Sources and References

These notes synthesize material on classical and chemical thermodynamics, solution theory, surface science, and chemical kinetics drawn from standard textbooks and open educational resources. They consolidate content from the two-term University of Waterloo Chemical Engineering physical chemistry sequence (CHE 230 together with its continuation CHE 231) into a single reference.

• P. Atkins and J. de Paula, Physical Chemistry, Oxford University Press (thermodynamic laws, phase equilibria, kinetics, surfaces).
• T. Engel and P. Reid, Physical Chemistry, Pearson (first and second laws, solutions, electrochemistry).
• I. N. Levine, Physical Chemistry, McGraw-Hill (chemical potential, activities, Debye–Huckel theory).
• D. A. McQuarrie and J. D. Simon, Physical Chemistry: A Molecular Approach, University Science Books (statistical underpinnings of entropy, partition functions).
• J. M. Smith, H. C. Van Ness, and M. M. Abbott, Introduction to Chemical Engineering Thermodynamics, McGraw-Hill (engineering formulation of the first law, fugacity, reaction equilibrium).
• MIT OpenCourseWare 5.60 Thermodynamics and Kinetics and 5.62 Physical Chemistry lecture material.
• Cambridge Part II Physical Chemistry lecture notes (surface phenomena, adsorption isotherms).

Chapter 1: Thermodynamic Foundations

1.1 Systems, States, and the Meaning of Work and Heat

Thermodynamics begins with an accounting problem. We draw a mental boundary around a region of matter, call it the system, and call everything outside the surroundings. The boundary may be rigid or flexible, insulating or conducting, permeable or impermeable. A system described by a small number of macroscopic variables—pressure \(p\), temperature \(T\), volume \(V\), amounts of substance \(n_i\)—is said to be in a well-defined thermodynamic state. State variables have the crucial property that their values depend only on the present condition of the system and not on the history that brought the system there.

Energy crosses the boundary in two recognizably different forms. Work is energy transferred through an organized displacement against a force, and heat is energy transferred by virtue of a temperature difference alone. Neither is a property of the system; both are properties of a process. This distinction is the reason we write \(\mathrm{d}U\) for an infinitesimal change in internal energy (a state function) but \(\delta w\) and \(\delta q\) for infinitesimal contributions of work and heat (path functions).

For a closed system undergoing a change in volume against an external pressure \(p_\mathrm{ex}\), the expansion work done on the system is

\[ \delta w = -p_\mathrm{ex}\,\mathrm{d}V. \]

When the process is carried out so slowly that the external pressure never differs measurably from the system pressure, the path is said to be reversible, and \(p_\mathrm{ex}=p\). A reversible expansion of an ideal gas between states 1 and 2 at constant temperature delivers the maximum possible work,

\[ w_\mathrm{rev} = -nRT\ln\frac{V_2}{V_1}. \]

Any irreversibility—a finite pressure drop, turbulence, friction at the piston—reduces the magnitude of the work actually obtained.

1.2 The First Law and Internal Energy

The first law of thermodynamics asserts that the internal energy of an isolated system is conserved. Written for a closed system that exchanges heat and work with its surroundings,

\[ \mathrm{d}U = \delta q + \delta w. \]

The sign convention used throughout these notes places heat absorbed by the system and work done on the system as positive. Although \(\delta q\) and \(\delta w\) depend separately on the path, their sum does not.

For a system whose only mode of work is pressure–volume work, the heat absorbed at constant volume equals the change in internal energy, \(q_V = \Delta U\), because \(\delta w = -p_\mathrm{ex}\,\mathrm{d}V = 0\). This is the fact exploited in bomb calorimetry, where a rigid reaction vessel is immersed in a water bath and the measured temperature rise is converted into \(\Delta U\) of combustion.

Because \(U\) is a state function, for an ideal gas it depends only on temperature. The constant-volume heat capacity is defined by \(C_V=\left(\partial U/\partial T\right)_V\), and integrating gives

\[ \Delta U = \int_{T_1}^{T_2} C_V(T)\,\mathrm{d}T \]

over any path connecting the two states, provided the gas remains ideal throughout.

