Comprehensive Physical Chemistry Notes

Kinetic Theory of Gases and the Gas Laws

• States and Characteristics of Matter:

  • Matter is defined as anything possessing mass and occupying space; it excludes a vacuum and is intrinsically associated with volume.
  • Solid State:
  • Possesses definite shape and definite volume.
  • Constituent species are arranged in a regular, repeating crystal lattice.
  • Intermolecular and interionic forces of attraction between constituent species are strong.
  • Constituent species undergo vibrational motion about fixed positions, with no individual translational or rotational motion.
  • Liquid State:
  • Possesses definite volume but takes the shape of the containing vessel.
  • Forces of attraction are weaker than those in solids, allowing molecular mobility.
  • Gaseous State:
  • Lacks both definite shape and definite volume, occupying the entire available volume of its container.
  • Constituent species possess extensive translational, rotational, and vibrational freedom.

• The Concept and Postulates of an Ideal Gas:

  • A gas consists of a very large collection of tiny particles (molecules) in continuous, random motion in all directions.
  • For a gas to be classified as thermodynamically ideal, three core conditions must be satisfied:
  • Molecules move randomly and independently of one another, with zero net attractive or repulsive forces existing between them.
  • The aggregate physical volume occupied by the gas molecules themselves is negligible compared to the total volume of the container ($V_{\text{molecules}} \ll V_{\text{container}}$).
  • All intermolecular collisions, as well as collisions with the container walls, are perfectly elastic, meaning that linear momentum and total kinetic energy are conserved.

• Gas Variables and Origins of Pressure:

  • The pressure ($P$) of a gas originates from the physical collisions of gas molecules against the walls of the containing vessel, applicable to both ideal and non-ideal gases.
  • The state of a gaseous system is defined by four measurable state variables:
  • Pressure ($P$
  • Volume ($V$
  • Temperature ($T$
  • Quantity of gas ($n$ in moles or mass $m$)

• Boyle’s Law (Isothermal Volume-Pressure Dependence):

  • At constant temperature ($T$) and constant quantity of gas ($n$), the volume of a definite mass of gas is inversely proportional to its pressure:     $V \propto \frac{1}{P}$$PV = k$
  • The value of the proportionality constant $k$ depends directly upon the temperature, mass, and chemical nature of the gas.
  • For an isothermal process (a process carried out at constant temperature) between initial and final states:     $P_1 V_1 = P_2 V_2$
  • Graphical representation of volume versus pressure produces a rectangular hyperbola known as an isotherm.

• Worked Numerical Problem:

  • Problem: A gas occupies $390\,\text{cm}^3$ at $1\,\text{atm}$ pressure. Calculate the volume occupied if the pressure increases to $1.5\,\text{atm}$ at constant temperature.

  • Solution:$P_1 = 1\,\text{atm}$$V_1 = 390\,\text{cm}^3$$P_2 = 1.5\,\text{atm}$$V_2 = \frac{P_1 V_1}{P_2} = \frac{1\,\text{atm} \times 390\,\text{cm}^3}{1.5\,\text{atm}} = 260\,\text{cm}^3$

  • Gay-Lussac’s Law and Charles’s Law (Isobaric Volume-Temperature Dependence):

• At constant pressure ($P$), the volume of a fixed mass of gas expands by a constant relative fraction for every degree Celsius rise in temperature:     $V_t = V_0(1 + \alpha t)$     where $V_0$ is the volume at $0^\circ\text{C}$, $V_t$ is the volume at temperature $t\,^\circ\text{C}$, and $\alpha$ is the thermal coefficient of expansion.

• Experimentally, $\alpha = \frac{1}{273.15}\,^\circ\text{C}^{-1}$, yielding:     $V_t = V_0\left(1 + \frac{t}{273.15}\right) = V_0\left(\frac{273.15 + t}{273.15}\right)$

• Introducing the Absolute Temperature Scale ($T = t + 273.15$$\frac{V_1}{T_1} = \frac{V_2}{T_2}$$\frac{V}{T} = \text{constant}$

• Charles’s Law Formulation: The volume of a given mass of gas at constant pressure is directly proportional to its absolute temperature ($V \propto T$). Constant-pressure processes are termed isobaric.

  • The Combined Equation of State and Isochoric Dependence:

• Derivation in two stages for a sample changing from $(P_1, V_1, T_1)$ to $(P_2, V_2, T_2)$

  • Stage 1: Isothermal step at $T_1$. Pressure changes from $P_1$ to $P_2$, volume changes from $V_1$ to an intermediate volume $V_i$:       $P_1 V_1 = P_2 V_i \implies V_i = \frac{P_1 V_1}{P_2}$
  • Stage 2: Isobaric step at $P_2$. Temperature changes from $T_1$ to $T_2$, volume changes from $V_i$ to $V_2$:       $\frac{V_i}{T_1} = \frac{V_2}{T_2} \implies V_i = \frac{V_2 T_1}{T_2}$
  • Equating expressions for $V_i$:       $\frac{P_1 V_1}{P_2} = \frac{V_2 T_1}{T_2} \implies \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} = \text{constant}$

• Isochoric Processes (Constant Volume):

  • When volume remains constant ($V_1 = V_2$), the equation reduces to:       $\frac{P_1}{T_1} = \frac{P_2}{T_2} \implies \frac{P}{T} = \text{constant}$

  • The pressure of a given mass of gas at constant volume is directly proportional to its absolute temperature.

  • Avogadro’s Principle and the Ideal Gas Equation:

• Avogadro’s Law: Equal volumes of all gases at the same temperature and pressure contain identical numbers of molecules.

• For one mole of any gas, the ratio $\frac{PV}{T}$ equals a universal constant $R$, the molar gas constant:     $PV_m = RT$

• For $n$ moles of an ideal gas:     $PV = nRT$

• Substituting $n = \frac{m}{M}$ (where $m$ is mass and $M$ is molar mass):     $PV = \frac{m}{M}RT \implies P = \frac{\rho RT}{M}$     where $\rho = \frac{m}{V}$ is the density of the gas.

