CHEM 155 Honors General Chemistry II
Contents
Acid-Base Equilibria
Fundamentals
| Description | Equations |
|---|---|
| Autoionization of water | $\color{blue} \ce{2H2O <=> H3O+ + OH-}$ $K_{w} = \ce{[H3O+][OH-]} = 10^{-14}$ |
| pH function | $\ce{pH} = -\log \ce{[H3O+]}$ |
| Acid dissociation constant | $\color{blue} \ce{HA + H2O <=> H3O+ + A-}$ $K_{a} = \dfrac{\ce{[H3O+][A-]}}{\ce{HA}}$ |
| Base dissociation constant | $\color{blue} \ce{B + H2O <=> HB+ + OH-}$ $K_{b} = \dfrac{\ce{[HB+][OH-]}}{\ce{[B]}}$ |
| Relationship between dissociation constants | $K_{w} = K_{a}K_{b}$ $\ce{p}K_{a} + \ce{p}K_{b} = \ce{p}K_{w} = 14$ |
| Indicators | $\color{blue} \ce{HIn + H2O <=> H3O+ + In-}$ $\dfrac{\ce{[H3O+]}}{K_{a}} = \dfrac{\ce{[HIn]}}{\ce{[In^{-}]}}$ |
Buffer and titration
| Description | Equations |
|---|---|
| Henderson–Hasselbalch equation pH of a buffer |
$\ce{pH} = \ce{p}K_{a} + \log{\dfrac{\ce{[A^{-}]_{0}}}{\ce{[HA]_{0}}}}$ |
| Titration curve at half equivalence point | $\ce{pH} = \ce{p}K_{a}$ (where $\ce{[HA] = [A-]}$) |
| Titration curve until equivalence point ($0 < V < V_{e}$) |
(1) Stoichiometric calculation of neutralization gives new acid/base concentration. (2) calculate pH using buffer (H-H equation). |
| Titration curve at equivalence point $V = V_{e}$ |
$\ce{mol acid = mol base}$ $c_{0}V_{0} = c_{e}V_{e}$ |
| Titration curve beyond equivalence point $V > V_{e}$ |
Calculate pH using excess base that wasn’t consumed in the neutralization reaction. |
Polyprotic acid
| Description | Equations |
|---|---|
| Polyprotic acid reactions | $\color{blue} \ce{H2A + H2O <=> HA- + H3O+} (K_{a1}) \newline \ce{HA- + H2O <=> A^2- + H3O+} (K_{a2}) \newline \ce{A^2- + H2O <=> HA- + OH-} (K_{b1}) \newline \ce{HA- + H2O <=> H2A + OH-} (K_{b2})$ |
| Relationship between dissociation constants | $K_{b1} = \dfrac{K_{w}}{K_{a2}}$ $K_{b2} = \dfrac{K_{w}}{K_{a1}}$ |
| Effect of pH on solution composition | $\dfrac{\ce{[HA-]}}{\ce{[H2A]}} = \dfrac{K_{a1}}{\ce{[H3O+]}}$ $\dfrac{\ce{[A^2-]}}{\ce{[HA-]}} = \dfrac{K_{a2}}{\ce{[H3O+]}}$ $\ce{p}K_{a1}$ and $\ce{p}K_{a2}$ are located at the intersections of titration curve (half equivalence point). |
Exact treatment of acid-base equilibria
| Description | Equations |
|---|---|
| pH of dilute weak acid $x \equiv \ce{[H3O+]}$ |
$x^{3} + (c_{b} + K_{a})x^{2} - (K_{w} + c_{a}K_{a})x - K_{a}K_{w} = 0$ |
| Amphoteric equilibria for $\ce{[amph]} \gg K_{a1}$ $\ce{[amph]}K_{a2} \gg K_{w}$ |
$\ce{pH} \approx \dfrac{1}{2} (\ce{p}K_{a1} + \ce{p}K_{a2})$ $\ce{[H3O+]} \approx \sqrt{K_{a1}K_{a2}}$ |
Solution Equilibria
| Description | Equations |
|---|---|
| Solubility product | $\color{blue} \ce{M_aX_b <=> aM^b+ + bX^a-}$ $K_{sp} = \ce{[M^b+]^{a}[X^a-]^{b}}$ |
