method: Direct observation of boundary motion: $t_+ = \frac{V C}{1000 Q}$ .
Conductance ( $L$ ): Reciprocal of resistance ( $L = 1/R$ ). * Specific conductance ( $L_s$ ): $L_s = (1/R) \cdot (l/A)$ . * Cell constant ( $K$ ): $K = l/A$ . $L_s = K / R$ . * Equivalent conductance ( $\Lambda$ ): $\Lambda = \frac{1000 L_s}{C}$ .
Wheatstone Bridge: Used for measuring resistance: $R_x = R_s \cdot (R_1 / R_2)$ .
Kohlrausch's Law (Infinite Dilution): $\Lambda_0 = l_+^0 + l_-^0$ . * Transition for strong electrolytes: $\Lambda = \Lambda_0 - b \sqrt{C}$ .
Effect of Temperature: $\Lambda_{0,T} = \Lambda_{0,25} [1 + \beta(T - 25)]$ .
Debye-Huckel-Onsager: $\Lambda = \Lambda_0 - [\theta \Lambda_0 + \sigma] \sqrt{C}$ . * For 1-1 electrolytes at $25^\circ C$ : $\theta = 0.2273, \sigma = 59.78$ .

Electrochemical Cells and The Phase Rule

Electrochemical Cell: Converts chemical energy to electrical (Galvanic) or vice-versa (Electrolytic).
Reversibility: Requires forces to be infinitesimally different; change must be reversible by infinitesimal force variation.
Weston Saturated Standard Cell: Cell reaction: $Cd(s) + Hg_2SO_4(s) + 8/3 H_2O(l) \rightleftharpoons CdSO_4 \cdot 8/3 H_2O(s) + 2Hg(l)$ . * EMF at $t(^\circ C)$ : $\xi_t = 1.01830 - 4.06 \times 10^{-5}(t-20) - 9.5 \times 10^{-7}(t-20)^2$
EMF Signs: $\Delta G = -nFE$ . Spontaneous reaction has E > 0 and \Delta G < 0.
Reference Electrodes: * Standard Hydrogen Electrode: Potential defined as $0.0000\,V$ at all temperatures. * Calomel Electrodes ( $Hg|Hg_2Cl_2, KCl$ ): $\text{Emf (0.1 N)} = 0.3338 - 7 \times 10^{-5}(t - 25)$ .
The Phase Rule (Gibbs): $F = C - P + 2$ . * $P$ : Number of phases (state of matter uniform in chemical and physical state). * $C$ : Number of components (minimum independent species to define composition). * $F$ : Variance (degrees of freedom; intensive variables like $P, T$ changeable independently). * One Component System: For 3 phases in equilibrium, $F = 1 - 3 + 2 = 0$ (Triple point).

Chemical Kinetics: Rates and Mechanisms

Kinetics vs Thermodynamics: Thermodynamics considers energy/equilibrium; kinetics considers rate/mechanism and the stages of conversion.
Classification: * Homogeneous/Heterogeneous; Reversible/Irreversible; Elementary (order = stoichiometry) / Non-elementary. * Molecularity: Number of atoms/ions/molecules in the rate-limiting step. * Multiple reactions: Consecutive ( $A \rightarrow R \rightarrow S$ ), Parallel/Competing, and Mixed.
Rate Definitions: For $A + 2B \rightarrow 3C + D$ * $\text{Rate} = -\frac{d[A]}{dt} = -1/2 \frac{d[B]}{dt} = 1/3 \frac{d[C]}{dt} = \frac{d[D]}{dt}$ .
Rate Law: $v = k [A]^a [B]^b$ . * Order: Sum of exponents ( $a + b$ ). * Determination: Isolation method (holding all but one reactant in excess); Method of Initial Rates ( $\log v_0 = \log k + a \log [A]_0$ ).
First Order Reactions: $\frac{dx}{dt} = k_1(a - x)$ . * Integrated form: $\ln\left(\frac{a}{a - x}\right) = k_1 t$ or $\log(a - x) = -\frac{k_1}{2.303} t + \log a$ . * Half-life ( $t_{1/2}$ ): $t_{1/2} = \frac{\ln 2}{k_1}$ ; independent of initial concentration.

Colloids and Quantum Theory

Colloid Size: Particles $10^{-7}$ to $10^{-3}\,cm$ ( $500\,nm$ ).
Systems: * Dispersions: Insoluble substances (Thermodynamically unstable, tend to coagulate). * Association Colloids: Low molecular weight aggregates (micelles).
Colloid Types (Phase/Medium): * Aerosol (Solid or Liquid in Gas), Sol (Solid in Liquid), Emulsion (Liquid in Liquid), Gel (Liquid in Solid), Foam (Gas in Liquid).
Optical Properties: Tyndall Effect (light scattering). Turbidity ( $\tau = 1/l \ln(I_0/I)$ ).
Quantum Origins: Classical physics failed at "Ultraviolet Catastrophe" (Black-body radiation) and Dulong-Petit Law (Heat capacity deviations at low $T$ ).
Wave-Particle Duality: Entities like photons/electrons exhibit both behaviors.
Schrödinger Equation: $-\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} + V(x)\psi = E\psi$ .
Born Interpretation: Square of wavefunction ( $|\psi|^2$ ) is proportional to the probability of finding the particle.
Heisenberg Uncertainty Principle: Impossible to specify both position and momentum with perfect accuracy ( $\Delta x \Delta p \ge \hbar/2$ ).