• At Standard Temperature and Pressure (STP: $0^\circ\text{C} = 273.15\,\text{K}$, $1\,\text{atm} = 101{,}325\,\text{N}\,\text{m}^{-2}$), one mole of any ideal gas occupies $22.414\,\text{L}$ ($22.414 \times 10^{-3}\,\text{m}^3$).

• Evaluation of $R$ in SI Units:$R = \frac{PV}{nT} = \frac{101{,}325\,\text{N}\,\text{m}^{-2} \times 22.414 \times 10^{-3}\,\text{m}^3}{1\,\text{mol} \times 273.15\,\text{K}} = 8.314\,\text{J}\,\text{K}^{-1}\,\text{mol}^{-1}$

• In alternate units: $R = 0.082057\,\text{L}\,\text{atm}\,\text{K}^{-1}\,\text{mol}^{-1} = 1.987\,\text{cal}\,\text{K}^{-1}\,\text{mol}^{-1}$.

• The Boltzmann Constant ($k$):

  • Defined as the gas constant per individual molecule:       $k = \frac{R}{N_A} = \frac{8.314\,\text{J}\,\text{K}^{-1}\,\text{mol}^{-1}}{6.022 \times 10^{23}\,\text{mol}^{-1}} = 1.38 \times 10^{-23}\,\text{J}\,\text{K}^{-1}$

  • Dalton’s Law of Partial Pressures:

• Partial pressure ($P_i$) is the pressure that a constituent gas in a mixture would exert if it alone occupied the entire volume of the container at the same temperature.

• The total pressure ($P_{\text{total}}$) of a mixture of chemically non-reacting gases is equal to the sum of the individual partial pressures:     $P_{\text{total}} = P_1 + P_2 + P_3 + \dots = \sum_{i} P_i$

• Applying the ideal gas law to each component:     $P_i = n_i \left(\frac{RT}{V}\right)$$P_{\text{total}} = \left(\sum n_i\right) \frac{RT}{V} = n_{\text{total}} \left(\frac{RT}{V}\right)$

• Ratio of partial pressure to total pressure defines the mole fraction ($X_i$$\frac{P_i}{P_{\text{total}}} = \frac{n_i}{n_{\text{total}}} = X_i \implies P_i = X_i P_{\text{total}}$

  • Graham’s Law of Diffusion and Effusion:

• Diffusion is the spontaneous, uniform dispersion of a substance throughout available space. Effusion is the escape of gas molecules through a pinhole into an evacuated chamber.

• Graham’s Law states that the rate of diffusion or effusion ($r$) of a gas is inversely proportional to the square root of its mass density ($\rho$), and consequently its molecular mass ($M$$r \propto \frac{1}{\sqrt{\rho}} \propto \frac{1}{\sqrt{M}}$$\frac{r_1}{r_2} = \sqrt{\frac{\rho_2}{\rho_1}} = \sqrt{\frac{M_2}{M_1}}$

• Because the time ($t$) required for effusion of a fixed volume is inversely proportional to rate ($t \propto 1/r$$\frac{t_1}{t_2} = \frac{r_2}{r_1} = \sqrt{\frac{M_1}{M_2}}$

• Worked Numerical Problem:

  • Problem: A sample of gas X diffuses through a membrane $2.92$ times slower than hydrogen gas ($\text{H}_2$). Calculate the relative molecular mass of gas X.

  • Solution:$\frac{r_{\text{H}_2}}{r_{\text{X}}} = 2.92 = \sqrt{\frac{M_{\text{X}}}{M_{\text{H}_2}}}$$M_{\text{X}} = (2.92)^2 \times M_{\text{H}_2} = 8.5264 \times 2.016\,\text{g}\,\text{mol}^{-1} = 17.19\,\text{g}\,\text{mol}^{-1}$

  • Kinetic Derivation of Gas Pressure:

• Consider a cubic container of side length $l$ containing $N$ identical molecules, each of mass $m$.

• The velocity vector $u$ of any molecule resolves into three orthogonal components:     $u^2 = u_x^2 + u_y^2 + u_z^2$

• Consider motion along the $x$-axis toward a face perpendicular to $x$:

  • Initial momentum before impact: $m u_x$
  • Elastic collision reverses direction without loss of speed: $-m u_x$
  • Momentum change per impact on the wall:       $\Delta p_x = m u_x - (-m u_x) = 2 m u_x$
  • Time between successive collisions on the same wall is the round-trip distance divided by velocity: $\Delta t = \frac{2l}{u_x}$
  • Collision frequency per unit time: $\frac{u_x}{2l}$
  • Rate of momentum change (force exerted by one molecule along $x$$f_x = 2 m u_x \times \frac{u_x}{2l} = \frac{m u_x^2}{l}$
  • Summing for all $N$ molecules:       $F_x = \sum_{i=1}^{N} \frac{m u_{xi}^2}{l} = \frac{m N \overline{u_x^2}}{l}$

• Because molecular motion is isotropic (completely random in three dimensions):     $\overline{u_x^2} = \overline{u_y^2} = \overline{u_z^2} = \frac{1}{3} \overline{u^2}$

• Total force on the wall:     $F = \frac{m N \overline{u^2}}{3l}$

• Pressure is force per unit area ($A = l^2$, volume $V = l^3$$P = \frac{F}{l^2} = \frac{m N \overline{u^2}}{3l^3} = \frac{1}{3} \frac{m N \overline{u^2}}{V}$$PV = \frac{1}{3} m N \overline{u^2} = \frac{1}{3} M_{\text{total}} \overline{u^2}$

  • Kinetic Energy and Root-Mean-Square Velocity:

• For one mole of gas ($N = N_A$, total mass $m N_A = M$$PV = \frac{1}{3} M \overline{u^2} = \frac{2}{3} \left(\frac{1}{2} M \overline{u^2}\right) = \frac{2}{3} E_k$

• Since $PV = RT$ for one mole:     $E_k = \frac{3}{2} RT$

• The total translational kinetic energy of one mole of an ideal gas depends solely upon absolute temperature, and is invariant for different gases at the same temperature.