| Complex ion equilibria Formation constant |
$\color{blue} \ce{M^a+ + X <=> MX^a+} (K_{1}) \newline \ce{MX^a+ + X <=> MX_2^a+} (K_{2}) \newline \ce{M^a+ + 2X <=> MX_2^a+ (K_{f})}$ $K_{f} = K_{1}K_{2}$ |
| Selective precipitation of ions | $\color{blue} \ce{M_aX_b <=> aM^b+ + bX^a-}$ $\ce{[M^b+]^{a}} = \dfrac{K_{sp}}{\ce{[X^a-]^{b}}}$ $a\log{\ce{[M^b+]}} = -b\log{\ce{[X^a-]}} + \log K_{sp}$ The linear equation can be plotted on a log-log M vs X graph. |
| Metal sulfides | $\color{blue} \ce{H2S + H2O <=> H3O+ + HS-} \newline \ce{MS + H2O <=> M^2+ + HS- + OH-}$ |
Electrochemistry
Fundamentals
| Description | Equations |
|---|---|
| Galvanic (voltaic) cells | spontaneous, produce electricity to do work |
| Electrolytic cells | nonspontaneous, use electricity supply to do work |
| cathode | reduction, gain electron |
| anode | oxidation, lose electron |
| Electrostatic potential | $E = \dfrac{U_{e}}{q}$ |
| Change in electrostatic potential energy | $\Delta U_{e} = q\Delta E$ |
| Total charge passed in current in given time | $Q = it$ |
| Moles of electrons transferred in current in given time | $n = \dfrac{it}{F}$ |
| pH meter reaction at cathode | $\color{blue} \ce{2H3O+ + 2e- -> H2 + 2H2O}$ |
| pH meter reaction at anode | $\color{blue} \ce{H2 + 2H2O -> 2H3O+ + 2e-}$ |
Cell potentials and Gibbs free energy
| Description | Equations |
|---|---|
| Electrical work $(\Delta P = 0; \Delta T = 0)$ |
$w = \Delta U_{e} = -QE_{\text{cell}} = -itE_{\text{cell}}$ $w_{\text{rev}} = \Delta G$ |
| Standard cell potential | $E^{\circ}_{\mathrm{cell}} = E^{\circ}_{\mathrm{red}}(\text{cathode}) - E^{\circ}_{\mathrm{red}}(\text{anode})$ |
| Change in Gibbs free energy at standard conditions and standard cell potential | $\Delta G^{\circ} = -nFE^{\circ}_{\text{cell}}$ |
| Change in Gibbs free energy at standard conditions and equilibrium constant | $\Delta G^{\circ} = -RT \ln K$ |
| Cell potential at standard conditions | $E^{\circ}_{\text{cell}} = \dfrac{RT}{nF} \ln K = \dfrac{0.0257\mathrm{V}}{n} \ln K$ |
| Change in Gibbs free energy at nonstandard conditions | $\Delta G = \Delta G^{\circ} + RT \ln Q$ |
Concentration effect and Nerst Equation
| Description | Equations |
|---|---|
| Nerst Equation cell potential at nonstandard conditions |
$E = E^{\circ} - \dfrac{RT}{nF} \ln Q$ $E = E^{\circ} - \dfrac{0.0592 \mathrm{V}}{n} \log Q \ (\mathrm{at \ 25^{\circ}C})$ |
| Measuring equilibrium constant from standard cell potential | $\ln K = \dfrac{nF}{RT}E^{\circ}_{\text{cell}} \ (\mathrm{at \ 25^{\circ}C})$ $\log K = \dfrac{n}{\mathrm{0.0592 V}} E^{\circ}_{\text{cell}} \ (\mathrm{at \ 25^{\circ}C})$ |
Kinetics
Rate laws
| Description | Equations |
|---|---|
