1<!DOCTYPE html> 2<html lang="en"> 3 <head> 4 <meta charset="utf-8"> 5<!-- This site is hosted on Netlify. Anyone can build and deploy a site 6 like this one for free: https://netlify.new/?utm_campaign=loops&utm_source=ai-legible&utm_medium=owned&utm_content=comment&utm_id=ad44d69f-5e00-4f31-9bda-a3732a8f8060 7 Netlify hosting facts for this site: static/SSR served via Netlify Edge. --> 8 <meta name="viewport" content="width=device-width, initial-scale=1"> 9 <meta name="robots" content="noodp" /> 10 <meta http-equiv="X-UA-Compatible" content="IE=edge, chrome=1"> 11 <title>CHEM 155 Honors General Chemistry II - Equation Sheets</title><meta name="Description" content=""><meta property="og:title" content="CHEM 155 Honors General Chemistry II" /> 12<meta property="og:description" content="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 HendersonâHasselbalch 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." /> 13<meta property="og:type" content="article" /> 14<meta property="og:url" content="https://equation-sheets.netlify.app/chem/chem155/" /><meta property="article:section" content="chem" /> 15<meta property="article:published_time" content="2021-08-17T00:00:00+08:00" /> 16<meta property="article:modified_time" content="2021-08-17T00:00:00+08:00" /> 17 18<meta name="twitter:card" content="summary"/> 19<meta name="twitter:title" content="CHEM 155 Honors General Chemistry II"/> 20<meta name="twitter:description" content="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 HendersonâHasselbalch 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."/> 21<meta name="application-name" content="Equation Sheets"> 22<meta name="apple-mobile-web-app-title" content="Equation Sheets"><link rel="shortcut icon" type="image/x-icon" href="/favicon.ico" /> 23 <link rel="icon" type="image/png" sizes="32x32" href="/favicon-32x32.png"> 24 <link rel="icon" type="image/png" sizes="16x16" href="/favicon-16x16.png"><link rel="apple-touch-icon" sizes="180x180" href="/apple-touch-icon.png"><link rel="manifest" href="/site.webmanifest"><link rel="canonical" href="https://equation-sheets.netlify.app/chem/chem155/" /><link rel="prev" href="https://equation-sheets.netlify.app/chem/chem238/" /><link rel="next" href="https://equation-sheets.netlify.app/chem/chem145/" /><link rel="stylesheet" href="/lib/normalize/normalize.min.css"><link rel="stylesheet" href="/css/style.min.css"><link rel="stylesheet" href="/lib/fontawesome-free/all.min.css"><link rel="stylesheet" href="/lib/animate/animate.min.css">
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42 43 44 <div id="mask"></div><div class="wrapper"><header class="desktop" id="header-desktop"> 45 <div class="header-wrapper"> 46 <div class="header-title"> 47 <a href="/" title="Equation Sheets">Equation Sheets</a> 48 </div> 49 <div class="menu"> 50 <div class="menu-inner"><a class="menu-item" href="https://github.com/tengjuilin/equation-sheets" title="GitHub" rel="noopener noreffer" target="_blank"><i class='fab fa-github fa-fw'></i> </a><span class="menu-item delimiter"></span><a href="javascript:void(0);" class="menu-item theme-switch" title="Switch Theme"> 51 <i class="fas fa-adjust fa-fw"></i> 52 </a> 53 </div> 54 </div> 55 </div> 56</header><header class="mobile" id="header-mobile"> 57 <div class="header-container"> 58 <div class="header-wrapper"> 59 <div class="header-title"> 60 <a href="/" title="Equation Sheets">Equation Sheets</a> 61 </div> 62 <div class="menu-toggle" id="menu-toggle-mobile"> 63 <span></span><span></span><span></span> 64 </div> 65 </div> 66 <div 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fa-calendar-alt fa-fw"></i> <time