1.3 Enthalpy and Constant-Pressure Processes

Most laboratory and industrial reactions occur at atmospheric pressure, not constant volume. Introducing the enthalpy

\[ H \equiv U + pV \]

converts the first law into a form adapted to open beakers and flow reactors. At constant pressure with only expansion work allowed,

\[ q_p = \Delta H, \]

so the heat released by a combustion carried out in a steady-flow calorimeter at 1 bar is the reaction enthalpy. The constant-pressure heat capacity \(C_p=\left(\partial H/\partial T\right)_p\) is always larger than \(C_V\) because at constant pressure some of the absorbed heat is spent pushing back the atmosphere. For an ideal gas, \(C_p-C_V=nR\).

Chapter 2: Thermochemistry

2.1 Standard Enthalpies and Hess’s Law

Because \(H\) is a state function, the enthalpy change of a reaction does not depend on whether the transformation is carried out in one step or in many. This observation, traditionally called Hess’s law, allows tabulated standard enthalpies of formation \(\Delta_\mathrm{f} H^\circ\)—the enthalpy of forming one mole of a substance from its elements in their standard states at 1 bar—to be combined algebraically:

\[ \Delta_\mathrm{r} H^\circ = \sum_i \nu_i\,\Delta_\mathrm{f} H^\circ(i), \]

where \(\nu_i\) are stoichiometric coefficients (positive for products, negative for reactants).

2.2 Temperature Dependence: Kirchhoff’s Law

A reaction enthalpy measured at one temperature can be corrected to another via Kirchhoff’s law,

\[ \Delta_\mathrm{r} H^\circ(T_2) = \Delta_\mathrm{r} H^\circ(T_1) + \int_{T_1}^{T_2} \Delta_\mathrm{r} C_p^\circ(T)\,\mathrm{d}T, \]

with \(\Delta_\mathrm{r} C_p^\circ = \sum_i \nu_i\,C_{p,i}^\circ\). For modest temperature ranges the heat capacities can often be treated as constants; for combustion temperatures or cryogenic work, empirical polynomials in \(T\) must be used.

2.3 Phase-Change and Bond Enthalpies

Phase transitions contribute their own enthalpy terms. Fusion, vaporization, and sublimation enthalpies are positive (energy must be supplied to break the cohesive forces of the condensed phase). Bond enthalpies, derived from gas-phase homolysis data, permit rough estimates of reaction enthalpies when calorimetric data are unavailable, though the approximation assumes a transferable bond energy that is only loosely true.

Chapter 3: The Second Law and Entropy

3.1 Spontaneity, Reversibility, and the Clausius Inequality

The first law tells us what changes are energetically possible; it says nothing about direction. A gas expands to fill its container, heat flows from hot to cold, and a dropped glass shatters rather than reassembles—none of these is forbidden by energy conservation, yet the reverse processes never occur spontaneously. The second law supplies the missing asymmetry.

Rudolf Clausius cast the second law as an inequality between the heat exchanged in a cyclic process and the temperature at which it is exchanged. For any cyclic process,

\[ \oint \frac{\delta q}{T_\mathrm{surr}} \le 0, \]

with equality if and only if the cycle is reversible. From this inequality follows the existence of a state function \(S\), the entropy, defined for a reversible infinitesimal change by

\[ \mathrm{d}S \equiv \frac{\delta q_\mathrm{rev}}{T}. \]

The second law then reads: for any process in an isolated system, \(\Delta S \ge 0\).

3.2 Carnot’s Theorem and the Thermodynamic Temperature Scale

Sadi Carnot analyzed an idealized heat engine that absorbs heat \(q_h\) from a hot reservoir at \(T_h\), rejects heat \(q_c\) to a cold reservoir at \(T_c\), and produces work \(w\). All such reversible engines have the same efficiency,

\[ \eta_\mathrm{Carnot} = 1 - \frac{T_c}{T_h}, \]

and no engine operating between these two temperatures can do better. Carnot’s result is simultaneously a bound on engineering performance and the definition of absolute temperature: the ratio \(T_c/T_h\) is fixed operationally by the ratio of heats exchanged in a reversible cycle.

