NE 121: Chemical Principles

Sources and References

• Atkins, P. W., and Jones, L. Chemical Principles: The Quest for Insight. W. H. Freeman.
• Zumdahl, S. S., and DeCoste, D. J. Chemical Principles. Cengage.
• Oxtoby, D. W., Gillis, H. P., and Butler, L. J. Principles of Modern Chemistry. Cengage.
• McQuarrie, D. A., Rock, P. A., and Gallogly, E. B. General Chemistry. University Science Books.
• MIT OpenCourseWare, 5.111 and 5.112 Principles of Chemical Science.
• LibreTexts, General Chemistry and Physical Chemistry libraries.
• Khan Academy, General Chemistry topic collection.
• Brus, L. E. “Electron-electron and electron-hole interactions in small semiconductor crystallites.” J. Chem. Phys. 80, 4403 (1984).
• LaMer, V. K., and Dinegar, R. H. “Theory, production and mechanism of formation of monodispersed hydrosols.” J. Am. Chem. Soc. 72, 4847 (1950).
• Turkevich, J., Stevenson, P. C., and Hillier, J. “A study of the nucleation and growth processes in the synthesis of colloidal gold.” Discuss. Faraday Soc. 11, 55 (1951).

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Why Chemistry for Nanotechnology Engineering

Nanotechnology lives at the scale where classical intuition breaks down and chemistry becomes unavoidable. A gold cube 10 cm on a side is a yellow metal; a gold nanoparticle 10 nm across is red, catalytically active, and behaves as a quantum-confined object. The difference is not mysterious once the underlying chemistry is in place: electronic structure, bonding, thermodynamics, and equilibrium. NE 121 is the course where a nanotechnology engineer assembles that toolkit. The goal is not memorizing reactions but learning how atoms organize themselves into molecules and phases, how energy flows through those systems, and how to predict whether a process will run.

This set of notes follows the traditional general-chemistry arc — atomic structure, bonding, stoichiometry, thermodynamics, equilibrium, acid/base, and redox — but keeps returning to the nanoscale framing. Quantum dots, self-assembled monolayers, carbon nanotubes, and nanoparticle synthesis serve as recurring illustrations. The mathematics is deliberately kept at the level of first-year calculus so that the physical reasoning stays visible.

1. Atomic Structure and Quantum Foundations

1.1 From Dalton to Rutherford

By 1910, three classical experiments had fixed the architecture of the atom. Thomson’s cathode-ray measurement gave the electron’s charge-to-mass ratio; Millikan’s oil-drop experiment fixed the electron charge at \(e = 1.602 \times 10^{-19}\ \mathrm{C}\); and Rutherford’s gold-foil scattering showed that positive charge is concentrated in a tiny, massive nucleus occupying roughly \(10^{-15}\ \mathrm{m}\) inside an atom of radius \(10^{-10}\ \mathrm{m}\). Nearly all of an atom is empty space — which is exactly why electron-electron interactions, not nuclear size, dominate chemistry.

1.2 The Bohr Model and the Spectroscopic Crisis

Classical electrodynamics predicts that an orbiting electron should radiate continuously and spiral into the nucleus in picoseconds. It does not. Hydrogen’s emission spectrum, instead of being continuous, shows sharp lines described by the Rydberg formula:

\[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]

with \(R_H = 1.0974 \times 10^{7}\ \mathrm{m^{-1}}\). Bohr (1913) salvaged stability by postulating quantized angular momentum \(L = n\hbar\), from which he derived allowed energies

\[ E_n = -\frac{13.6\ \mathrm{eV}}{n^2} \]

for the hydrogen atom. Photons are emitted when an electron drops from \(n_2\) to \(n_1\), carrying \(h\nu = E_{n_2} - E_{n_1}\). The Bohr model succeeds brilliantly for one-electron systems and fails for everything else, but it cements two permanent ideas: energies are quantized, and transitions release or absorb discrete photons.

1.3 Wave-Particle Duality and the de Broglie Wavelength

De Broglie (1924) proposed that any particle of momentum \(p\) carries a wavelength

\[ \lambda = \frac{h}{p} \]

Davisson and Germer confirmed this by diffracting electrons off a nickel crystal. A 1 kg baseball at 30 m/s has \(\lambda \approx 2 \times 10^{-35}\ \mathrm{m}\) — undetectable. A 1 eV electron has \(\lambda \approx 1.2\ \mathrm{nm}\), comparable to atomic spacings. Matter waves are a phenomenon of the small.