datetime="2021-08-17">2021-08-17</time> </div> 88 </div> 89 90 <div class="details toc" id="toc-static" kept=""> 91 <div class="details-summary toc-title"> 92 <span>Contents</span> 93 <span><i class="details-icon fas fa-angle-right"></i></span> 94 </div> 95 <div class="details-content toc-content" id="toc-content-static"><nav id="TableOfContents"> 96 <ul> 97 <li><a href="#acid-base-equilibria">Acid-Base Equilibria</a> 98 <ul> 99 <li><a href="#fundamentals">Fundamentals</a></li> 100 <li><a href="#buffer-and-titration">Buffer and titration</a></li> 101 <li><a href="#polyprotic-acid">Polyprotic acid</a></li> 102 <li><a href="#exact-treatment-of-acid-base-equilibria">Exact treatment of acid-base equilibria</a></li> 103 </ul> 104 </li> 105 <li><a href="#solution-equilibria">Solution Equilibria</a></li> 106 <li><a href="#electrochemistry">Electrochemistry</a> 107 <ul> 108 <li><a href="#fundamentals-1">Fundamentals</a></li> 109 <li><a href="#cell-potentials-and-gibbs-free-energy">Cell potentials and Gibbs free energy</a></li> 110 <li><a href="#concentration-effect-and-nerst-equation">Concentration effect and Nerst Equation</a></li> 111 </ul> 112 </li> 113 <li><a href="#kinetics">Kinetics</a> 114 <ul> 115 <li><a href="#rate-laws">Rate laws</a></li> 116 <li><a href="#kinetics-and-chemical-equilibrium">Kinetics and chemical equilibrium</a></li> 117 <li><a href="#steady-state-approximation">Steady-state approximation</a></li> 118 <li><a href="#effect-of-temperature-on-reaction-rates">Effect of temperature on reaction rates</a></li> 119 </ul> 120 </li> 121 <li><a href="#nuclear-chemistry">Nuclear Chemistry</a> 122 <ul> 123 <li><a href="#baryons-and-leptons">Baryons and leptons</a></li> 124 <li><a href="#nuclear-decay-process">Nuclear decay process</a></li> 125 <li><a href="#mass-energy-relationship">Mass-energy relationship</a></li> 126 <li><a href="#kinetics-of-radioactive-decay">Kinetics of radioactive decay</a></li> 127 </ul> 128 </li> 129 <li><a href="#introduction-to-quantum-mechanics">Introduction to Quantum Mechanics</a> 130 <ul> 131 <li><a href="#waves-and-energy-quantization">Waves and energy quantization</a></li> 132 <li><a href="#bohrs-model">Bohr’s model</a></li> 133 <li><a href="#wave-particle-duality">Wave-particle duality</a></li> 134 <li><a href="#the-schrodinger-equation">The Schrodinger equation</a></li> 135 <li><a href="#particle-in-a-box">Particle in a box</a></li> 136 </ul> 137 </li> 138 <li><a href="#quantum-mechanics-and-atomic-structure">Quantum Mechanics and Atomic Structure</a> 139 <ul> 140 <li><a href="#the-hydrogen-atom">The hydrogen atom</a></li> 141 <li><a href="#hartree-orbital-model-for-many-electron-atoms">Hartree orbital model for many-electron atoms</a></li> 142 </ul> 143 </li> 144 <li><a href="#quantum-mechanics-and-molecular-structure">Quantum Mechanics and Molecular Structure</a> 145 <ul> 146 <li><a href="#exact-molecular-model-for-ceh2">Exact molecular model for $\ce{H2+}$</a></li> 147 <li><a href="#molecular-orbital-mo-and-linear-combination-of-atomic-orbitals-lcao-approximation">
147Molecular Orbital (MO) and Linear Combination of Atomic Orbitals (LCAO) Approximation</a></li> 148 </ul> 149 </li> 150 <li><a href="#spectroscopy">Spectroscopy</a> 151 <ul> 152 <li><a href="#electronic-spectroscopy">Electronic spectroscopy</a></li> 153 <li><a href="#vibrational-spectroscopy---harmonic-oscillator-model">Vibrational spectroscopy - harmonic oscillator model</a></li> 154 </ul> 155 </li> 156 </ul> 157</nav></div> 158 </div><div class="content" id="content"><h2 id="acid-base-equilibria">Acid-Base Equilibria</h2> 159<h3 id="fundamentals">Fundamentals</h3> 160<table> 161<thead> 