• Root-Mean-Square Velocity ($u_{\text{rms}}$):$\overline{u^2} = \frac{3RT}{M} \implies u_{\text{rms}} = \sqrt{\frac{3RT}{M}}$

• Worked Numerical Problem:

  • Problem: Calculate $u_{\text{rms}}$ of oxygen molecules ($\text{O}_2$) at $25^\circ\text{C}$.
  • Solution:$T = 25 + 273.15 = 298.15\,\text{K}$$M = 32.00 \times 10^{-3}\,\text{kg}\,\text{mol}^{-1}$$R = 8.314\,\text{J}\,\text{K}^{-1}\,\text{mol}^{-1}$$u_{\text{rms}} = \sqrt{\frac{3 \times 8.314 \times 298.15}{32.00 \times 10^{-3}}} = \sqrt{232{,}399} = 482.08\,\text{m/s}$

Real Gases and Deviations from Ideality

• The van der Waals Equation of State:

  • Real gases deviate from the ideal gas law at high pressures and low temperatures (near condensation).
  • In 1873, J. D. van der Waals modified the ideal equation by incorporating two corrections:
  • Intermolecular Attractions (Pressure Correction):
    • A molecule in the interior of a gas experiences isotropic attractions from all sides.
    • A molecule colliding with the container wall experiences an unbalanced net inward pull from bulk molecules, reducing the force of impact.
    • Measured pressure $P_{\text{real}}$ is therefore lower than ideal pressure $P_{\text{ideal}}$.
    • The inward force is proportional to the concentration of striking molecules ($n/V$) and the concentration of pulling molecules ($n/V$), making the correction proportional to $(n/V)^2$.
    • Correction term is $\frac{a n^2}{V^2}$, where $a$ is a constant specific to the attractive forces of the gas:         $P_{\text{ideal}} = P + \frac{a n^2}{V^2}$
  • Excluded Volume / Covolume (Volume Correction):
    • Gas molecules possess finite physical volumes; the free volume available for motion is smaller than container volume $V$.
    • Effective covolume is subtracted: $nb$, where $b$ is the effective volume occupied by one mole of molecules:         $V_{\text{ideal}} = V - nb$
  • Combined van der Waals Equation:
    • For 1 mole:         $\left(P + \frac{a}{V_m^2}\right)(V_m - b) = RT$
    • For $n$ moles:         $\left(P + \frac{a n^2}{V^2}\right)(V - nb) = nRT$

• Representative van der Waals Constants:

  • Hydrogen ($\text{H}_2$
  • $a = 0.245\,\text{atm}\,\text{L}^2\,\text{mol}^{-2}$
  • $b = 0.0266\,\text{L}\,\text{mol}^{-1}$
  • Helium ($\text{He}$
  • $a = 0.034\,\text{atm}\,\text{L}^2\,\text{mol}^{-2}$
  • $b = 0.0237\,\text{L}\,\text{mol}^{-1}$
  • Nitrogen ($\text{N}_2$
  • $a = 1.39\,\text{atm}\,\text{L}^2\,\text{mol}^{-2}$
  • $b = 0.0391\,\text{L}\,\text{mol}^{-1}$
  • Oxygen ($\text{O}_2$
  • $a = 1.36\,\text{atm}\,\text{L}^2\,\text{mol}^{-2}$
  • $b = 0.0318\,\text{L}\,\text{mol}^{-1}$
  • Carbon dioxide ($\text{CO}_2$
  • $a = 3.59\,\text{atm}\,\text{L}^2\,\text{mol}^{-2}$
  • $b = 0.0427\,\text{L}\,\text{mol}^{-1}$

• Compressibility Factor ($Z$):

  • Quantifies the extent of deviation from ideality:     $Z = \frac{P V_m}{RT} = \frac{V_{m,\text{real}}}{V_{m,\text{ideal}}}$
  • For an ideal gas, $Z = 1$ under all conditions.
  • Pressure Regimes:
  • Low to Moderate Pressures ($Z < 1$): Long-range attractive intermolecular forces dominate. Gas molecules are pulled closer together, making real molar volume smaller than ideal molar volume ($V_{m,\text{real}} < V_{m,\text{ideal}}$).
  • High Pressures ($Z > 1$): Short-range repulsive forces and molecular excluded volume dominate. Crowded molecules cannot compress further, making $V_{m,\text{real}} > V_{m,\text{ideal}}$.
  • Extreme low-temperature conditions or proximity to the critical point increase deviations from $Z = 1$.

Chemical Equilibrium Principles and Calculations

• The Nature of Dynamic Chemical Equilibrium:

  • Chemical equilibrium is established in a closed system when opposing forward and reverse reactions proceed at identical rates ($r_1 = r_2$).
  • Concentrations of all reactants and products remain invariant over time, while molecular interconversion continues actively.
  • Illustrated by the esterification of acetic acid and ethanol:     $\text{CH}_3\text{COOH} + \text{C}_2\text{H}_5\text{OH} \rightleftharpoons \text{CH}_3\text{COOC}_2\text{H}_5 + \text{H}_2\text{O}$
  • Initiating the reaction with pure acid and alcohol produces ester and water at forward rate $r_1$.
  • As products accumulate, the reverse reaction proceeds at rate $r_2$.
  • Dynamic equilibrium is established when $r_1 = r_2$.

• The Law of Mass Action and the Equilibrium Constant ($K_c$):

  • For the general homogeneous chemical reaction:     $aA + bB \rightleftharpoons yY + zZ$
  • The equilibrium expression is given by:     $K_c = \frac{[Y]^y [Z]^z}{[A]^a [B]^b}$
  • The equilibrium expression depends strictly upon reaction stoichiometry, regardless of the reaction mechanism.
  • The numerical value of $K_c$ varies with temperature but is independent of initial concentrations.
  • Magnitude Interpretation:
  • $K \gg 1$: Equilibrium lies far to the right; products predominate.
  • $K \ll 1$: Equilibrium lies far to the left; reactants predominate.