3.3 Entropy from the Molecular Side

Ludwig Boltzmann gave entropy a statistical meaning. For an isolated system having \(W\) equally probable microscopic configurations consistent with its macroscopic state,

\[ S = k_\mathrm{B}\ln W, \]

where \(k_\mathrm{B}\) is Boltzmann’s constant. Entropy increases because the macroscopic state compatible with the largest number of microstates is overwhelmingly the most probable. The statistical view recovers the third law of thermodynamics: as a perfect crystalline solid is cooled toward 0 K, the number of accessible microstates tends to unity, so \(S\to 0\).

3.4 Computing Entropy Changes

For an ideal gas undergoing a reversible change,

\[ \Delta S = nC_V\ln\frac{T_2}{T_1} + nR\ln\frac{V_2}{V_1}, \]

equivalent forms follow by substituting \(pV=nRT\). For a phase transition at the transition temperature,

\[ \Delta_\mathrm{trs} S = \frac{\Delta_\mathrm{trs} H}{T_\mathrm{trs}}, \]

the empirical observation that \(\Delta_\mathrm{vap} S\approx 85\) J K\(^{-1}\) mol\(^{-1}\) for many nonpolar liquids is known as Trouton’s rule.

Chapter 4: Free Energies and the Direction of Change

4.1 Helmholtz and Gibbs Functions

The second law in the form \(\Delta S_\mathrm{tot}\ge 0\) is awkward because it requires tracking the surroundings. By combining first- and second-law information into a new state function, we can express spontaneity in terms of system properties alone. Define

\[ A \equiv U - TS, \qquad G \equiv H - TS. \]

\(A\) is the Helmholtz energy, useful for closed systems at constant \(T\) and \(V\); \(G\) is the Gibbs energy, useful at constant \(T\) and \(p\). At constant temperature and volume a spontaneous process satisfies \(\mathrm{d}A\le 0\); at constant temperature and pressure, \(\mathrm{d}G\le 0\). Equilibrium corresponds to the minimum of the appropriate free energy.

The differential forms

\[ \mathrm{d}A = -S\,\mathrm{d}T - p\,\mathrm{d}V, \qquad \mathrm{d}G = -S\,\mathrm{d}T + V\,\mathrm{d}p \]

lead directly to the Maxwell relations, of which the most heavily used in this course is

\[ \left(\frac{\partial S}{\partial p}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_p. \]

These cross-derivative identities let us exchange experimentally inconvenient derivatives (like entropy changes with pressure) for easily measured ones (like thermal expansion).

4.2 Gibbs Energy and Non-Expansion Work

A subtle but practically central result says that at constant \(T\) and \(p\), the change in Gibbs energy equals the maximum non-expansion work available from a process,

\[ \Delta G = w_\mathrm{non-exp,max}. \]

This is the thermodynamic basis for connecting chemical reactions to electrical work in galvanic cells, where the cell potential is linked to \(\Delta_\mathrm{r} G\) by \(\Delta_\mathrm{r} G = -nFE_\mathrm{cell}\).

4.3 The Gibbs–Helmholtz Equation

Temperature dependence of \(\Delta G\) is captured compactly by

\[ \left(\frac{\partial\,(\Delta G / T)}{\partial T}\right)_p = -\frac{\Delta H}{T^2}, \]

which, once \(\Delta H\) is known, lets the free energy at one temperature be propagated to another without repeating the entropy accounting.

Chapter 5: Variable Composition and Chemical Potential

5.1 Partial Molar Quantities

When several substances share a phase, each contributes to extensive properties in a way that depends on the composition, not just on pure-component values. The partial molar Gibbs energy of species \(i\),

\[ \mu_i \equiv \left(\frac{\partial G}{\partial n_i}\right)_{T,p,n_{j\ne i}}, \]

is called the chemical potential. The Gibbs energy of a mixture is then \(G=\sum_i n_i\mu_i\), and the fundamental equation for an open system becomes

\[ \mathrm{d}G = -S\,\mathrm{d}T + V\,\mathrm{d}p + \sum_i \mu_i\,\mathrm{d}n_i. \]

5.2 The Gibbs–Duhem Relation

Because \(G\) is homogeneous of degree one in the \(n_i\) at fixed \(T\) and \(p\), Euler’s theorem combined with the fundamental equation gives the constraint

\[ S\,\mathrm{d}T - V\,\mathrm{d}p + \sum_i n_i\,\mathrm{d}\mu_i = 0. \]

At constant \(T\) and \(p\), changes in chemical potential across the species are not independent; this Gibbs–Duhem relation is what makes thermodynamic consistency tests on experimental activity data possible.