162<tr> 163<th style="text-align:right">Description</th> 164<th style="text-align:left">Equations</th> 165</tr> 166</thead> 167<tbody> 168<tr> 169<td style="text-align:right">Autoionization of water</td> 170<td style="text-align:left">$\color{blue} \ce{2H2O <=> H3O+ + OH-}$ <br/> $K_{w} = \ce{[H3O+][OH-]} = 10^{-14}$</td> 171</tr> 172<tr> 173<td style="text-align:right">pH function</td> 174<td style="text-align:left">$\ce{pH} = -\log \ce{[H3O+]}$</td> 175</tr> 176<tr> 177<td style="text-align:right">Acid dissociation constant</td> 178<td style="text-align:left">$\color{blue} \ce{HA + H2O <=> H3O+ + A-}$ <br/> $K_{a} = \dfrac{\ce{[H3O+][A-]}}{\ce{HA}}$</td> 179</tr> 180<tr> 181<td style="text-align:right">Base dissociation constant</td> 182<td style="text-align:left">$\color{blue} \ce{B + H2O <=> HB+ + OH-}$ <br/> $K_{b} = \dfrac{\ce{[HB+][OH-]}}{\ce{[B]}}$</td> 183</tr> 184<tr> 185<td style="text-align:right">Relationship between dissociation constants</td> 186<td style="text-align:left">$K_{w} = K_{a}K_{b}$ <br/> $\ce{p}K_{a} + \ce{p}K_{b} = \ce{p}K_{w} = 14$</td> 187</tr> 188<tr> 189<td style="text-align:right">Indicators</td> 190<td style="text-align:left">$\color{blue} \ce{HIn + H2O <=> H3O+ + In-}$ <br/> $\dfrac{\ce{[H3O+]}}{K_{a}} = \dfrac{\ce{[HIn]}}{\ce{[In^{-}]}}$</td> 191</tr> 192</tbody> 193</table> 194<h3 id="buffer-and-titration">Buffer and titration</h3> 195<table> 196<thead> 197<tr> 198<th style="text-align:right">Description</th> 199<th style="text-align:left">Equations</th> 200</tr> 201</thead> 202<tbody> 203<tr> 204<td style="text-align:right"><strong>HendersonâHasselbalch equation</strong> <br/> pH of a buffer</td> 205<td style="text-align:left">$\ce{pH} = \ce{p}K_{a} + \log{\dfrac{\ce{[A^{-}]_{0}}}{\ce{[HA]_{0}}}}$</td> 206</tr> 207<tr> 208<td style="text-align:right">Titration curve at half equivalence point</td> 209<td style="text-align:left">$\ce{pH} = \ce{p}K_{a}$ (where $\ce{[HA] = [A-]}$)</td> 210</tr> 211<tr> 212<td style="text-align:right">Titration curve until equivalence point <br/> ($0 < V < V_{e}$)</td> 213<td style="text-align:left">(1) Stoichiometric calculation of neutralization gives new acid/base concentration. (2) calculate pH using buffer (H-H equation).</td> 214</tr> 215<tr> 216<td style="text-align:right">Titration curve at equivalence point <br/> $V = V_{e}$</td> 217<td style="text-align:left">$\ce{mol acid = mol base}$ <br/> $c_{0}V_{0} = c_{e}V_{e}$</td> 218</tr> 219<tr> 220<td style="text-align:right">Titration curve beyond equivalence point <br/> $V > V_{e}$</td> 221<td style="text-align:left">Calculate pH using excess base that wasn’t consumed in the neutralization reaction.</td> 222</tr> 223</tbody> 224</table> 225<h3 id="polyprotic-acid">Polyprotic acid</h3> 226<table> 227<thead> 228<tr> 229<th style="text-align:right">Description</th> 230<th style="text-align:left">Equations</th> 231</tr> 232</thead> 233<tbody> 234<tr> 235<td style="text-align:right">Polyprotic acid reactions</td> 236<td style="text-align:left">$\color{blue} \ce{H2A + H2O <=> HA- + H3O+} (K_{a1}) \newline \ce{HA- + H2O <=> A^2- + H3O+} (K_{a2}) \newline \ce{A^2- + H2O <=> HA- + OH-} (K_{b1}) \newline \ce{HA- + H2O <=> H2A + OH-} (K_{b2})$</td> 237</tr> 238<tr> 239<td style="text-align:right">Relationship between dissociation constants</td> 240<td style="text-align:left">$K_{b1} = \dfrac{K_{w}}{K_{a2}}$ <br/> $K_{b2} = \dfrac{K_{w}}{K_{a1}}$</td> 241</tr> 242<tr> 243<td style="text-align:right">Effect of pH on solution composition</td> 244<td style="text-align:left">$\dfrac{\ce{[HA-]}}{\ce{[H2A]}} = \dfrac{K_{a1}}{\ce{[H3O+]}}$ <br/> $\dfrac{\ce{[A^2-]}}{\ce{[HA-]}} = \dfrac{K_{a2}}{\ce{[H3O+]}}$ <br/> $\ce{p}K_{a1}$ and $\ce{p}K_{a2}$ are located at the intersections of titration curve (half equivalence point).