• Reversibility and Reaction Direction:

  • Reversing the chemical equation inverts the equilibrium constant:     $\text{N}_2\text{O}_4(g) \rightleftharpoons 2\text{NO}_2(g) \implies K_c = \frac{[\text{NO}_2]^2}{[\text{N}_2\text{O}_4]}$$2\text{NO}_2(g) \rightleftharpoons \text{N}_2\text{O}_4(g) \implies K_c' = \frac{[\text{N}_2\text{O}_4]}{[\text{NO}_2]^2} = \frac{1}{K_c}$

• Gaseous Equilibria ($K_p$) and Relationship to $K_c$:

  • For a gas-phase reaction:     $eE + fF \rightleftharpoons gG + hH$$K_p = \frac{P_G^g P_H^h}{P_E^e P_F^f}$
  • Expressing partial pressure via the ideal gas law: $P_i = [i]RT$$K_p = \frac{([G]RT)^g ([H]RT)^h}{([E]RT)^e ([F]RT)^f} = \frac{[G]^g [H]^h}{[E]^e [F]^f} (RT)^{(g+h)-(e+f)}$$K_p = K_c (RT)^{\Delta n}$     where $\Delta n = \sum n_{\text{products, gas}} - \sum n_{\text{reactants, gas}}$.
  • If $\Delta n = 0$, then $K_p = K_c$ (e.g., $\text{H}_2(g) + \text{I}_2(g) \rightleftharpoons 2\text{HI}(g)$).

• Heterogeneous Equilibrium:

  • When reacting species reside in multiple phases, pure solids and pure liquids have constant active masses (thermodynamic activities equal to $1$).
  • Their concentrations are omitted from the equilibrium expression:     $\text{CaCO}_3(s) \rightleftharpoons \text{CaO}(s) + \text{CO}_2(g)$$K_c = [\text{CO}_2]$$K_p = P_{\text{CO}_2}$

• Le Chatelier’s Principle and Response to Disturbances:

  • If a system at chemical equilibrium experiences a change in concentration, pressure, or temperature, it shifts its equilibrium position in the direction that counteracts the applied stress.
  • Concentration Effects:
  • Adding a component shifts the position to consume that component.
  • Removing a component shifts the position to regenerate that component.
  • Pressure and Volume Effects:
  • Increasing external pressure (reducing volume) shifts the equilibrium toward the side with fewer moles of gas.
  • If $\Delta n = 0$, pressure changes have no effect on the equilibrium position.
  • Temperature Effects:
  • For endothermic reactions ($\Delta H > 0$), heat acts as a reactant:       $\text{Reactants} + \text{Heat} \rightleftharpoons \text{Products}$       Increasing temperature shifts equilibrium to the right, increasing $K$.
  • For exothermic reactions ($\Delta H < 0$), heat acts as a product:       $\text{Reactants} \rightleftharpoons \text{Products} + \text{Heat}$       Increasing temperature shifts equilibrium to the left, decreasing $K$.

Phase Equilibria, Solution Thermodynamics, and Colligative Properties

• Phase Definition and the Single-Component Water Phase Diagram:
  • A phase is a physically distinct, homogeneous state of aggregation with uniform chemical composition and physical properties.

• Key Boundaries:

  • Curve AT: Sublimation curve (solid-gas equilibrium).

  • Curve TC: Melting/fusion curve (solid-liquid equilibrium). For water, the slope is negative, indicating that the melting temperature decreases as pressure increases.

  • Curve TTc: Vaporization/boiling curve (liquid-vapor equilibrium).

  • Triple Point ($T$$T = 0.01^\circ\text{C}$$P = 0.006\,\text{atm}$       All three phases coexist in invariant dynamic equilibrium.

  • Critical Point ($T_c$$T_c = 374^\circ\text{C}$$P_c = 218\,\text{atm}$       Beyond $T_c$, liquid cannot exist regardless of applied pressure.

  • Thermodynamics of Solutions and Colligative Properties:

• An ideal solution exhibits uniform intermolecular forces: cohesive interactions between like molecules ($A-A$ and $B-B$) equal adhesive interactions between unlike molecules ($A-B$).

• Colligative Properties: Intensive properties that depend solely upon the ratio of the number of solute particles to solvent molecules, independent of solute chemical identity.

• Solute addition lowers the chemical potential of the liquid solvent ($\mu_A < \mu_A^\circ$), stabilizing the liquid phase relative to gas and solid phases.

• Colligative Relationships:

  • Vapor Pressure Lowering (Raoult’s Law):$P_A = X_A P_A^\circ$$\Delta P = P_A^\circ - P_A = X_B P_A^\circ$

  • Boiling Point Elevation (Ebullioscopy):$\Delta T_b = i K_b m$$K_b = \frac{R M_{\text{solvent}} T_b^2}{\Delta H_{\text{vap}}}$       For water, $K_b = 0.512\,^\circ\text{C}\,\text{kg}\,\text{mol}^{-1}$.

  • Freezing Point Depression (Cryoscopy):$\Delta T_f = i K_f m$$K_f = \frac{R M_{\text{solvent}} T_f^2}{\Delta H_{\text{fus}}}$       For water, $K_f = 1.86\,^\circ\text{C}\,\text{kg}\,\text{mol}^{-1}$.

  • Osmotic Pressure ($\Pi$ - van ’t Hoff Equation):$\Pi = \left(\frac{n}{V}\right)RT = M R T$

  • Binary Liquid-Vapor Mixtures:

• Total vapor pressure of an ideal binary solution:     $P_{\text{total}} = P_A + P_B = X_A P_A^\circ + X_B P_B^\circ = P_B^\circ + X_A(P_A^\circ - P_B^\circ)$
• Vapor composition ($Y_A$) is determined by Dalton's law:     $Y_A = \frac{P_A}{P_{\text{total}}} = \frac{X_A P_A^\circ}{P_{\text{total}}}$

• The vapor phase is always richer in the more volatile component (lower boiling point). Successive vaporization-condensation cycles in a fractionating column allow separation of components.