Heisenberg’s uncertainty relation, \(\Delta x \, \Delta p \ge \hbar/2\), follows from the wave description. Confining an electron to a smaller box raises its momentum spread and therefore its kinetic energy. This single fact underlies quantum confinement in semiconductor nanocrystals.

1.4 The Schrödinger Equation

For a stationary state of energy \(E\), the time-independent Schrödinger equation reads

\[ -\frac{\hbar^2}{2m}\,\nabla^2 \psi + V(\mathbf{r})\,\psi = E\,\psi \]

The wavefunction \(\psi(\mathbf{r})\) has no direct physical meaning; the Born interpretation identifies \(\lvert \psi \rvert^2\) as a probability density. For an electron in a three-dimensional Coulomb potential \(V(r) = -e^2/(4\pi\varepsilon_0 r)\), separation of variables in spherical coordinates yields wavefunctions labelled by three quantum numbers: principal \(n = 1, 2, 3, \ldots\); angular \(\ell = 0, 1, \ldots, n-1\); magnetic \(m_\ell = -\ell, \ldots, +\ell\). Spin \(m_s = \pm 1/2\) is added by hand (or derived relativistically).

1.5 Atomic Orbitals

Orbitals \(\ell = 0, 1, 2, 3\) are labelled \(s, p, d, f\). Each \(s\) orbital is spherical. Each \(p\) orbital has a nodal plane through the nucleus, giving the familiar dumbbell. \(d\) orbitals have two angular nodes and come in five shapes. The radial wavefunction of hydrogen \(1s\) decays as \(e^{-r/a_0}\) with Bohr radius \(a_0 = 5.29 \times 10^{-11}\ \mathrm{m}\).

1.6 Many-Electron Atoms and the Aufbau Principle

Electron-electron repulsion breaks the \(n\)-only degeneracy. Orbitals fill according to increasing \(n + \ell\) with ties broken by lower \(n\), giving the familiar sequence \(1s\ 2s\ 2p\ 3s\ 3p\ 4s\ 3d\ 4p\ 5s\ 4d\ 5p\ 6s\ 4f\ 5d\ 6p\). Three rules govern the ground state:

• The Pauli exclusion principle: no two electrons share all four quantum numbers.
• The Aufbau principle: electrons occupy the lowest-energy available orbital.
• Hund’s rule: within a degenerate set, unpaired spins align before pairing.

The configuration of iron is \([\mathrm{Ar}]\,3d^6\,4s^2\), with four unpaired \(3d\) electrons — a fact that determines iron’s magnetism and the ferromagnetism of iron-oxide nanoparticles used in MRI contrast.

2. Periodicity

2.1 Effective Nuclear Charge

An outer electron does not feel the full nuclear charge \(Z\) because inner electrons screen it. The effective charge \(Z_{\text{eff}} = Z - \sigma\), where \(\sigma\) is Slater’s shielding, rises across a period and only slowly down a group. This single quantity drives almost every periodic trend.

2.2 Trends

• Atomic radius decreases left-to-right (rising \(Z_{\text{eff}}\) pulls the shell inward) and increases top-to-bottom (new shells dominate).
• Ionization energy rises left-to-right and falls down a group, with kinks at filled/half-filled subshells.
• Electron affinity is most exothermic near the halogens; it is positive or near zero for noble gases and the \(s^2\) alkaline earths.
• Electronegativity (Pauling scale) rises toward fluorine (4.0) and falls toward cesium (0.7).

3. Chemical Bonding

3.1 Ionic Bonding and Lattice Energy

When the electronegativity difference exceeds roughly 1.7, electrons transfer and a lattice of cations and anions forms. The Born-Landé equation estimates lattice energy:

\[ U = -\frac{N_A M z^+ z^- e^2}{4\pi\varepsilon_0 r_0}\left(1 - \frac{1}{n}\right) \]

where \(M\) is the Madelung constant, \(n\) a Born repulsion exponent, and \(r_0\) the equilibrium ionic separation. Lattice energies run from roughly 600 kJ/mol (NaI) to several thousand (MgO), and they set melting points and solubilities.

3.2 Covalent Bonding: Lewis and VSEPR

Lewis structures distribute valence electrons so that every atom (except hydrogen and the expanded-octet third-row and beyond) achieves an octet. Formal charge \(= \text{valence electrons} - \text{nonbonding} - \tfrac{1}{2}\text{bonding}\). The best structure minimizes formal-charge magnitudes and places negative formal charge on the most electronegative atom.