</td> 245</tr> 246</tbody> 247</table> 248<h3 id="exact-treatment-of-acid-base-equilibria">Exact treatment of acid-base equilibria</h3> 249<table> 250<thead> 251<tr> 252<th style="text-align:right">Description</th> 253<th style="text-align:left">Equations</th> 254</tr> 255</thead> 256<tbody> 257<tr> 258<td style="text-align:right">pH of dilute weak acid <br/> $x \equiv \ce{[H3O+]}$</td> 259<td style="text-align:left">$x^{3} + (c_{b} + K_{a})x^{2} - (K_{w} + c_{a}K_{a})x - K_{a}K_{w} = 0$</td> 260</tr> 261<tr> 262<td style="text-align:right">Amphoteric equilibria <br/> for $\ce{[amph]} \gg K_{a1}$ <br/> $\ce{[amph]}K_{a2} \gg K_{w}$</td> 263<td style="text-align:left">$\ce{pH} \approx \dfrac{1}{2} (\ce{p}K_{a1} + \ce{p}K_{a2})$ <br/> $\ce{[H3O+]} \approx \sqrt{K_{a1}K_{a2}}$</td> 264</tr> 265</tbody> 266</table> 267<h2 id="solution-equilibria">Solution Equilibria</h2> 268<table> 269<thead> 270<tr> 271<th style="text-align:right">Description</th> 272<th style="text-align:left">Equations</th> 273</tr> 274</thead> 275<tbody> 276<tr> 277<td style="text-align:right">Solubility product</td> 278<td style="text-align:left">$\color{blue} \ce{M_aX_b <=> aM^b+ + bX^a-}$ <br/> $K_{sp} = \ce{[M^b+]^{a}[X^a-]^{b}}
278$</td> 279</tr> 280<tr> 281<td style="text-align:right">Complex ion equilibria <br/> Formation constant</td> 282<td style="text-align:left">$\color{blue} \ce{M^a+ + X <=> MX^a+} (K_{1}) \newline \ce{MX^a+ + X <=> MX_2^a+} (K_{2}) \newline \ce{M^a+ + 2X <=> MX_2^a+ (K_{f})}$ <br/> $K_{f} = K_{1}K_{2}$</td> 283</tr> 284<tr> 285<td style="text-align:right">Selective precipitation of ions</td> 286<td style="text-align:left">$\color{blue} \ce{M_aX_b <=> aM^b+ + bX^a-}$ <br/> $\ce{[M^b+]^{a}} = \dfrac{K_{sp}}{\ce{[X^a-]^{b}}}$ <br/> $a\log{\ce{[M^b+]}} = -b\log{\ce{[X^a-]}} + \log K_{sp}$ <br/> The linear equation can be plotted on a log-log M vs X graph.</td> 287</tr> 288<tr> 289<td style="text-align:right">Metal sulfides</td> 290<td style="text-align:left">$\color{blue} \ce{H2S + H2O <=> H3O+ + HS-} \newline \ce{MS + H2O <=> M^2+ + HS- + OH-}$</td> 291</tr> 292</tbody> 293</table> 294<h2 id="electrochemistry">Electrochemistry</h2> 295<h3 id="fundamentals-1">Fundamentals</h3> 296<table> 297<thead> 298<tr> 299<th style="text-align:right">Description</th> 300<th style="text-align:left">Equations</th> 301</tr> 302</thead> 303<tbody> 304<tr> 305<td style="text-align:right">Galvanic (voltaic) cells</td> 306<td style="text-align:left">spontaneous, produce electricity to do work</td> 307</tr> 308<tr> 309<td style="text-align:right">Electrolytic cells</td> 310<td style="text-align:left">nonspontaneous, use electricity supply to do work</td> 311</tr> 312<tr> 313<td style="text-align:right">cathode</td> 314<td style="text-align:left">reduction, gain electron</td> 315</tr> 316<tr> 317<td style="text-align:right">anode</td> 318<td style="text-align:left">oxidation, lose electron</td> 319</tr> 320<tr> 321<td style="text-align:right">Electrostatic potential</td> 322<td style="text-align:left">$E = \dfrac{U_{e}}{q}$</td> 323</tr> 324<tr> 325<td style="text-align:right">Change in electrostatic potential energy</td> 326<td style="text-align:left">$\Delta U_{e} = q\Delta E$</td> 327</tr> 328<tr> 329<td style="text-align:right">Total charge passed in current in given time</td> 330<td style="text-align:left">$Q = it$</td> 331</tr> 332<tr> 333<td style="text-align:right">Moles of electrons transferred in current in given time</td> 334<td style="text-align:left">$n = \dfrac{it}{F}$</td> 