  • Non-Ideal Solutions and Deviations from Raoult’s Law:

• Negative Deviations ($A-B$ attraction > $A-A, B-B$):

  • Vapor pressure is lower than ideal; boiling point is higher than ideal.
  • Forms a maximum-boiling azeotrope (e.g., chloroform and acetone, stabilized by hydrogen bonding).

• Positive Deviations ($A-B$ attraction < $A-A, B-B$):
  • Vapor pressure is higher than ideal; boiling point is lower than ideal.
  • Forms a minimum-boiling azeotrope (e.g., ethanol and water).

• Henry’s Law for Real Solutions:
  • In dilute non-ideal solutions, the solvent follows Raoult's Law ($X_{\text{solvent}} \approx 1$), while the volatile solute obeys Henry's Law ($X_{\text{solute}} \ll 1$$P_2 = k_{H,2} X_2$     where $k_{H,2}$ is the Henry’s law constant.

• Immiscible Liquids and Steam Distillation:

  • For two immiscible liquids, each exerts its full saturated vapor pressure independently:     $P_{\text{total}} = P_1^\circ + P_2^\circ$
  • The mixture boils when $P_{\text{total}}$ equals atmospheric pressure, which occurs below the boiling point of either pure component.
  • Mass ratio in the distillate:     $\frac{w_1}{w_2} = \frac{n_1 M_1}{n_2 M_2} = \frac{P_1^\circ M_1}{P_2^\circ M_2}$

• Liquid-Liquid Partition:

  • When a solute distributes between two immiscible liquid phases (A and B), the distribution coefficient ($K_D$) is constant at equilibrium:     $K_D = \frac{C_A}{C_B}$
  • Multiple small-volume extractions remove more solute than a single extraction using the entire volume of extracting solvent.

Chemical Kinetics and Reaction Mechanisms

• Reaction Rates and Stoichiometry:

  • Chemical kinetics evaluates reaction speeds and their mechanisms.
  • Rates depend on reactant concentrations, temperature, presence of catalysts, and reactant surface area.
  • For the general reaction:     $aA + bB \rightarrow cC + dD$$\text{Rate} = -\frac{1}{a}\frac{d[A]}{dt} = -\frac{1}{b}\frac{d[B]}{dt} = \frac{1}{c}\frac{d[C]}{dt} = \frac{1}{d}\frac{d[D]}{dt}$

• Rate Laws and Reaction Order:

  • General rate law:     $\text{Rate} = k [A]^m [B]^n$     where $m$ and $n$ are partial orders determined experimentally, and $m+n$ is the overall reaction order.

• Integrated Rate Laws:

  • Zero-Order Reactions:$-\frac{d[A]}{dt} = k_0$$[A] = [A]_0 - k_0 t$$t_{1/2} = \frac{[A]_0}{2k_0}$     Units of $k_0$: $\text{mol}\,\text{L}^{-1}\,\text{s}^{-1}$.     For gas-phase zero-order reactions:     $k_0 = \frac{1}{t}(p_0 - p)$$t_{1/2} = \frac{p_0}{2k_0}$
  • First-Order Reactions:$-\frac{d[A]}{dt} = k_1 [A]$$\ln[A] = \ln[A]_0 - k_1 t \implies [A] = [A]_0 e^{-k_1 t}$$k_1 = \frac{2.303}{t}\log_{10}\left(\frac{[A]_0}{[A]}\right)$$t_{1/2} = \frac{\ln 2}{k_1} = \frac{0.693}{k_1}$     First-order half-life is completely independent of initial concentration.     Amount remaining after $n$ half-lives:     $[A] = \frac{[A]_0}{2^n}$     Specific intervals:     $t_{75\%} = 2 \times t_{50\%}$$t_{99.9\%} = 10 \times t_{1/2}$     For gaseous reactions ($A(g) \rightarrow B(g) + C(g)$$k_1 = \frac{2.303}{t}\log_{10}\left(\frac{p_0}{2p_0 - p_t}\right)$
  • Second-Order Reactions ($\text{Rate} = k [A]^2$):$\frac{1}{[A]} = \frac{1}{[A]_0} + kt$$t_{1/2} = \frac{1}{k [A]_0}$
  • General Half-Life Dependence:$t_{1/2} \propto \frac{1}{[A]_0^{n-1}}$

• Special Chemical Reaction Cases:

  • Ester Hydrolysis (Acid-Catalyzed):$\text{CH}_3\text{COOC}_2\text{H}_5 + \text{H}_2\text{O} \xrightarrow{\text{H}^+} \text{CH}_3\text{COOH} + \text{C}_2\text{H}_5\text{OH}$     Titrating with $\text{NaOH}$ allows tracking of rate constant via volume of titrant:     $k_1 = \frac{2.303}{t}\log_{10}\left(\frac{V_\infty - V_0}{V_\infty - V_t}\right)$
  • Inversion of Cane Sugar:$\text{C}_{12}\text{H}_{22}\text{O}_{11} + \text{H}_2\text{O} \xrightarrow{\text{H}^+} \text{C}_6\text{H}_{12}\text{O}_6 (\text{glucose}) + \text{C}_6\text{H}_{12}\text{O}_6 (\text{fructose})$     Using polarimetric angle readings ($r$$k_1 = \frac{2.303}{t}\log_{10}\left(\frac{r_0 - r_\infty}{r_t - r_\infty}\right)$

• Temperature Dependence and the Arrhenius Equation:

  • Reaction rates typically double for every $10^\circ\text{C}$ rise in temperature.
  • Temperature coefficient:     $\frac{k_{T+10}}{k_T} \approx 2 \text{ to } 3$

• Catalysts provide an alternative reaction pathway with a lower activation energy ($E_a(c) < E_a$), leaving $\Delta H$ unchanged.
• Arrhenius Equation:$k = A e^{-E_a / RT}$$\ln k = \ln A - \frac{E_a}{RT}$$\log_{10} k = \log_{10} A - \frac{E_a}{2.303 RT}$

• Two-temperature relationship:     $\ln\left(\frac{k_2}{k_1}\right) = \frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)$

  • Reaction Mechanisms and Molecularity:

• Reactions proceed through elementary steps characterized by molecularity (unimolecular, bimolecular, termolecular).