VSEPR theory assigns molecular geometry from the number of electron domains around the central atom:

Domains	Shape	Bond angle	Example
2	linear	180°	\(\mathrm{CO_2}\)
3	trigonal planar	120°	\(\mathrm{BF_3}\)
4	tetrahedral	109.5°	\(\mathrm{CH_4}\)
4 (1 lone pair)	trigonal pyramidal	~107°	\(\mathrm{NH_3}\)
4 (2 lone pairs)	bent	~104.5°	\(\mathrm{H_2O}\)
5	trigonal bipyramidal	90° and 120°	\(\mathrm{PCl_5}\)
6	octahedral	90°	\(\mathrm{SF_6}\)

Lone pairs occupy more angular space than bonding pairs, squeezing bond angles below the idealized values.

3.3 Valence Bond Theory and Hybridization

Valence bond theory assembles molecular orbitals from hybridized atomic orbitals. Mixing one \(s\) and \(n\) \(p\) orbitals gives \(sp^{n}\) hybrids pointing toward bond partners: \(sp\) (linear), \(sp^2\) (trigonal planar), \(sp^3\) (tetrahedral). Carbon’s flexibility between \(sp^3\) (diamond), \(sp^2\) (graphite, graphene, carbon nanotubes), and \(sp\) (carbyne) is why the element underwrites organic chemistry and nanocarbon materials.

Sigma (\(\sigma\)) bonds are cylindrically symmetric about the internuclear axis; pi (\(\pi\)) bonds overlap above and below. A double bond is \(\sigma + \pi\); a triple bond is \(\sigma + 2\pi\).

3.4 Molecular Orbital Theory

MO theory builds delocalized orbitals from linear combinations of atomic orbitals across the whole molecule. For a diatomic \(\mathrm{A_2}\), two atomic orbitals produce a bonding combination \(\sigma\) (lower energy, constructive interference) and an antibonding \(\sigma^*\) (higher energy, node between nuclei). Bond order:

\[ \text{BO} = \tfrac{1}{2}\left(n_\text{bonding} - n_\text{antibonding}\right) \]

\(\mathrm{O_2}\) has bond order 2 with two unpaired electrons in \(\pi^*\) — correctly predicting its paramagnetism, which Lewis theory cannot. Extended MO reasoning gives band structure: in a solid, \(N\) atoms contribute \(N\) orbitals that merge into a near-continuous band. The filled band is the valence band, the empty one above it the conduction band, separated by a bandgap \(E_g\). Metals have no gap; semiconductors have small gaps (Si: 1.12 eV); insulators have large gaps.

4. Intermolecular Forces

No real condensed phase is held together by covalent bonds alone. The weaker, longer-range intermolecular forces explain boiling points, surface tension, solubility, and self-assembly.

• London dispersion — instantaneous dipole-induced dipole. Scales as \(\alpha^2 / r^6\); grows with polarizability and therefore with molar mass.
• Dipole-dipole — permanent dipoles aligning head-to-tail.
• Hydrogen bonding — special strong dipole interaction when H is bonded to N, O, or F and near another lone pair on N, O, or F. Responsible for water’s anomalous density maximum, for DNA base pairing, and for the pH buffering of protein solutions.
• Ion-dipole — dominant when ions dissolve in a polar solvent.

5. Stoichiometry and Balances

5.1 Moles, Mass, and the Balanced Equation

A balanced chemical equation is an expression of atomic conservation; each element must appear on both sides in equal number. The mole (\(N_A = 6.022 \times 10^{23}\)) is the conversion between microscopic and macroscopic. Molar mass relates grams to moles: \(n = m/M\).

Limiting reagent, theoretical yield, and percent yield follow from stoichiometric ratios. For \(\mathrm{2\,H_2 + O_2 \to 2\,H_2O}\), 4 g H\(_2\) and 4 g O\(_2\) produce water limited by oxygen: \(n_{\mathrm{O_2}} = 0.125\) mol, giving at most 0.25 mol (4.5 g) water and leaving 3.75 g H\(_2\) unreacted.

5.2 Mass and Charge Balance

Whenever two or more reactions share a solution, two bookkeeping equations close the system:

• Mass balance: the total amount of each element, summed across every species that contains it, equals what was added.
• Charge balance: the sum of positive charges equals the sum of negative charges, weighted by the magnitude of each ion’s charge.