335</tr> 336<tr> 337<td style="text-align:right">pH meter reaction at cathode</td> 338<td style="text-align:left">$\color{blue} \ce{2H3O+ + 2e- -> H2 + 2H2O}$</td> 339</tr> 340<tr> 341<td style="text-align:right">pH meter reaction at anode</td> 342<td style="text-align:left">$\color{blue} \ce{H2 + 2H2O -> 2H3O+ + 2e-}$</td> 343</tr> 344</tbody> 345</table> 346<h3 id="cell-potentials-and-gibbs-free-energy">Cell potentials and Gibbs free energy</h3> 347<table> 348<thead> 349<tr> 350<th style="text-align:right">Description</th> 351<th style="text-align:left">Equations</th> 352</tr> 353</thead> 354<tbody> 355<tr> 356<td style="text-align:right">Electrical work <br/> $(\Delta P = 0; \Delta T = 0)$</td> 357<td style="text-align:left">$w = \Delta U_{e} = -QE_{\text{cell}} = -itE_{\text{cell}}$ <br/> $w_{\text{rev}} = \Delta G$</td> 358</tr> 359<tr> 360<td style="text-align:right">Standard cell potential</td> 361<td style="text-align:left">$E^{\circ}_{\mathrm{cell}} = E^{\circ}_{\mathrm{red}}(\text{cathode}) - E^{\circ}_{\mathrm{red}}(\text{anode})$</td> 362</tr> 363<tr> 364<td style="text-align:right">Change in Gibbs free energy at standard conditions and standard cell potential</td> 365<td style="text-align:left">$\Delta G^{\circ} = -nFE^{\circ}_{\text{cell}}$</td> 366</tr> 367<tr> 368<td style="text-align:right">Change in Gibbs free energy at standard conditions and equilibrium constant</td> 369<td style="text-align:left">$\Delta G^{\circ} = -RT \ln K$</td> 370</tr> 371<tr> 372<td style="text-align:right">Cell potential at standard conditions</td> 373<td style="text-align:left">$E^{\circ}_{\text{cell}} = \dfrac{RT}{nF} \ln K = \dfrac{0.0257\mathrm{V}}{n} \ln K$</td> 374</tr> 375<tr> 376<td style="text-align:right">Change in Gibbs free energy at nonstandard conditions</td> 377<td style="text-align:left">$\Delta G = \Delta G^{\circ} + RT \ln Q$</td> 378</tr> 379</tbody> 380</table> 381<h3 id="concentration-effect-and-nerst-equation">Concentration effect and Nerst Equation</h3> 382<table> 383<thead> 384<tr> 385<th style="text-align:right">Description</th> 386<th style="text-align:left">Equations</th> 387</tr> 388</thead> 389<tbody> 390<tr> 391<td style="text-align:right"><strong>Nerst Equation</strong> <br/> cell potential at nonstandard conditions</td> 392<td style="text-align:left">$E = E^{\circ} - \dfrac{RT}{nF} \ln Q$ <br/><br/> $E = E^{\circ} - \dfrac{0.0592 \mathrm{V}}{n} \log Q \ (\mathrm{at \ 25^{\circ}C})$</td> 393</tr> 394<tr> 395<td style="text-align:right">Measuring equilibrium constant from standard cell potential</td> 396<td style="text-align:left">$\ln K = \dfrac{nF}{RT}E^{\circ}_{\text{cell}} \ (\mathrm{at \ 25^{\circ}C})$ <br/> $\log K = \dfrac{n}{\mathrm{0.0592 V}} E^{\circ}_{\text{cell}} \ (\mathrm{at \ 25^{\circ}C})$</td> 397</tr> 398</tbody> 399</table> 400<h2 id="kinetics">Kinetics</h2> 401<h3 id="rate-laws">Rate laws</h3> 402<table> 403<thead> 404<tr> 405<th style="text-align:right">Description</th> 406<th style="text-align:left">Equations</th> 407</tr> 408</thead> 409<tbody> 410<tr> 411<td style="text-align:right">Rate of reaction</td> 412<td style="text-align:left">$\color{blue}\ce{aA + b B -> cC + dD}$ <br/> $\text{rate} = -\dfrac{1}{a}\dfrac{d\ce{[A]}}{dt} = -\dfrac{1}{b}\dfrac{d\ce{[B]}}{dt}$ <br/><br/> $\text{rate} = \dfrac{1}{c}\dfrac{d\ce{[C]}}{dt} = \dfrac{1}{d}\dfrac{d\ce{[D]}}{dt}$</td> 413</tr> 414<tr> 415<td style="text-align:right"><strong>First order rate law</strong> <br/> Negative line in ln[A] vs t graph.