5.3 The Chemical Potential of Ideal and Real Mixtures

For an ideal gas mixture,

\[ \mu_i(T,p) = \mu_i^\circ(T) + RT\ln\frac{p_i}{p^\circ}, \]

where \(p_i\) is the partial pressure. For real gases, the partial pressure is replaced by the fugacity \(f_i\), a pressure-like quantity defined so that the ideal-gas form is preserved while all non-ideality is absorbed into the relation between \(f\) and \(p\). For condensed-phase mixtures the analogous role is played by the activity \(a_i\),

\[ \mu_i = \mu_i^\star + RT\ln a_i, \]

with \(\mu_i^\star\) a reference potential whose meaning depends on whether the Raoult (pure-component) or Henry (infinite-dilution) standard state is adopted.

Chapter 6: Phase Equilibria and the Phase Rule

6.1 Condition for Phase Equilibrium

When two phases \(\alpha\) and \(\beta\) of the same pure substance coexist at equilibrium, the chemical potentials must match: \(\mu^\alpha(T,p)=\mu^\beta(T,p)\). Equating differentials gives the Clapeyron equation,

\[ \frac{\mathrm{d}p}{\mathrm{d}T} = \frac{\Delta_\mathrm{trs} S}{\Delta_\mathrm{trs} V} = \frac{\Delta_\mathrm{trs} H}{T\,\Delta_\mathrm{trs} V}, \]

which prescribes the slope of a coexistence curve on a \(p\)-\(T\) diagram. For vaporization, where the vapor is approximately ideal and \(V_\mathrm{gas}\gg V_\mathrm{liq}\), the Clapeyron equation reduces to the Clausius–Clapeyron form,

\[ \frac{\mathrm{d}\ln p}{\mathrm{d}T} = \frac{\Delta_\mathrm{vap} H}{RT^2}, \]

from which vapor-pressure curves and latent heats of vaporization are extracted by plotting \(\ln p\) against \(1/T\).

6.2 The Phase Rule

Josiah Willard Gibbs showed that for a system of \(C\) components distributed among \(P\) phases at equilibrium,

\[ F = C - P + 2, \]

where \(F\) is the number of degrees of freedom—independent intensive variables that can be varied without changing the number of phases. A single-component, single-phase system has two degrees of freedom (e.g., \(T\) and \(p\)); at a triple point, \(P=3\), and \(F=0\).

6.3 Two-Component Phase Diagrams

Binary systems display a richer vocabulary of diagrams: liquid–vapor envelopes for partially miscible solutions, eutectic points in metallic and salt–water systems, azeotropes where the boiling composition equals the condensing composition, and congruent or incongruent melting of intermediate compounds. The lever rule computes the relative amounts of two coexisting phases from the distances along a tie line,

\[ \frac{n^\alpha}{n^\beta} = \frac{x_\beta - x_0}{x_0 - x_\alpha}, \]

where \(x_0\) is the overall composition and \(x_\alpha\), \(x_\beta\) are the phase compositions.

System type	Key feature	Typical example
Ideal binary liquid	Linear Raoult’s-law envelope	Benzene and toluene
Positive deviation	Minimum-boiling azeotrope	Ethanol and water
Negative deviation	Maximum-boiling azeotrope	Nitric acid and water
Partially miscible	Liquid–liquid coexistence loop	Phenol and water
Binary eutectic solid	Melting-point minimum	Lead and tin

Chapter 7: Ideal Solutions and Colligative Properties

7.1 Raoult’s Law and the Ideal Solution

An ideal solution is one in which every component obeys Raoult’s law over the entire composition range,

\[ p_i = x_i\,p_i^\star, \]

where \(p_i^\star\) is the vapor pressure of pure \(i\) at the system temperature. This implies \(\mu_i = \mu_i^\star + RT\ln x_i\). Mixing two ideal liquids produces no enthalpy change and no volume change; the Gibbs energy of mixing per mole is \(RT(x_1\ln x_1+x_2\ln x_2)<0\), purely entropic.