6. Gases

6.1 The Ideal Gas Law

\[ PV = nRT \]

captures Boyle, Charles, Avogadro, and Gay-Lussac in one expression. Partial pressures add: Dalton’s law gives \(P_\text{total} = \sum_i P_i\) with mole fractions \(x_i = P_i / P_\text{total}\). Graham’s law of effusion follows from equal kinetic energies: \(r_1/r_2 = \sqrt{M_2/M_1}\).

6.2 Kinetic-Molecular Theory

Treating gas as point particles in elastic random motion derives pressure and identifies temperature with average kinetic energy:

\[ \left\langle \tfrac{1}{2} m v^2 \right\rangle = \tfrac{3}{2} k_B T \]

The Maxwell-Boltzmann speed distribution peaks at most-probable speed \(v_p = \sqrt{2 k_B T / m}\), with mean \(\bar v = \sqrt{8 k_B T/(\pi m)}\) and root-mean-square \(v_\text{rms} = \sqrt{3 k_B T/m}\).

6.3 Real Gases

The van der Waals equation introduces attractions and excluded volume:

\[ \left(P + \frac{a n^2}{V^2}\right)\left(V - n b\right) = n R T \]

Near the critical point the isotherms kink; below \(T_c\) a liquid-vapour coexistence curve appears. In nanoparticle synthesis by vapour condensation, understanding deviation from ideality matters because precursor partial pressures sit in the range where attractive interactions can no longer be ignored.

7. The First Law of Thermodynamics

7.1 Energy, Work, Heat

The first law is conservation: \(\Delta U = q + w\), where \(q\) is heat absorbed by the system and \(w\) is work done on it. Pressure-volume work at constant external pressure is \(w = -P_\text{ext}\,\Delta V\).

Defining enthalpy \(H = U + PV\) makes constant-pressure heat exchanges directly measurable: \(q_P = \Delta H\). Enthalpies of reaction, formation, combustion, and phase change are tabulated as \(\Delta H^\circ\) at standard state.

7.2 Hess’s Law and Standard Enthalpies of Formation

Because \(H\) is a state function, any thermochemical cycle sums to zero. Hess’s law gives

\[ \Delta H^\circ_\text{rxn} = \sum_\text{products} \nu_i \Delta H^\circ_{f,i} - \sum_\text{reactants} \nu_j \Delta H^\circ_{f,j} \]

with \(\Delta H^\circ_f = 0\) for elements in their reference state. Calorimetry — coffee-cup for constant-pressure reactions, bomb for constant-volume combustions — provides the experimental numbers.

7.3 Heat Capacity

\(C_V = (\partial U/\partial T)_V\) and \(C_P = (\partial H/\partial T)_P\); for an ideal gas \(C_P - C_V = nR\). Monatomic gases have \(C_V = \tfrac{3}{2}R\); diatomics rise to \(\tfrac{5}{2}R\) once rotations activate. The Dulong-Petit rule gives solids \(C_V \approx 3R\) per mole of atoms at high temperature, and Debye/Einstein corrections handle the low-temperature drop.

8. The Second Law and Entropy

8.1 Statements of the Second Law

Classically: heat does not spontaneously flow from cold to hot (Clausius), and no cyclic engine converts heat entirely to work (Kelvin-Planck). Both are captured by a single state function, entropy:

\[ dS = \frac{\delta q_\text{rev}}{T} \]

For any process, \(\Delta S_\text{universe} \ge 0\), with equality only for reversible processes.

8.2 Statistical Interpretation

Boltzmann identified entropy with the logarithm of microstate count:

\[ S = k_B \ln W \]

Systems spontaneously evolve toward macrostates of highest \(W\). Mixing increases \(S\) because there are vastly more ways to arrange mixed particles than segregated ones. Phase transitions, reactions, and even self-assembly all obey this counting.