</td> 416<td style="text-align:left">$\text{rate} = -\dfrac{1}{a}\dfrac{d\ce{[A]}}{dt} = k\ce{[A]}$ <br/> $\ln\ce{[A]}_{t} = -akt + \ln\ce{[A]}_{0}$ <br/> $\ce{[A]}_{t} = \ce{[A]}_{0} e^{-akt}$</td> 417</tr> 418<tr> 419<td style="text-align:right"><strong>Second order rate law</strong> <br/> Positive line in 1/[A] vs t graph.</td> 420<td style="text-align:left">$\text{rate} = -\dfrac{1}{a}\dfrac{d\ce{[A]}}{dt} = k\ce{[A]}^{2}$ <br/><br/> $\dfrac{1}{\ce{[A]}_{t}} = \dfrac{1}{\ce{[A]}_{0}} + akt$</td> 421</tr> 422<tr> 423<td style="text-align:right"><strong>Zeroth order rate law</strong> <br/> Negative line in [A] vs t graph.</td> 424<td style="text-align:left">$\text{rate} = -\dfrac{1}{a}\dfrac{d\ce{[A]}}{dt} = k$ <br/> $\ce{[A]}_{t} = -akt + \ce{[A]}_{0}$</td> 425</tr> 426</tbody> 427</table> 428<h3 id="kinetics-and-chemical-equilibrium">Kinetics and chemical equilibrium</h3> 429<table> 430<thead> 431<tr> 432<th style="text-align:right">Description</th> 433<th style="text-align:left">Equations</th> 434</tr> 435</thead> 436<tbody> 437<tr> 438<td style="text-align:right"><strong>Principle of detailed balance</strong> <br/> equilibrium rate of elementary reaction is balanced by (equal to) the rate of reverse reaction</td> 439<td style="text-align:left">$\ce{aA + bB <=>[\mathit{k_{1}}][\mathit{k_{-1}}] cC + dD}$ <br/> $k_{1} \ce{[A]^{a}[B]^{b}} = k_{-1} \ce{[C]^{c}[D]^{d}}$ <br/> $K_{1} = \dfrac{\ce{[C]^{c}[D]^{d}}}{\ce{[A]^{a}[B]^{b}}}$ <br/> $\boxed{K_{1} = \dfrac{k_{1}}{k_{-1}}}$</td> 440</tr> 441<tr> 442<td style="text-align:right">Overall relationship between equilibrium constant and rate constant</td> 443<td style="text-align:left">$K = \prod\limits_{i} K_{i} = \dfrac{\prod\limits_{i}k_{i}}{\prod\limits_{i}k_{-i}}$</td> 444</tr> 445</tbody> 446</table> 447<h3 id="steady-state-approximation">Steady-state approximation</h3> 448<table> 449<thead> 450<tr> 451<th style="text-align:right">Description</th> 452<th style="text-align:left">Equations</th> 453</tr> 454</thead> 455<tbody> 456<tr> 457<td style="text-align:right"><strong>Steady-state approximation</strong> <br/> no single step in the reaction is much slower than the others; assume concentration of intermediates remain constant throughout reaction (B is neighboring molecule) <br/> (<em>Specific reaction may differ</em>)</td> 458<td style="text-align:left">$\color{blue} \ce{A -> C} \ (\text{overall}) \newline \ce{A + B <->[\mathit{k_{1}}][\mathit{k_{-1}}] I + B} \newline \ce{I ->[\mathit{k_{2}}] C}$ <br/> $\dfrac{d\ce{[I]}}{dt} = 0 = k_{1}\ce{[A][B]} - k_{-1}\ce{[I][B]} - k_{2} \ce{[I]}$ <br/> $[I] = \dfrac{k_{1}\ce{[A][B]}}{k_{-1}\ce{[B]} + k_{2}}$ <br/> $\mathrm{rate} = \dfrac{d\ce{[C]}}{dt} = k_{2}\ce{[I]} = \dfrac{k_{1}k_{2}\ce{[A][B]}}{k_{-1}\ce{[B]} + k_{2}}$</td> 459</tr> 460<tr> 461<td style="text-align:right"><strong>Enzyme Kinetics (Michaelis-Menten)</strong> <br/> follows analysis from steady-state approximation: intermediate $\ce{[ES]}$ remains constant; total enzyme $\ce{[E_{T}]}$ remains constant</td> 462<td style="text-align:left">$\color{blue} \ce{E + S -> P} \ (\text{overall}) \newline \ce{E + S <=>[\mathit{k_{1}}][\mathit{k_{-1}}] ES} \newline \ce{ES ->[\mathit{k_{2}}] E + P}$ <br/> $\ce{[E_{T}] \equiv [E] + [ES]} \implies \ce{[E] = [E_{T}] - [ES]}$ <br/><br/> $\dfrac{d\ce{[ES]}}{dt} = 0 = k_{1} \ce{[E][S]} - k_{-1}\ce{[ES]} - k_{2}\ce{[ES]}$ <br/><br/> $0 = k_{1}\ce{[E_{T}][S]} - k_{1}\ce{[ES][S]} - k_{-1}\ce{[ES]} - k_{2}\ce{[ES]}$ <br/><br/> $\ce{[ES]} = \dfrac{k_{1}\ce{[E_{T}][S]}}{k_{1}\ce{[S]} + k_{-1} + k_{2}} = \dfrac{\ce{[E_{T}][S]}}{\ce{[S]} + K_{m}}$</td> 463</tr> 464<tr> 465<td style="text-align:right"><strong>Michaelis-Menten equation</strong> <br/> rate of enzyme catalysis</td> 466<td style="text-align:left">$\mathrm{rate} = \dfrac{d\ce{[P]}}{dt} = k_{2}\ce{[ES]} = \dfrac{k_{2}\ce{[E_{T}][S]}}{\ce{[S]} + K_{m}}$</td> 