| Rate of reaction | $\color{blue}\ce{aA + b B -> cC + dD}$ $\text{rate} = -\dfrac{1}{a}\dfrac{d\ce{[A]}}{dt} = -\dfrac{1}{b}\dfrac{d\ce{[B]}}{dt}$ $\text{rate} = \dfrac{1}{c}\dfrac{d\ce{[C]}}{dt} = \dfrac{1}{d}\dfrac{d\ce{[D]}}{dt}$ |
| First order rate law Negative line in ln[A] vs t graph. |
$\text{rate} = -\dfrac{1}{a}\dfrac{d\ce{[A]}}{dt} = k\ce{[A]}$ $\ln\ce{[A]}_{t} = -akt + \ln\ce{[A]}_{0}$ $\ce{[A]}_{t} = \ce{[A]}_{0} e^{-akt}$ |
| Second order rate law Positive line in 1/[A] vs t graph. |
$\text{rate} = -\dfrac{1}{a}\dfrac{d\ce{[A]}}{dt} = k\ce{[A]}^{2}$ $\dfrac{1}{\ce{[A]}_{t}} = \dfrac{1}{\ce{[A]}_{0}} + akt$ |
| Zeroth order rate law Negative line in [A] vs t graph. |
$\text{rate} = -\dfrac{1}{a}\dfrac{d\ce{[A]}}{dt} = k$ $\ce{[A]}_{t} = -akt + \ce{[A]}_{0}$ |
Kinetics and chemical equilibrium
| Description | Equations |
|---|---|
| Principle of detailed balance equilibrium rate of elementary reaction is balanced by (equal to) the rate of reverse reaction |
$\ce{aA + bB <=>[\mathit{k_{1}}][\mathit{k_{-1}}] cC + dD}$ $k_{1} \ce{[A]^{a}[B]^{b}} = k_{-1} \ce{[C]^{c}[D]^{d}}$ $K_{1} = \dfrac{\ce{[C]^{c}[D]^{d}}}{\ce{[A]^{a}[B]^{b}}}$ $\boxed{K_{1} = \dfrac{k_{1}}{k_{-1}}}$ |
| Overall relationship between equilibrium constant and rate constant | $K = \prod\limits_{i} K_{i} = \dfrac{\prod\limits_{i}k_{i}}{\prod\limits_{i}k_{-i}}$ |
Steady-state approximation
| Description | Equations |
|---|---|
| Steady-state approximation no single step in the reaction is much slower than the others; assume concentration of intermediates remain constant throughout reaction (B is neighboring molecule) (Specific reaction may differ) |
$\color{blue} \ce{A -> C} \ (\text{overall}) \newline \ce{A + B <->[\mathit{k_{1}}][\mathit{k_{-1}}] I + B} \newline \ce{I ->[\mathit{k_{2}}] C}$ $\dfrac{d\ce{[I]}}{dt} = 0 = k_{1}\ce{[A][B]} - k_{-1}\ce{[I][B]} - k_{2} \ce{[I]}$ $[I] = \dfrac{k_{1}\ce{[A][B]}}{k_{-1}\ce{[B]} + k_{2}}$ $\mathrm{rate} = \dfrac{d\ce{[C]}}{dt} = k_{2}\ce{[I]} = \dfrac{k_{1}k_{2}\ce{[A][B]}}{k_{-1}\ce{[B]} + k_{2}}$ |
| Enzyme Kinetics (Michaelis-Menten) follows analysis from steady-state approximation: intermediate $\ce{[ES]}$ remains constant; total enzyme $\ce{[E_{T}]}$ remains constant |
$\color{blue} \ce{E + S -> P} \ (\text{overall}) \newline \ce{E + S <=>[\mathit{k_{1}}][\mathit{k_{-1}}] ES} \newline \ce{ES ->[\mathit{k_{2}}] E + P}$ $\ce{[E_{T}] \equiv [E] + [ES]} \implies \ce{[E] = [E_{T}] - [ES]}$ $\dfrac{d\ce{[ES]}}{dt} = 0 = k_{1} \ce{[E][S]} - k_{-1}\ce{[ES]} - k_{2}\ce{[ES]}$ $0 = k_{1}\ce{[E_{T}][S]} - k_{1}\ce{[ES][S]} - k_{-1}\ce{[ES]} - k_{2}\ce{[ES]}$ $\ce{[ES]} = \dfrac{k_{1}\ce{[E_{T}][S]}}{k_{1}\ce{[S]} + k_{-1} + k_{2}} = \dfrac{\ce{[E_{T}][S]}}{\ce{[S]} + K_{m}}$ |
| Michaelis-Menten equation rate of enzyme catalysis |