7.2 Colligative Properties

Colligative properties depend only on the number of solute particles, not on their chemical identity. For a dilute solution of a non-volatile, non-electrolyte solute, the main colligative effects are:

• Vapor-pressure lowering. \(\Delta p = -x_2 p_1^\star\) for mole fraction \(x_2\) of solute.
• Boiling-point elevation. \(\Delta T_b = K_b\,m\), where \(K_b=RT_b^{\star\,2}M_1/\Delta_\mathrm{vap} H\) is the ebullioscopic constant and \(m\) the molality.
• Freezing-point depression. \(\Delta T_f = -K_f\,m\) by analogous derivation.
• Osmotic pressure. For a dilute solution separated from pure solvent by a semipermeable membrane, \(\Pi = cRT\), where \(c\) is the molar concentration of solute (van ’t Hoff equation).

All four expressions come from a single move: equate the chemical potential of the solvent in the mixed phase to that of the pure phase on the other side of a phase boundary, expand the logarithm for small mole fractions, and read off the shift in the intensive variable that restores equilibrium.

Chapter 8: Non-Ideal and Electrolyte Solutions

8.1 Activity Coefficients and Excess Functions

Real solutions deviate from Raoult’s law because unlike molecules exert different forces on each other than like molecules do. The deviation is quantified by the activity coefficient \(\gamma_i\), defined by \(a_i = \gamma_i x_i\) (Raoult convention) or \(a_i=\gamma_i^{(H)} x_i\) (Henry convention for dilute solutes). The excess Gibbs energy,

\[ G^\mathrm{E} = RT\sum_i n_i\ln\gamma_i, \]

measures the total deviation from ideal-solution mixing. Engineering correlations such as the Margules, van Laar, Wilson, and NRTL equations provide explicit models for \(G^\mathrm{E}\) that are then differentiated to obtain the individual \(\gamma_i\) required for design calculations.

8.2 Henry’s Law for Dilute Solutes

At high dilution, the solute no longer “sees” other solute molecules, and its partial pressure becomes proportional to mole fraction,

\[ p_2 = K_H\,x_2, \]

with a Henry’s-law constant \(K_H\) specific to the solute–solvent pair and to temperature. Raoult’s law describes the solvent end of the same diagram; Henry’s law describes the solute end.

8.3 Electrolyte Solutions and Debye–Huckel Theory

Ionic solutions are intrinsically non-ideal because long-range Coulomb forces between ions never average out. Peter Debye and Erich Huckel developed a statistical treatment in which each ion is surrounded by a diffuse cloud of opposite charge that partially screens its field. The resulting limiting law, valid at low ionic strength,

\[ \log\gamma_\pm = -A\,|z_+ z_-|\sqrt{I}, \]

predicts a square-root dependence on the ionic strength

\[ I = \tfrac{1}{2}\sum_i m_i z_i^2. \]

Here \(A\approx 0.509\) kg\(^{1/2}\) mol\(^{-1/2}\) for water at 25 degrees C, and \(z_\pm\) are the charge numbers of cation and anion. At higher ionic strength, extended forms—Davies equation, specific-ion-interaction theories, Pitzer equations—correct for finite ion size and short-range interactions.

Chapter 9: Reaction Equilibrium

9.1 Extent of Reaction and the Equilibrium Condition

For a chemical reaction \(0 = \sum_i \nu_i A_i\), define the extent \(\xi\) by \(\mathrm{d}n_i=\nu_i\,\mathrm{d}\xi\). At constant \(T\) and \(p\),

\[ \left(\frac{\partial G}{\partial \xi}\right)_{T,p} = \sum_i \nu_i\mu_i \equiv \Delta_\mathrm{r} G. \]

Equilibrium corresponds to the composition that minimizes \(G\), i.e. to \(\Delta_\mathrm{r} G=0\).