467</tr> 468<tr> 469<td style="text-align:right">Michaelis-Menten constant</td> 470<td style="text-align:left">$K_{m} = \dfrac{k_{-1}+k_{2}}{k_{1}}$</td> 471</tr> 472<tr> 473<td style="text-align:right">Maximum rate of enzyme catalysis <br/> $(\ce{[S]} \gg K_{m})$</td> 474<td style="text-align:left">$\dfrac{d\ce{[P]}}{dt} = \dfrac{k_{2}\ce{[E_{T}][S]}}{\ce{[S]} + K_{m}} = \dfrac{V_{\text{max}}\ce{[S]}}{\ce{[S]} + K_{m}}$ <br/> $V_{\text{max}} = k_{2}\ce{[E_{T}]}$</td> 475</tr> 476<tr> 477<td style="text-align:right">Experimental determination of Michaelis-Menten constant</td> 478<td style="text-align:left">$K_{m} = \ce{[S]} \left( \left( \dfrac{V_{\text{max}}}{dP/dt} \right) - 1 \right)$</td> 479</tr> 480<tr> 481<td style="text-align:right">Observation from dP/dt vs. [S] graph</td> 482<td style="text-align:left">When $\ce{[S]} = K_{m}$, $\dfrac{d\ce{[P]}}{dt} = \dfrac{1}{2}V_{\text{max}}$ <br/><br/> When $\ce{[S]} \rightarrow \infty$, $\dfrac{d\ce{[P]}}{dt} \rightarrow V_{\text{max}}$</td> 483</tr> 484<tr> 485<td style="text-align:right">Turnover number of enzyme <br/> (when saturated, $\ce{[E_{T}] = [ES]}$)</td> 486<td style="text-align:left">$k_{\text{cat}} \equiv k_{2} = \dfrac{V_{\text{max}}}{\ce{[E_{T}]}}$</td> 487</tr> 488<tr> 489<td style="text-align:right">Linearization of Michaelis-Menten equation</td> 490<td style="text-align:left">$\dfrac{1}{dP/dt}
490 = \left( \dfrac{K_{m}}{V_{\text{max}}} \right) \dfrac{1}{\ce{[S]}} + \dfrac{1}{V_{\text{max}}}$</td> 491</tr> 492</tbody> 493</table> 494<h3 id="effect-of-temperature-on-reaction-rates">Effect of temperature on reaction rates</h3> 495<table> 496<thead> 497<tr> 498<th style="text-align:right">Description</th> 499<th style="text-align:left">Equations</th> 500</tr> 501</thead> 502<tbody> 503<tr> 504<td style="text-align:right"><strong>Arrhenius equation</strong> <br/> temperature dependence of reaction rate</td> 505<td style="text-align:left">$k = Ae^{-E_{a}/RT}$</td> 506</tr> 507<tr> 508<td style="text-align:right"><strong>Linearization of Arrhenius equation</strong> <br/> experimental determination of activation energy</td> 509<td style="text-align:left">$\ln k = \ln A - \dfrac{E_{a}}{RT}$ <br/> $\ln\dfrac{k_{2}}{k_{1}} = -\dfrac{E_{a}}{R} \left( \dfrac{1}{T_{2}} - \dfrac{1}{T_{1}} \right)$</td> 510</tr> 511</tbody> 512</table> 513<h2 id="nuclear-chemistry">Nuclear Chemistry</h2> 514<h3 id="baryons-and-leptons">Baryons and leptons</h3> 515<p>Baryon number is conserved.</p> 516<table> 517<thead> 518<tr> 519<th style="text-align:right">Types of Baryon</th> 520<th style="text-align:left">Symbol</th> 521<th style="text-align:left">Baryon Number</th> 522<th style="text-align:left">Charge</th> 523</tr> 524</thead> 525<tbody> 526<tr> 527<td style="text-align:right">Proton</td> 528<td style="text-align:left">$\ce{p+}$</td> 529<td style="text-align:left">+1</td> 530<td style="text-align:left">+1</td> 531</tr> 532<tr> 533<td style="text-align:right">Antiproton</td> 534<td style="text-align:left">$\bar{\ce{p}}$</td> 535<td style="text-align:left">-1</td> 536<td style="text-align:left">-1</td> 537</tr> 538<tr> 539<td style="text-align:right">Neutron</td> 540<td style="text-align:left">$\ce{n}$</td> 541<td style="text-align:left">+1</td> 542<td style="text-align:left">0</td> 543</tr> 544<tr> 545<td style="text-align:right">Antineutron</td> 546<td style="text-align:left">$\bar{\ce{n}}$</td> 547<td style="text-align:left">-1</td> 548<td style="text-align:left">0</td> 549</tr> 550</tbody> 551</table> 552<p>Lepton number is conserved.