• The slowest elementary step is the rate-determining step (RDS).

• Intermediates formed in fast pre-equilibrium steps must be eliminated from the overall rate expression using equilibrium relations.

Electrolytic Conductance and Solution Ionic Transport

• Conduction Mechanisms and Definitions:

  • Electronic (Metallic) Conductors: Conduction occurs via mobile valence electrons without chemical change or matter transfer.
  • Electrolytic Conductors: Conduction occurs via migration of ions in solution or molten salts, accompanied by chemical decomposition (electrolysis).
  • Ohm's Law:$I = \frac{E}{R}$
  • **Conductance ($G$$G = \frac{1}{R}\quad (\text{Siemens, } \text{S})$
  • **Specific Resistance ($\rho$$R = \rho \frac{l}{A}$
  • **Specific Conductance / Conductivity ($\kappa$$\kappa = \frac{1}{\rho} = \frac{l}{A} \frac{1}{R} = x \cdot G$     where $x = \frac{l}{A}$ is the cell constant ($\text{cm}^{-1}$ or $\text{m}^{-1}$
  • **Molar Conductivity ($\Lambda_m$$\Lambda_m = \frac{1000 \kappa}{c}\quad (\text{S}\,\text{cm}^2\,\text{mol}^{-1})$     where $c$ is concentration in $\text{mol}\,\text{L}^{-1}$.

• Conductance Measurement (Kohlrausch Bridge):

• Conductance is measured using an AC Wheatstone bridge operating at approximately $1000\,\text{Hz}$ to eliminate electrode polarization and concentration changes.

• A variable capacitor parallel to resistor $R_3$ balances residual cell capacitance.

  • Dilution Effects and Kohlrausch’s Law:

• On dilution, conductivity ($\kappa$) decreases because the number of current-carrying ions per unit volume drops.

• Molar conductivity ($\Lambda_m$) increases on dilution because the increase in volume ($V$) outweighs the decrease in $\kappa$.

• Strong electrolytes approach a limiting molar conductivity ($\Lambda_m^\infty$) by extrapolation to zero concentration.

• Weak electrolytes dissociate sharply only at high dilution, preventing direct extrapolation.

• Kohlrausch's Law of Independent Migration of Ions:$\Lambda_m^\infty = \nu_+ \lambda_+^\infty + \nu_- \lambda_-^\infty$

• Applications:

  • Calculating $\Lambda_m^\infty$ of weak electrolytes by combining strong electrolyte data:       $\Lambda_m^\infty(\text{CH}_3\text{COOH}) = \Lambda_m^\infty(\text{HCl}) + \Lambda_m^\infty(\text{CH}_3\text{COONa}) - \Lambda_m^\infty(\text{NaCl})$

  • Determining degree of dissociation: $\alpha = \frac{\Lambda_m^c}{\Lambda_m^\infty}$

  • Determining solubility ($s$) of sparingly soluble salts:       $s = \frac{1000 \kappa}{\Lambda_m^\infty}\quad (\text{mol}\,\text{L}^{-1})$

  • Ostwald’s Dilution Law:

• For a binary weak electrolyte ($MA \rightleftharpoons M^+ + A^-$):     $K_a = \frac{c \alpha^2}{1 - \alpha} = \frac{c (\Lambda_m^c / \Lambda_m^\infty)^2}{1 - (\Lambda_m^c / \Lambda_m^\infty)}$

• When $\alpha \ll 1$:     $K_a \approx c \alpha^2 \implies \alpha = \sqrt{\frac{K_a}{c}}$

  • Debye-Hückel-Onsager (DHO) Theory for Strong Electrolytes:

• Strong electrolytes are completely dissociated; decreases in molar conductivity at higher concentrations result from interionic forces.

• Relaxation Effect (Asymmetry Effect): The ionic atmosphere surrounding a central moving ion distorts, creating a retardation force.

• Electrophoretic Effect: Solvent molecules associated with the counter-ion atmosphere move in the opposite direction, creating frictional drag.

  • **Ionic Mobility ($u$) and Transference Numbers ($t$

• Mobility is ionic velocity under unit potential gradient ($1\,\text{V}\,\text{cm}^{-1}$$\lambda = z u F$

• Transference number is the fraction of total current carried by an ion:     $t_+ = \frac{u_+}{u_+ + u_-} = \frac{\Lambda_+}{\Lambda_m}$$t_- = \frac{u_-}{u_+ + u_-} = \frac{\Lambda_-}{\Lambda_m}$$t_+ + t_- = 1$

• Conductometric Titrations:
  • Endpoints are determined from intersections of straight-line branches:
  • Strong Acid with Strong Base ($\text{HCl} + \text{NaOH}$): Conductance drops steeply as fast $H^+$ ($\lambda = 350$) is replaced by $Na^+$ ($\lambda = 50$), then rises sharply post-endpoint due to excess $OH^-$ ($\lambda = 198$).
  • Strong Acid with Weak Base ($\text{HCl} + \text{NH}_4\text{OH}$): Conductance falls until equivalence, then remains flat because excess $\text{NH}_4\text{OH}$ is poorly ionized.
  • Weak Acid with Strong Base ($\text{CH}_3\text{COOH} + \text{NaOH}$): Initial slight dip, steady rise due to salt formation, then sharp rise post-endpoint.
  • Mixture of Strong and Weak Acids with Strong Base: Yields two distinct breaks.

Electrochemistry and Galvanic Cells

• Galvanic Cells and Components:
  • A galvanic (voltaic) cell generates electrical energy from spontaneous redox reactions.