9.2 The Equilibrium Constant

Substituting \(\mu_i = \mu_i^\circ + RT\ln a_i\) into the equilibrium condition gives

\[ \Delta_\mathrm{r} G^\circ = -RT\ln K, \qquad K = \prod_i a_i^{\nu_i}\bigg|_\mathrm{eq}. \]

For gaseous reactions, activities are replaced by \(f_i/p^\circ\); for dilute aqueous species, by activity-coefficient-corrected molalities; for pure solids or liquids, by unity. The equilibrium constant is a pure number; care must be taken that the standard states adopted in the tabulation of \(\Delta_\mathrm{f} G^\circ\) match those assumed in the activity expression.

9.3 Temperature and Pressure Dependence

Differentiation of \(\ln K\) with respect to temperature yields the van ’t Hoff equation,

\[ \frac{\mathrm{d}\ln K}{\mathrm{d}T} = \frac{\Delta_\mathrm{r} H^\circ}{RT^2}, \]

whose integrated form (assuming a temperature-independent reaction enthalpy) is

\[ \ln\frac{K_2}{K_1} = -\frac{\Delta_\mathrm{r} H^\circ}{R}\left(\frac{1}{T_2}-\frac{1}{T_1}\right). \]

Le Chatelier’s principle emerges as a corollary: an exothermic reaction’s equilibrium constant falls with increasing temperature; a reaction that produces fewer moles of gas is favored by increased pressure.

Chapter 10: Surface Phenomena

10.1 Surface Tension and the Young–Laplace Equation

Molecules at a liquid’s surface have fewer neighbors of their own kind than molecules in the bulk, and therefore a higher Gibbs energy per molecule. The reversible work required to extend a surface by an area \(\mathrm{d}A\) at constant \(T\) and \(p\) is

\[ \delta w = \gamma\,\mathrm{d}A, \]

where \(\gamma\), the surface tension, has units of energy per area or equivalently force per length. A curved interface supports a pressure difference between the two sides, given by the Young–Laplace equation,

\[ \Delta p = \gamma\left(\frac{1}{R_1}+\frac{1}{R_2}\right), \]

with \(R_1\) and \(R_2\) the principal radii of curvature. For a spherical droplet of radius \(r\), this reduces to \(\Delta p = 2\gamma/r\); for a cylindrical jet, \(\Delta p = \gamma/r\).

10.2 Capillarity and the Kelvin Equation

When a liquid wets a narrow capillary of radius \(r\) with contact angle \(\theta\), it rises to a height

\[ h = \frac{2\gamma\cos\theta}{\rho g r}, \]

determined by the balance between surface tension and gravitational potential energy. The Kelvin equation relates the vapor pressure over a curved surface to that over a flat one,

\[ \ln\frac{p(r)}{p^\star} = \frac{2\gamma V_\mathrm{m}}{rRT}, \]

explaining why small droplets evaporate preferentially in a fog and why a liquid condenses in narrow pores at subsaturated pressures (capillary condensation).

10.3 Properties of Small Particles

Surface-to-volume ratios scale as \(1/r\), so nanoscale particles have a larger fraction of their molecules at the surface than bulk material does. This pushes chemical potential up for small particles, lowers their melting points (Gibbs–Thomson effect), enhances their solubility, and makes them more reactive than their bulk counterparts. Catalysis, adsorption, and colloid stability all ride on these size-dependent corrections.

10.4 Adsorption Isotherms

Adsorption is the accumulation of molecules at an interface. The Langmuir isotherm models the simplest case: a uniform surface, one adsorbate molecule per site, no lateral interactions, and dynamic equilibrium between adsorption and desorption. The result is

\[ \theta = \frac{K p}{1 + K p}, \]

where \(\theta\) is the fractional surface coverage and \(K\) is a temperature-dependent equilibrium constant. At low pressure \(\theta\approx Kp\) (linear Henry regime); at high pressure \(\theta\to 1\) (monolayer saturation).