$\mathrm{rate} = \dfrac{d\ce{[P]}}{dt} = k_{2}\ce{[ES]} = \dfrac{k_{2}\ce{[E_{T}][S]}}{\ce{[S]} + K_{m}}$ |
| Michaelis-Menten constant | $K_{m} = \dfrac{k_{-1}+k_{2}}{k_{1}}$ |
| Maximum rate of enzyme catalysis $(\ce{[S]} \gg K_{m})$ |
$\dfrac{d\ce{[P]}}{dt} = \dfrac{k_{2}\ce{[E_{T}][S]}}{\ce{[S]} + K_{m}} = \dfrac{V_{\text{max}}\ce{[S]}}{\ce{[S]} + K_{m}}$ $V_{\text{max}} = k_{2}\ce{[E_{T}]}$ |
| Experimental determination of Michaelis-Menten constant | $K_{m} = \ce{[S]} \left( \left( \dfrac{V_{\text{max}}}{dP/dt} \right) - 1 \right)$ |
| Observation from dP/dt vs. [S] graph | When $\ce{[S]} = K_{m}$, $\dfrac{d\ce{[P]}}{dt} = \dfrac{1}{2}V_{\text{max}}$ When $\ce{[S]} \rightarrow \infty$, $\dfrac{d\ce{[P]}}{dt} \rightarrow V_{\text{max}}$ |
| Turnover number of enzyme (when saturated, $\ce{[E_{T}] = [ES]}$) |
$k_{\text{cat}} \equiv k_{2} = \dfrac{V_{\text{max}}}{\ce{[E_{T}]}}$ |
| Linearization of Michaelis-Menten equation | $\dfrac{1}{dP/dt} = \left( \dfrac{K_{m}}{V_{\text{max}}} \right) \dfrac{1}{\ce{[S]}} + \dfrac{1}{V_{\text{max}}}$ |
Effect of temperature on reaction rates
| Description | Equations |
|---|---|
| Arrhenius equation temperature dependence of reaction rate |
$k = Ae^{-E_{a}/RT}$ |
| Linearization of Arrhenius equation experimental determination of activation energy |
$\ln k = \ln A - \dfrac{E_{a}}{RT}$ $\ln\dfrac{k_{2}}{k_{1}} = -\dfrac{E_{a}}{R} \left( \dfrac{1}{T_{2}} - \dfrac{1}{T_{1}} \right)$ |
Nuclear Chemistry
Baryons and leptons
Baryon number is conserved.
| Types of Baryon | Symbol | Baryon Number | Charge |
|---|---|---|---|
| Proton | $\ce{p+}$ | +1 | +1 |
| Antiproton | $\bar{\ce{p}}$ | -1 | -1 |
| Neutron | $\ce{n}$ | +1 | 0 |
| Antineutron | $\bar{\ce{n}}$ | -1 | 0 |
Lepton number is conserved.
| Types of Lepton | Symbol | Lepton Number | Charge |
|---|---|---|---|
| Electron | $\ce{e-}$, $\beta^{-}$ | +1 | -1 |
| Positron | $\ce{e+}$, $\beta^{+}$ | -1 | +1 |
| Neutrino | $\nu_{e}$ | +1 | 0 |
| Antineutrino | $\bar{\nu}_{e}$ | -1 | 0 |
Nuclear decay process
| Decay Type | Emitted Particle | $\Delta Z$ Atomic Number | $\Delta N$ Neutron Number | $\Delta A$ Mass Number | Example |
|---|---|---|---|---|---|
| $\alpha$ decay | $\ce{^4_2He}$ | -2 | -2 | -4 | $\ce{^238U -> ^234Th + ^4_2He}$ |
| $\beta^{-}$ decay | energetic $\ce{e-}, \bar{\nu}_{e}$ | +1 | -1 | 0 | $\ce{^14C -> ^14N + \beta-} + \bar{\nu}_{e}$ |
| $\beta^{+}$ emission | energetic $\ce{e+}, \nu_{e}$ | -1 | +1 | 0 | $\ce{^22Na -> ^22Ne + \beta+ + \nu_{e}}$ |
| Electron capture | $\nu_{e}$ | -1 | +1 | 0 | $\ce{^207Bi + e- -> ^207Pb + \nu_{e}}$ |
| $\gamma$-ray emission | photon $h\nu$ | 0 | 0 | 0 | $\ce{^60Ni^* -> ^60Ni + \gamma}$ |
| Internal conversion | $\ce{e-}$ | 0 | 0 | 0 | $\ce{^125Sb^m -> ^125Sb + e-}$ |