8.3 Gibbs Free Energy

Coupling the first and second laws at constant \(T\) and \(P\) gives

\[ \Delta G = \Delta H - T\,\Delta S \]

A process is spontaneous iff \(\Delta G < 0\). The sign of \(\Delta H\) and \(\Delta S\) yield four regimes:

\(\Delta H\)	\(\Delta S\)	Spontaneous?
negative	positive	always
positive	negative	never
negative	negative	at low \(T\)
positive	positive	at high \(T\)

9. The Third Law and Standard Entropies

Nernst’s third law states that the entropy of a perfect crystal approaches zero as \(T \to 0\ \mathrm{K}\). This is a genuine zero, not a convention, and it makes absolute entropies \(S^\circ\) measurable — unlike \(H\), which is only known relative to a reference state. Reaction entropies use the usual sum rule:

\[ \Delta S^\circ_\text{rxn} = \sum \nu_i S^\circ_i(\text{products}) - \sum \nu_j S^\circ_j(\text{reactants}) \]

Standard Gibbs energies of formation \(\Delta G^\circ_f\) are tabulated likewise, again with zero for elements in their reference states.

10. Chemical Equilibrium

10.1 The Equilibrium Constant

For a generic reaction \(aA + bB \rightleftharpoons cC + dD\) in the gas phase,

\[ K_P = \frac{(P_C/P^\circ)^c (P_D/P^\circ)^d}{(P_A/P^\circ)^a (P_B/P^\circ)^b} \]

In solution, replace partial pressures by concentrations divided by \(c^\circ = 1\ \text{mol/L}\) to obtain \(K_c\). \(K\) is dimensionless by construction, though the \(c^\circ\) or \(P^\circ\) factors are often omitted.

The thermodynamic link is

\[ \Delta G^\circ = -RT \ln K \]

An exergonic reaction has \(K > 1\); endergonic reactions have \(K < 1\). The van ’t Hoff equation ties temperature dependence to \(\Delta H^\circ\):

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

10.2 Le Chatelier’s Principle

If a stress is applied to a system at equilibrium, the equilibrium shifts to partially relieve that stress. Added reactant pushes forward; added product pushes back; increased pressure (for gas reactions) shifts toward fewer moles of gas; increased temperature shifts in the endothermic direction. Catalysts speed both directions equally and leave \(K\) unchanged.

10.3 Reaction Quotient and Direction

For arbitrary concentrations, compute \(Q\) with the same algebra as \(K\). If \(Q < K\) the reaction proceeds forward; if \(Q > K\), backward. The instantaneous free energy is \(\Delta G = \Delta G^\circ + RT \ln Q\); \(Q = K\) exactly when \(\Delta G = 0\).

11. Acids and Bases

11.1 Brønsted-Lowry and Lewis Definitions

A Brønsted-Lowry acid donates a proton; a base accepts one. Every acid has a conjugate base (\(\mathrm{HA} \to \mathrm{A^-}\)) and vice versa. Lewis broadened the definition to electron pair donation/acceptance, which accommodates \(\mathrm{BF_3}\) and \(\mathrm{AlCl_3}\) as acids without a proton in sight.

Water’s autoionization is the anchor:

\[ 2\,\mathrm{H_2O} \rightleftharpoons \mathrm{H_3O^+} + \mathrm{OH^-} \qquad K_w = 10^{-14}\ \text{at 25 } \degree\mathrm{C} \]

so \(\mathrm{pH} + \mathrm{pOH} = 14\) at this temperature.

11.2 \(K_a\), \(K_b\), and Conjugate Pairs

For \(\mathrm{HA} \rightleftharpoons \mathrm{H^+} + \mathrm{A^-}\),

\[ K_a = \frac{\left[\mathrm{H^+}\right]\left[\mathrm{A^-}\right]}{\left[\mathrm{HA}\right]} \]

and \(K_a K_b = K_w\) for any conjugate pair. Strong acids (\(K_a \gg 1\)): \(\mathrm{HCl}\), \(\mathrm{HBr}\), \(\mathrm{HI}\), \(\mathrm{HNO_3}\), \(\mathrm{HClO_4}\), \(\mathrm{H_2SO_4}\) (first proton). Weak acids obey the equilibrium expression and are solved with an ICE table.

11.3 pH Calculations

For a weak acid of formal concentration \(C_0\), if dissociation is small,

\[ \left[\mathrm{H^+}\right] \approx \sqrt{K_a C_0} \]

The approximation fails when \(K_a / C_0\) is not small; then solve the quadratic exactly.

11.4 Buffers and the Henderson-Hasselbalch Equation

A buffer is a mixture of a weak acid and its conjugate base. Rearranging the \(K_a\) expression:

\[ \mathrm{pH} = \mathrm{p}K_a + \log\!\left(\frac{\left[\mathrm{A^-}\right]}{\left[\mathrm{HA}\right]}\right) \]

Maximum buffering capacity occurs at \(\mathrm{pH} = \mathrm{p}K_a\). Biological buffers (phosphate \(\mathrm{p}K_a = 7.2\), bicarbonate ~6.1 with physiological \(\mathrm{CO_2}\)) illustrate the chemistry that keeps cells alive.