Real surfaces depart from this ideal in familiar ways. The Freundlich isotherm \(\theta=k p^{1/n}\) fits heterogeneous surfaces empirically; the BET isotherm extends Langmuir to multilayer adsorption and underlies the standard measurement of catalyst surface areas from nitrogen physisorption.

Isotherm	Expression	Assumptions
Henry	\(\theta = Kp\)	Low coverage, linear limit
Langmuir	\(\theta = Kp/(1+Kp)\)	Uniform sites, monolayer, no interactions
Freundlich	\(\theta = kp^{1/n}\)	Heterogeneous surface, empirical
BET	Multilayer extension of Langmuir	Layers beyond the first behave like bulk liquid

Chapter 11: Chemical Kinetics

11.1 Rates, Rate Laws, and Orders

Thermodynamics tells us which way a reaction proceeds; kinetics tells us how fast. For a reaction \(aA+bB\to cC+dD\), the rate is defined so that it is independent of which species we monitor,

\[ r = -\frac{1}{a}\frac{\mathrm{d}[A]}{\mathrm{d}t} = \frac{1}{c}\frac{\mathrm{d}[C]}{\mathrm{d}t}. \]

An experimental rate law is an empirical relation

\[ r = k\,[A]^\alpha[B]^\beta, \]

where \(\alpha\) and \(\beta\) are the reaction orders with respect to \(A\) and \(B\). Orders are determined by experiment and need not equal the stoichiometric coefficients. The rate constant \(k\) has temperature-dependent units that depend on the overall order.

11.2 Integrated Rate Laws

For simple orders, the rate law integrates to familiar forms.

Determining order experimentally usually proceeds by the method of initial rates (compare rates at different initial concentrations) or by fitting the integrated form that linearizes the data.

11.3 Temperature Dependence: The Arrhenius Equation

Rate constants rise steeply with temperature. Svante Arrhenius proposed

\[ k(T) = A\exp\!\left(-\frac{E_a}{RT}\right), \]

with pre-exponential factor \(A\) and activation energy \(E_a\). A plot of \(\ln k\) against \(1/T\) yields a straight line of slope \(-E_a/R\). Modern transition-state theory supplies a molecular interpretation: molecules must surmount an energy barrier at a transition-state configuration, and \(A\) reflects the frequency and steric accessibility of such encounters.

11.4 Mechanisms and Elementary Steps

Most reactions are not elementary. A mechanism is a proposed sequence of elementary steps whose summed stoichiometry reproduces the overall reaction and whose kinetics reproduce the observed rate law. Common simplifying assumptions include:

• Rate-determining step. When one step is much slower than the others, the overall rate equals that of the slow step, and preceding fast equilibria appear as equilibrium constants in the rate law.
• Steady-state approximation. For reactive intermediates \(I\) present at low concentration, \(\mathrm{d}[I]/\mathrm{d}t\approx 0\) simplifies the system of differential equations.
• Pre-equilibrium. A fast reversible step establishes \([I] = K_1[A][B]\) before the slow productive step consumes \(I\).

11.5 Chain and Parallel Reactions

Radical chain mechanisms—such as the hydrogen–bromine reaction—exhibit initiation, propagation, and termination steps whose relative rates give rise to non-integer apparent orders. Parallel first-order reactions from a common reactant partition the reactant in a ratio \(k_1:k_2\) that is independent of the initial concentration; consecutive reactions \(A\to B\to C\) yield a transient maximum in \([B]\) at time

\[ t_\mathrm{max} = \frac{\ln(k_1/k_2)}{k_1-k_2}. \]

Chapter 12: Catalysis

12.1 What a Catalyst Does, and Does Not Do

A catalyst accelerates a reaction by providing an alternative mechanism with a lower activation energy; it is regenerated in the course of the reaction and does not appear in the overall stoichiometry. A catalyst cannot shift the equilibrium position, because it changes neither \(\Delta G\) nor the equilibrium constant. It reduces the time to reach equilibrium, not the equilibrium itself.