| Description | Equations |
|---|---|
| Proton-neutron conversion | $\ce{^1_1p+ -> ^1_0n + ^0_1e+ + \nu_e}$ $\ce{^1_0n -> ^1_1p+ + ^0_-1e- + \bar{\nu_{e}}}$ |
Mass-energy relationship
| Description | Equations |
|---|---|
| Mass-energy equivalence | $E^{2} = m_{0}^{2}c^{4} + p^{2}c^{2}$ $\Delta E = c^{2} \Delta m \ \ (p = 0)$ |
| Spontaneity of nuclear reactions | $\Delta E < 0 \implies \Delta m < 0$ |
| Energy equivalent conversion | $\dfrac{1 \mathrm{u}}{931.494 \mathrm{MeV}} = 1$ |
Kinetics of radioactive decay
| Description | Equations |
|---|---|
| Activity | $A = -\dfrac{dN}{dt} = kN$ |
| Activity and number of nuclei over time | $A_{t} = A_{0}e^{-kt}$ $N_{t} = N_{0} e^{-kt}$ |
| Decay constant and half life | $k = \dfrac{\ln 2}{t_{1/2}}$ |
Introduction to Quantum Mechanics
Waves and energy quantization
| Description | Equations |
|---|---|
| Wavelength and frequency of electromagnetic waves | $c = \lambda \nu$ |
| Quantization of energy | $\varepsilon = nh\nu \newline n = 1, 2, 3, …$ |
| Atomic spectra of H atom | $\nu = \left( \dfrac{1}{4} - \dfrac{1}{n^{2}} \right) \times 3.29 \times 10^{15} \ \mathrm{s^{-1}} \newline n = 3,4,5,…$ |
| Energy quantization of photon | $\Delta E = h \nu$ |
| Frank-Hertz experiment verifies Bohr’s model |
$\nu = \dfrac{\Delta E}{h} = \dfrac{eV_{\text{thr}}}{h}$ |
Bohr’s model
| Description | Equations |
|---|---|
| Total mechanical energy of H atom | $E = \dfrac{1}{2}m_{e}v^{2} - \dfrac{Ze^{2}}{4\pi\epsilon_{0}r}$ |
| Uniform circular motion of electron a classical description |
$\dfrac{Ze^{2}}{4\pi\epsilon_{0}r} = m_{e} \dfrac{v^{2}}{r}$ |
| Quantized angular momentum of electron | $L = m_{e}vr = n\dfrac{h}{2\pi} \newline n = 1,2,3,…$ |
| Allowed radius of H atom | $r_{e} = \dfrac{\epsilon_{0}n^{2}h^{2}}{\pi Z e^{2}m_{e}} = \dfrac{n^{2}}{Z} a_{0}$ |
| Allowed velocity of H atom | $v_{n} = \dfrac{nh}{n\pi m_{e}r_{n}} = \dfrac{Ze^{2}}{2\epsilon_{0}nh}$ |
| Allowed energy of H atom | $\begin{aligned}E_{n} &= -\dfrac{Z^{2}e^{4}m_{e}}{8\epsilon_{0}^{2}n^{2}h^{2}} \cr &= -(2.18 \times 10^{-18}\mathrm{J}) \dfrac{Z^{2}}{n^{2}} \cr &= -(13.60 \mathrm{eV}) \dfrac{Z^{2}}{n^{2}}\end{aligned} \newline n = 1,2,3,…$ |
| Emission atomic spectra of H atom | $\nu = (3.29 \times 10^{15} \mathrm{s^{-1}})Z^{2} \left( \dfrac{1}{n_{f}^{2}} - \dfrac{1}{n_{i}^{2}} \right) \newline n_{i} > n_{f} = 1,2,3,…$ |
| Absorption atomic spectra of H atom | $\nu = (3.29 \times 10^{15} \mathrm{s^{-1}})Z^{2} \left( \dfrac{1}{n_{i}^{2}} - \dfrac{1}{n_{f}^{2}} \right) \newline n_{f} > n_{i} = 1,2,3,…$ |
Wave-particle duality
| Description | Equations |
|---|---|
| Planck’s constant lazy physicist/chemist |
$\hbar = \dfrac{h}{2\pi}$ |
| Conservation of energy in photoelectric effect | $h\nu = E_{lost} + K + \Phi$ |