11.5 Titration

Plotting pH vs titrant volume gives sigmoidal curves. The equivalence point is the volume at which moles of titrant equal moles of analyte; the half-equivalence point of a weak-acid titration sits at \(\mathrm{pH} = \mathrm{p}K_a\). Polyprotic titrations show multiple inflections at successive \(\mathrm{p}K_a\) values.

12. Solubility and Precipitation

For a sparingly soluble salt \(\mathrm{M_a X_b}\), dissolution follows

\[ \mathrm{M_a X_b}(s) \rightleftharpoons a\,\mathrm{M^{n+}} + b\,\mathrm{X^{m-}} \qquad K_{sp} = \left[\mathrm{M^{n+}}\right]^a \left[\mathrm{X^{m-}}\right]^b \]

A solution precipitates when \(Q_{sp} > K_{sp}\). Common-ion effect: adding \(\mathrm{X^{m-}}\) shifts equilibrium toward the solid, lowering solubility. Selective precipitation separates ions by choosing a counterion whose \(K_{sp}\) with one cation is much smaller than with another.

13. Complex-Ion Equilibria

Many metal cations bind Lewis-base ligands to form complex ions, described by a stepwise or cumulative formation constant:

\[ \mathrm{M^{n+}} + j\,\mathrm{L} \rightleftharpoons \mathrm{ML_j^{n+}} \qquad \beta_j = \frac{\left[\mathrm{ML_j^{n+}}\right]}{\left[\mathrm{M^{n+}}\right]\left[\mathrm{L}\right]^j} \]

Silver with ammonia forms \(\mathrm{Ag(NH_3)_2^+}\) (\(\beta_2 \approx 1.6 \times 10^7\)); EDTA binds nearly every divalent cation with \(\beta\) in the range \(10^8\text{–}10^{18}\). Adding a strong complexing agent dissolves salts that would otherwise be insoluble by pulling \(\mathrm{M^{n+}}\) out of the \(K_{sp}\) equilibrium.

14. Oxidation-Reduction

14.1 Oxidation States and Balancing

Assign oxidation numbers by the familiar rules: free elements zero; group-1 ions +1; oxygen −2 (with peroxide exceptions); hydrogen +1 with nonmetals; overall charge sums consistent. A redox reaction is any reaction in which oxidation states change.

Balance by half-reactions:

1. Split into oxidation and reduction half-reactions.
2. Balance elements other than O and H.
3. Balance O with \(\mathrm{H_2O}\), H with \(\mathrm{H^+}\).
4. Balance charge with electrons.
5. Multiply half-reactions by integers so that electrons cancel.
6. In basic solution, add \(\mathrm{OH^-}\) to both sides to neutralize \(\mathrm{H^+}\).

14.2 Electrochemical Potentials

Standard reduction potentials \(E^\circ\) are tabulated against the standard hydrogen electrode (\(E^\circ = 0\)). For a full cell,

\[ E^\circ_\text{cell} = E^\circ_\text{cathode} - E^\circ_\text{anode} \]

and the free energy is \(\Delta G^\circ = -n F E^\circ\) with Faraday’s constant \(F = 96{,}485\ \mathrm{C/mol}\). The Nernst equation gives the non-standard potential:

\[ E = E^\circ - \frac{RT}{nF} \ln Q \]

A battery is a spontaneous redox cell; electrolysis is the opposite, driven by an external voltage.

15. Nanoscale-Specific Chemistry

15.1 Surface-Area-to-Volume Scaling

A sphere of radius \(R\) has surface \(4\pi R^2\) and volume \(\tfrac{4}{3}\pi R^3\), giving \(A/V = 3/R\). Slice bulk silver (say \(R = 1\ \mathrm{cm}\)) into 10 nm cubes and the specific surface grows by six orders of magnitude. Surface atoms dominate reactivity: heterogeneous catalysis, sensor response, and toxicology all scale with \(A/V\).