• Anode: Negative electrode where oxidation occurs:     $\text{Zn}(s) \rightarrow \text{Zn}^{2+}(aq) + 2e^-$

• Cathode: Positive electrode where reduction occurs:     $\text{Cu}^{2+}(aq) + 2e^- \rightarrow \text{Cu}(s)$

• Salt Bridge: Contains an inert electrolyte (e.g., $\text{KNO}_3$ or $\text{KCl}$ in agar) to maintain electroneutrality by ion migration, preventing bulk solution mixing.

  • Cell Line Notation:

• Anode is written on the left, cathode on the right.

• Single vertical bar ($|$) indicates a phase boundary; double vertical bars ($||$) indicate a salt bridge:     $\text{Zn}(s) \mid \text{ZnSO}_4(1.0\,\text{M}) \parallel \text{CuSO}_4(1.0\,\text{M}) \mid \text{Cu}(s)$

  • Standard Potentials and the Standard Hydrogen Electrode (SHE):

• The absolute potential of an isolated half-cell cannot be measured; potentials are measured against the Standard Hydrogen Electrode (SHE), assigned $E^\circ = 0.00\,\text{V}$ at all temperatures.

• Half-cell reaction:     $2\text{H}^+(aq, 1\,\text{M}) + 2e^- \rightleftharpoons \text{H}_2(g, 1\,\text{atm})$

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

• If $E_{\text{cell}}^\circ > 0$, the reaction is spontaneous as written.

  • The Nernst Equation:

• Relates cell potential to non-standard concentrations/activities:     $E = E^\circ - \frac{RT}{nF} \ln Q$

• At $298.15\,\text{K}$ ($25^\circ\text{C}$$E = E^\circ - \frac{0.05916}{n} \log_{10} Q$

  • Equilibrium Constants and Thermodynamic Relations:

• At equilibrium, $E_{\text{cell}} = 0$ and $Q = K_{\text{eq}}$$\Delta G^\circ = -nFE^\circ$$E^\circ = \frac{RT}{nF} \ln K_{\text{eq}} = \frac{0.05916}{n} \log_{10} K_{\text{eq}}$

Electroanalytical Techniques

• Classification of Electroanalytical Methods:

• Bulk Methods: Measure properties of the whole solution (e.g., conductometry).

• Interfacial Methods: Measure phenomena occurring at the electrode-solution interface.

  • Static Methods ($I = 0$): Potentiometry.

  • Dynamic Methods ($I > 0$): Controlled potential (voltammetry, amperometry, constant-potential coulometry) or controlled current (coulometric titrations, electrogravimetry).

  • Three-Electrode Cells and Liquid Junction Potential:

• Three-Electrode Setup:

  • Working Electrode: Site of the reaction of interest.
  • Reference Electrode: Non-polarizable electrode providing a stable reference potential with negligible current flow.
  • Auxiliary (Counter) Electrode: Current-supporting partner to the working electrode.

• **Liquid Junction Potential ($E_{lj}$

  • Develops at the interface between solutions of differing composition due to unequal ionic diffusion rates ($u_{\text{H}^+} > u_{\text{Cl}^-}$).

  • Minimized by using concentrated $\text{KCl}$ salt bridges.     $E_{\text{cell}} = E_{\text{ind}} - E_{\text{ref}} + E_{lj}$

  • Reference and Indicator Electrodes in Potentiometry:

• Reference Electrodes:

  • Saturated Calomel Electrode (SCE): $\text{Hg}_2\text{Cl}_2(s) + 2e^- \rightleftharpoons 2\text{Hg}(l) + 2\text{Cl}^-(aq)$
  • Silver/Silver Chloride Electrode (SSCE): $\text{AgCl}(s) + e^- \rightleftharpoons \text{Ag}(s) + \text{Cl}^-(aq)$

• Indicator Electrodes:

  • Metallic Electrodes of the 1st Kind: Metal in equilibrium with its cation ($\text{Ag}, \text{Cu}, \text{Zn}$

  • Metallic Electrodes of the 2nd Kind: Metal responding to anion activity forming an insoluble salt (calomel, SSCE).

  • Redox (Inert) Electrodes: Electron sink/source ($\text{Pt}, \text{Au}$

  • Ion-Selective Electrodes (ISEs): Potential develops across an ion-selective membrane:       $E_{\text{mem}} = E_{\text{asym}} - \frac{0.05916}{z} \log_{10}\left(\frac{[A]_{\text{int}}}{[A]_{\text{samp}}}\right)$

    • Glass pH electrode: Corning 015 glass ($22\%\,\text{Na}_2\text{O}, 6\%\,\text{CaO}, 72\%\,\text{SiO}_2$
    • Solid-state ISE: $\text{Ag}_2\text{S}$ polycrystalline pellet for $\text{Ag}^+, \text{S}^{2-}$.
    • Liquid-based ISE: $\text{Ca}^{2+}$ electrode using di-($n$-decyl) phosphate.
    • Gas-sensing electrodes: Permeable membrane for $\text{CO}_2, \text{NH}_3$.
    • Potentiometric biosensors: Enzyme-coated membranes (e.g., urease for urea).

  • Coulometric Methods of Analysis:

• Based on exhaustive electrolysis requiring $100\%$ current efficiency.

• Faraday’s Law:$Q = nFN = I t_e$

• Controlled-Potential Coulometry: Applied potential is held constant; current decays exponentially with time:     $Q = \int_0^{t_e} i(t)\,dt$

• Controlled-Current Coulometry (Coulometric Titration): Current is held constant using a galvanostat; charge is calculated directly:     $Q = I t_e$

• Voltammetric Methods of Analysis:
  • Measures current as a function of applied potential under conditions promoting concentration polarization.
  • Mass Transport Regimes:
  • Diffusion: Movement down a chemical concentration gradient.
  • Migration: Electrostatic movement of ions in an electric field (suppressed by excess supporting electrolyte).
  • Convection: Mechanical transport by stirring or hydrodynamic flow.
  • Polarography: Voltammetry using a Dropping Mercury Electrode (DME).
  • Current-voltage curve is a polarogram.
  • Limiting diffusion current is governed by the Ilkovic Equation:       $i_d = 607 n D^{1/2} m^{2/3} t^{1/6} C$       where $m$ is mercury flow rate, $t$ is drop time, and $D$ is diffusion coefficient.
  • The half-wave potential ($E_{1/2}$) identifies species qualitatively; diffusion current ($i_d$) determines concentration quantitatively.