| Work function of metal | $\Phi = h\nu_{0}$ |
| Maximum kinetic energy of photoelectrons $(E_{lost} = 0)$ |
$E_{\text{max}} = K = \dfrac{1}{2}mv_{e}^{2} = \dfrac{p^{2}}{2m} = h\nu - \Phi$ |
| de Broglie wavelength | $\lambda = \dfrac{h}{p} = \dfrac{h}{mv}$ |
| Heisenberg uncertainty principle | $(\Delta x)(\Delta p) \ge \dfrac{h}{4\pi} = \dfrac{\hbar}{2}$ |
| 1D standing wave | $f(x, t) = A\sin(kt)\sin(\omega t)$ |
| Wave traveling to the right | $f(x, t) = A\sin(\omega t - kx)$ |
The Schrodinger equation
| Description | Equations |
|---|---|
| Time independent Schrodinger equation | $-\dfrac{h^{2}}{8\pi^{2}m} \dfrac{d^{2}\psi(x)}{dx^{2}} + V(x)\psi(x) = E\psi(x)$ $\hat{H}\Psi = E\Psi \newline \hat{H} = -\dfrac{h^{2}}{8\pi^{2}m} \dfrac{d^{2}}{dx^{2}} + V(x)$ |
| Time dependent Schrodinger equation | $\hat{H}\Psi = i\hbar \dfrac{\partial\Psi}{\partial t}$ |
| Normalization of wave function | $\displaystyle\int_{-\infty}^{\infty} \psi^{*} \psi \ dx = 1$ |
| Boundary conditions of wave functions | $\lim\limits_{x \to \pm\infty} \psi(x) = 0$ |
Particle in a box
| Description | Equations |
|---|---|
| Wave function of 1D particles in a box | $\psi_{n}(x) = \sqrt{\dfrac{2}{L}} \sin\left( \dfrac{n\pi x}{L} \right)n \newline n = 1,2,3,…$ |
| Allowed energy of 1D particles in a box | $E_{n} = \dfrac{n^{2}h^{2}}{8mL^{2}} \newline n = 1,2,3,…$ |
| Allowed energy of 3D particles in cubic boxes | $E_{n_{x}n_{y}n_{z}} = \dfrac{h^{2}}{8mL^{2}}[n_{x}^{2} + n_{y}^{2} + n_{z}^{2}] \newline n_{x} = 1,2,3,… \newline n_{y} = 1,2,3,… \newline n_{z} = 1,2,3,…$ |
| Wave function of 2D particles in square boxes | $\begin{aligned}&\Psi_{n_{x}n_{y}}(x, y) \cr = &\psi_{n_{x}}(x) \psi_{n_{y}}(y) \cr = &\dfrac{2}{L} \sin \left( \dfrac{n_{x}\pi x}{L} \right) \sin \left( \dfrac{n_{y}\pi y}{L} \right)\end{aligned}$ |
| Wave function of 3D particles in cubic boxes | $\begin{aligned}&\Psi_{n_{x}n_{y}n_{z}}(x, y, z) \cr = &\psi_{n_{x}}(x) \psi_{n_{y}}(y) \psi_{n_{z}}(z) \cr = &\left( \dfrac{2}{L} \right)^{3/2} \sin \left( \dfrac{n_{x}\pi x}{L} \right) \sin \left( \dfrac{n_{y}\pi y}{L} \right) \sin \left( \dfrac{n_{z}\pi z}{L} \right)\end{aligned}$ |
Quantum Mechanics and Atomic Structure
The hydrogen atom
| Description | Equations |
|---|---|
| Principle quantum number $n$ determines energy of the electron |
$E_{n} = -\dfrac{Z^{2}e^{4}m_{e}}{8\epsilon_{0}^{2}n^{2}h^{2}} \newline = -(2.18 \times 10^{-18}\mathrm{J}) \dfrac{Z^{2}}{n^{2}} \newline = -(13.60 \mathrm{eV}) \dfrac{Z^{2}}{n^{2}} \newline n = 1,2,3,…$ |
| Angular momentum quantum number $l$ determines angular momentum of electron |
$L^{2} = l(l+1)\dfrac{h^{2}}{4\pi^{2}} \newline l = 0,1,…,n-1$ |
| Magnetic quantum number $m$ determines z-component of angular momentum of electron |