12.2 Homogeneous Catalysis and Acid–Base Mechanisms

In homogeneous catalysis, the catalyst and reactants share a single phase. Aqueous acid-catalyzed ester hydrolysis is a familiar example: the proton activates the carbonyl oxygen, a water molecule attacks the carbonyl carbon, and the tetrahedral intermediate collapses to the carboxylic acid. The empirical rate law \(r=k_\mathrm{H^+}[\mathrm{ester}][\mathrm{H^+}]\) is first order in catalyst.

12.3 Enzyme Kinetics: Michaelis–Menten

Enzymes are nature’s catalysts, and the simplest kinetic model for a single-substrate enzymatic reaction is due to Leonor Michaelis and Maud Menten. Assuming a rapid equilibrium (or, more generally, steady state) for the enzyme–substrate complex \(ES\),

\[ E + S \rightleftharpoons ES \to E + P, \]

the rate of product formation is

\[ r = \frac{V_\mathrm{max}[S]}{K_\mathrm{M} + [S]}, \]

with \(V_\mathrm{max}=k_\mathrm{cat}[E]_\mathrm{tot}\) and the Michaelis constant \(K_\mathrm{M}=(k_{-1}+k_\mathrm{cat})/k_1\). At \([S]\ll K_\mathrm{M}\) the rate is first order in substrate; at \([S]\gg K_\mathrm{M}\) it saturates at \(V_\mathrm{max}\). The Lineweaver–Burk plot \((1/r\) vs \(1/[S])\) linearizes the data and has been superseded in practice by nonlinear least-squares fitting of the hyperbolic form.

12.4 Heterogeneous Catalysis

In heterogeneous catalysis the catalyst and reactants occupy different phases, almost always a solid catalyst contacting gaseous or liquid reactants. The canonical mechanism is Langmuir–Hinshelwood:

1. Adsorption of each reactant onto the surface, governed by Langmuir isotherms.
2. Surface reaction between adsorbed species, usually rate-determining.
3. Desorption of products.

For a bimolecular surface reaction with both species obeying Langmuir adsorption,

\[ r = k\,\theta_A\theta_B = k\,\frac{K_A p_A\,K_B p_B}{(1 + K_A p_A + K_B p_B)^2}. \]

This form captures the experimentally observed maxima in reaction rate versus partial pressure: when one species saturates the surface, it crowds out the other, and the rate falls.

An alternative is the Eley–Rideal mechanism, in which one species reacts directly from the gas phase with an adsorbed partner. The resulting rate law is \(r = k\,\theta_A p_B\), first order in gas-phase \(B\) at fixed coverage. Distinguishing Langmuir–Hinshelwood from Eley–Rideal experimentally is a classic exercise in pressure-dependence fitting.

12.5 Catalyst Deactivation and Selectivity

Industrial catalysts rarely retain constant activity indefinitely. Sintering reduces surface area by coalescence of small particles; poisoning blocks active sites with strongly adsorbing impurities (sulfur on platinum is the textbook example); coking deposits carbonaceous residues on surfaces during hydrocarbon processing. Selectivity, the ratio of desired to undesired product rates, is often the economically decisive performance metric and is tuned by the choice of support, promoter, particle size, and operating conditions.

Chapter 13: Synthesis and Looking Ahead

The physical chemistry of this course is organized around two great unifying ideas. The first is that energy and entropy, taken together, dictate both the direction of spontaneous change (through the free energies) and the position of equilibrium (through equality of chemical potentials across phases and reactions). The second is that approach to equilibrium is governed by mechanism-dependent rates, captured empirically by rate laws and rationalized molecularly by collision and transition-state theories.

Every topic treated here—from bomb-calorimeter thermochemistry to Debye–Huckel activity coefficients, from the Clausius–Clapeyron equation to Langmuir–Hinshelwood kinetics—is an application of those two ideas to a specific class of systems. Later courses in transport phenomena, reactor design, separation processes, and electrochemical engineering all presume fluent use of chemical potential, activity, equilibrium constant, and rate law. Time invested now in seeing the derivations from the Gibbs equation \(\mathrm{d}G=-S\,\mathrm{d}T+V\,\mathrm{d}p+\sum_i\mu_i\,\mathrm{d}n_i\) as a single master relation will repay itself many times over in the engineering subjects to follow.