• Stripping Voltammetry: Analyte is preconcentrated cathodically or anodically onto an electrode, then stripped off electrochemically; achieves detection limits in the parts-per-trillion ($\text{pptr}$) range.

• Amperometry: Potential is fixed, and current is measured as a function of time (e.g., the Clark oxygen sensor for dissolved $\text{O}_2$).

  • Electrogravimetric Analysis:

• Quantitative deposition of a solid metal coating on a pre-weighed platinum gauze cathode.

• Performed under constant-current or controlled-cathode-potential conditions.

• Current densities are maintained below $0.1\,\text{A}\,\text{cm}^{-2}$ with vigorous stirring to produce smooth, adherent deposits.

Classical Thermodynamics and Energetics

• Definitions and Terminology:

  • System Types:
  • Open System: Exchanges both matter and energy with surroundings.
  • Closed System: Exchanges energy but not matter.
  • Isolated System: Exchanges neither energy nor matter.
  • Properties:
  • Extensive: Depends on total quantity of matter (mass, volume, internal energy, enthalpy, entropy).
  • Intensive: Independent of sample size (temperature, pressure, density, molar volume).
  • Processes:
  • Isothermal: $\Delta T = 0$
  • Adiabatic: $q = 0$
  • Isobaric: $\Delta P = 0$
  • Isochoric: $\Delta V = 0$
  • Reversible Process: Proceeds through a continuum of equilibrium states at an infinitesimal rate; produces maximum work.
  • Irreversible Process: Real, spontaneous change proceeding in one direction at a finite rate.

• The First Law of Thermodynamics:

  • Conservation of energy: Energy can neither be created nor destroyed.
  • Mathematical formulations:     $\Delta U = q + w$$dU = dq + dw$     For expansion work ($dw = -P_{\text{ext}} dV$$dU = dq - P_{\text{ext}} dV$
  • Expansion Work Calculations:
  • Free expansion into a vacuum: $P_{\text{ext}} = 0 \implies w = 0$.
  • Reversible isothermal expansion of an ideal gas:       $w_{\text{rev}} = -\int_{V_1}^{V_2} P dV = -nRT \ln\left(\frac{V_2}{V_1}\right) = -nRT \ln\left(\frac{P_1}{P_2}\right)$
  • Irreversible expansion against constant external pressure:       $w_{\text{irr}} = -P_{\text{ext}} (V_2 - V_1)$
  • Comparison confirms: $|w_{\text{rev}}| > |w_{\text{irr}}|$.

• Enthalpy and Heat Capacities:

  • **Enthalpy ($H$$H = U + PV$$\Delta H = q_p$
  • Heat Capacities:$C_v = \left(\frac{\partial U}{\partial T}\right)_V$$C_p = \left(\frac{\partial H}{\partial T}\right)_P$
  • For an ideal gas:     $C_p - C_v = nR$$C_{p,m} - C_{v,m} = R = 8.314\,\text{J}\,\text{K}^{-1}\,\text{mol}^{-1}$
  • Adiabatic Reversible Expansion of an Ideal Gas:$dq = 0 \implies dU = dw = -P dV$$C_v dT = -\frac{nRT}{V} dV$$T V^{\gamma - 1} = \text{constant}$$P V^\gamma = \text{constant}$     where $\gamma = \frac{C_p}{C_v}$.

• The Second Law of Thermodynamics and Entropy:

  • Natural, spontaneous processes proceed toward an increase in the total entropy of the universe:     $\Delta S_{\text{universe}} = \Delta S_{\text{system}} + \Delta S_{\text{surroundings}} > 0$
  • **Definition of Entropy ($S$$dS = \frac{dq_{\text{rev}}}{T}$
  • Entropy Calculations:
  • Phase Transitions:$\Delta S_{\text{vap}} = \frac{\Delta H_{\text{vap}}}{T_b}$$\Delta S_{\text{fus}} = \frac{\Delta H_{\text{fus}}}{T_m}$
  • Isothermal Gas Expansion:$\Delta S = nR \ln\left(\frac{V_2}{V_1}\right) = nR \ln\left(\frac{P_1}{P_2}\right)$
  • Isobaric Heating:$\Delta S = n C_p \ln\left(\frac{T_2}{T_1}\right)$
  • Isochoric Heating:$\Delta S = n C_v \ln\left(\frac{T_2}{T_1}\right)$

• The Carnot Heat Engine Cycle:

  • Four reversible stages operating between heat source ($T_2$) and heat sink ($T_1$
  • Stage I: Isothermal reversible expansion at $T_2$ ($V_1 \rightarrow V_2$$Q_2 = -W_1 = nRT_2 \ln\left(\frac{V_2}{V_1}\right)$
  • Stage II: Adiabatic reversible expansion ($V_2 \rightarrow V_3$, temperature falls from $T_2$ to $T_1$$W_2 = C_v(T_2 - T_1)$
  • Stage III: Isothermal reversible compression at $T_1$ ($V_3 \rightarrow V_4$$Q_1 = -W_3 = nRT_1 \ln\left(\frac{V_4}{V_3}\right)$
  • Stage IV: Adiabatic reversible compression ($V_4 \rightarrow V_1$, temperature rises from $T_1$ to $T_2$$W_4 = C_v(T_1 - T_2)$
  • Net work:     $W_{\text{net}} = nR(T_2 - T_1) \ln\left(\frac{V_2}{V_1}\right)$
  • **Thermodynamic Efficiency ($\epsilon$$\epsilon = \frac{W_{\text{net}}}{Q_2} = \frac{T_2 - T_1}{T_2} = 1 - \frac{T_1}{T_2}$