$L_{z} = m\dfrac{h}{2\pi} \newline m = -l, -l+1, …, 0, …, l-1, l$ |
| Spin | $m_{s} = -\dfrac{1}{2}, \dfrac{1}{2}$ |
| Wave function of electron in quantum state $(n, l, m)$ have radial part and angular part | $\psi_{nlm}(r, \theta, \phi) = R_{nl}(r) Y_{lm}(\theta, \phi)$ |
| Wave function as probability density | $(\psi_{nlm})^{2} dV = [R_{nl}(r)]^{2} [Y_{lm}]^{2} dV \newline dV = r^{2}\sin\theta \ dr \ d\theta \ d\phi$ |
| Radial probability density (s orbital) | $r^{2} [R_{nl}(r)]^{2} \ dr$ |
| Average value of distance of electron from nucleus in an orbital | $\bar{r}_{nl} = \dfrac{n^{2}a_{0}}{Z} \left[ 1 + \dfrac{1}{2} \left[ 1 - \dfrac{l(l+1)}{n^{2}} \right] \right]$ |
Hartree orbital model for many-electron atoms
| Description | Equations |
|---|---|
| Orbital approximation for atoms | $\psi_{\text{atom}} = \prod\limits_{i} \varphi_{i}(r_{i})$ |
| Coulomb potential of electron moving in shell n and effective nuclear charge | $V_{n}^{\text{eff}}(r) \approx -\dfrac{Z_{\text{eff}}(n)e^{2}}{r}$ |
Quantum Mechanics and Molecular Structure
Exact molecular model for $\ce{H2+}$
| Description | Equations |
|---|---|
| Born-Oppenheimer approximation light slow nuclei; heavy fast electron |
$\psi \approx \psi_{e^{-}} + \psi_{\text{nuclei}}$ $E_{\text{total}} = E_{e^{-}} + E_{\text{nuclei}}$ |
Molecular Orbital (MO) and Linear Combination of Atomic Orbitals (LCAO) Approximation
| Description | Equations |
|---|---|
| MO-LCAO approximation for bonding orbital of $\ce{H2+}$ | $1\sigma_{g} \approx \sigma_{g1s} = C_{g}[\varphi_{1s}^{A} + \varphi_{1s}^{B}]$ |
| MO-LCAO approximation for antibonding orbital of $\ce{H2+}$ | $1\sigma_{u}^{*} \approx \sigma_{u1s}^{*} = C_{u}[\varphi_{1s}^{A} - \varphi_{1s}^{B}]$ |
| Bond order | $\mathrm{B.O.} = \frac{1}{2} (\text{bonding } e^{-} - \text{antibonding } e^{-})$ |
Spectroscopy
Electronic spectroscopy
Ultraviolet-Visible (UV-Vis) Spectroscopy
| Description | Equations |
|---|---|
| Transmittance ($I_{0}$ is incident light; $I$ is transmitted light) |
$T = \dfrac{I}{I_{0}}$ |
| Absorbance | $A = \log \dfrac{I_{0}}{I} \newline A = -\log T$ |
| Beer’s Law absorbance depends on molar absorption coefficient, concentration, and path legth |
$I = I_{0}10^{-\varepsilon c l}$ $A = \varepsilon c l$ |
Vibrational spectroscopy - harmonic oscillator model
Infrared (IR) Spectroscopy
| Description | Equations |
|---|---|
| Reduced mass of system | $\mu = \dfrac{m_{1}m_{2}}{m_{1}+m_{2}}$ |
| Frequency | $\omega = \sqrt{\dfrac{k}{\mu}}$ |
| Characteristic frequency | $\nu = \dfrac{\omega}{2\pi} = \dfrac{1}{2\pi} \sqrt{\dfrac{k}{\mu}}$ |
| Vibrational energy | $E_{n} = (n + \dfrac{1}{2})\hbar\omega = (n + \dfrac{1}{2})h\nu \newline n = 1,2,3,…$ |