15.2 Quantum Confinement

For a cubic box of side \(L\) containing a single electron, the allowed energies are

\[ E_{n_x n_y n_z} = \frac{\hbar^2 \pi^2}{2 m L^2}\left(n_x^2 + n_y^2 + n_z^2\right) \]

At \(L = 3\ \mathrm{nm}\) and using a semiconductor effective mass, the ground-state energy sits in the hundreds of meV — comparable to bulk bandgaps. Quantum dots, nanowires (confinement in two dimensions), and quantum wells (one dimension) each show a characteristic spectroscopic fingerprint.

15.3 Molecular Self-Assembly

Self-assembly is equilibrium bottom-up construction: weak interactions (hydrogen bonds, van der Waals, hydrophobic, electrostatic) collectively find a global or near-global free-energy minimum. Examples include lipid bilayers, block-copolymer microphases, DNA origami (using Watson-Crick base pairing as an addressable code), and alkanethiol SAMs on gold. The design principle — choose interactions whose strength sits just above \(k_B T\) so the system can anneal — turns chemistry into a manufacturing technology.

16. Worked-Example Compilation

(i) Bandgap of a 4 nm CdSe dot. Using the Brus equation with \(E_g^\text{bulk} = 1.74\ \mathrm{eV}\), effective masses \(m_e^* = 0.13\,m_0\) and \(m_h^* = 0.45\,m_0\), reduced mass \(\mu = 0.101\,m_0\). The confinement term \(\hbar^2 \pi^2/(2\mu R^2)\) at \(R = 2\ \mathrm{nm}\) gives roughly 0.47 eV, plus a small Coulomb correction, predicting an effective gap near 2.2 eV — consistent with yellow-orange emission.

(ii) pH of 0.0500 M \(\mathrm{NH_3}\). \(K_b = 1.8 \times 10^{-5}\). Approximating \(\left[\mathrm{OH^-}\right] \approx \sqrt{K_b C_0} = \sqrt{(1.8\times 10^{-5})(0.0500)} = 9.5 \times 10^{-4}\ \mathrm{M}\). \(\mathrm{pOH} = 3.02\); \(\mathrm{pH} = 10.98\).

(iii) Equilibrium shift on compression. For \(\mathrm{N_2(g)} + 3\,\mathrm{H_2(g)} \rightleftharpoons 2\,\mathrm{NH_3(g)}\), four moles of gas on the left become two on the right. Increasing pressure shifts the equilibrium toward ammonia — the Haber-Bosch industrial lever. High temperature works against \(K\) (the forward reaction is exothermic), so the compromise operating point is ~450 \(\degree\)C at ~200 bar.

(iv) Cell potential for zinc-copper. \(E^\circ_\text{cell} = E^\circ(\mathrm{Cu^{2+}/Cu}) - E^\circ(\mathrm{Zn^{2+}/Zn}) = 0.34 - (-0.76) = 1.10\ \mathrm{V}\). \(\Delta G^\circ = -(2)(96485)(1.10) = -212\ \mathrm{kJ/mol}\) — which is why a Daniell cell is spontaneous.

(v) Hess cycle for methane combustion. From formation enthalpies \(\mathrm{CH_4}\ (-74.8)\), \(\mathrm{CO_2}\ (-393.5)\), \(\mathrm{H_2O(\ell)}\ (-285.8)\), with \(\mathrm{O_2} = 0\):

\[ \Delta H^\circ_\text{rxn} = (-393.5) + 2(-285.8) - (-74.8) = -890.3\ \mathrm{kJ/mol} \]

(vi) Solubility of \(\mathrm{AgCl}\) in 0.10 M NaCl. \(K_{sp}(\mathrm{AgCl}) = 1.8 \times 10^{-10}\). With \(\left[\mathrm{Cl^-}\right] \approx 0.10\), \(\left[\mathrm{Ag^+}\right] = K_{sp}/\left[\mathrm{Cl^-}\right] = 1.8 \times 10^{-9}\ \mathrm{M}\) — four orders of magnitude below the pure-water solubility. That is the common-ion effect made numerical.

17. Synthesis: Seeing the Course as One Object

The intellectual move of NE 121 is repeated at each scale: write down the relevant conservation (of atoms, of charge, of energy, of microstates), impose the relevant equilibrium relation, and solve. Quantum mechanics gives the energy levels and bond angles. Thermodynamics says whether a process can run at all. Kinetics (treated later in NE 221) says how fast. Equilibrium chemistry says where it stops. For a nanotechnology engineer, this arc is the grammar; the remaining curriculum is vocabulary piled on top.