</p> 553<table> 554<thead> 555<tr> 556<th style="text-align:right">Types of Lepton</th> 557<th style="text-align:left">Symbol</th> 558<th style="text-align:left">Lepton Number</th> 559<th style="text-align:left">Charge</th> 560</tr> 561</thead> 562<tbody> 563<tr> 564<td style="text-align:right">Electron</td> 565<td style="text-align:left">$\ce{e-}$, $\beta^{-}$</td> 566<td style="text-align:left">+1</td> 567<td style="text-align:left">-1</td> 568</tr> 569<tr> 570<td style="text-align:right">Positron</td> 571<td style="text-align:left">$\ce{e+}$, $\beta^{+}$</td> 572<td style="text-align:left">-1</td> 573<td style="text-align:left">+1</td> 574</tr> 575<tr> 576<td style="text-align:right">Neutrino</td> 577<td style="text-align:left">$\nu_{e}$</td> 578<td style="text-align:left">+1</td> 579<td style="text-align:left">0</td> 580</tr> 581<tr> 582<td style="text-align:right">Antineutrino</td> 583<td style="text-align:left">$\bar{\nu}_{e}$</td> 584<td style="text-align:left">-1</td> 585<td style="text-align:left">0</td> 586</tr> 587</tbody> 588</table> 589<h3 id="nuclear-decay-process">Nuclear decay process</h3> 590<table> 591<thead> 592<tr> 593<th>Decay Type</th> 594<th>Emitted Particle</th> 595<th style="text-align:center">$\Delta Z$ Atomic Number</th> 596<th style="text-align:center">$\Delta N$ Neutron Number</th> 597<th style="text-align:center">$\Delta A$ Mass Number</th> 598<th style="text-align:center">Example</th> 599</tr> 600</thead> 601<tbody> 602<tr> 603<td>$\alpha$ decay</td> 604<td>$\ce{^4_2He}$</td> 605<td style="text-align:center">-2</td> 606<td style="text-align:center">-2</td> 607<td style="text-align:center">-4</td> 608<td style="text-align:center">$\ce{^238U -> ^234Th + ^4_2He}$</td> 609</tr> 610<tr> 611<td>$\beta^{-}$ decay</td> 612<td>energetic $\ce{e-}, \bar{\nu}_{e}$</td> 613<td style="text-align:center">+1</td> 614<td style="text-align:center">-1</td> 615<td style="text-align:center">0</td> 616<td style="text-align:center">$\ce{^14C -> ^14N + \beta-} + \bar{\nu}_{e}$</td> 617</tr> 618<tr> 619<td>$\beta^{+}$ emission</td> 620<td>energetic $\ce{e+}, \nu_{e}$</td> 621<td style="text-align:center">-1</td> 622<td style="text-align:center">+1</td> 623<td style="text-align:center">0</td> 624<td style="text-align:center">$\ce{^22Na -> ^22Ne + \beta+ + \nu_{e}}$</td> 625</tr> 626<tr> 627<td>Electron capture</td> 628<td>$\nu_{e}$</td> 629<td style="text-align:center">-1</td> 630<td style="text-align:center">+1</td> 631<td style="text-align:center">0</td> 632<td style="text-align:center">$\ce{^207Bi + e- -> ^207Pb + \nu_{e}}$</td> 633</tr> 634<tr> 635<td>$\gamma$-ray emission</td> 636<td>photon $h\nu$</td> 637<td style="text-align:center">0</td> 638<td style="text-align:center">0</td> 639<td style="text-align:center">0</td> 640<td style="text-align:center">$\ce{^60Ni^* -> ^60Ni + \gamma}$</td> 641</tr> 642<tr> 643<td>
643Internal conversion</td> 644<td>$\ce{e-}$</td> 645<td style="text-align:center">0</td> 646<td style="text-align:center">0</td> 647<td style="text-align:center">0</td> 648<td style="text-align:center">$\ce{^125Sb^m -> ^125Sb + e-}$</td> 649</tr> 650</tbody> 651</table> 652<table> 653<thead> 654<tr> 655<th style="text-align:right">Description</th> 656<th style="text-align:left">Equations</th> 657</tr> 658</thead> 659<tbody> 660<tr> 661<td style="text-align:right">Proton-neutron conversion</td> 662<td style="text-align:left">$\ce{^1_1p+ -> ^1_0n + ^0_1e+ + \nu_e}$ <br/> $\ce{^1_0n -> ^1_1p+ + ^0_-1e- + \bar{\nu_{e}}}$</td> 663</tr> 664</tbody> 665</table> 666<h3 id="mass-energy-relationship">Mass-energy relationship</h3> 667<table> 668<thead> 669<tr> 670<th style="text-align:right">Description</th> 671<th style="text-align:left">Equations</th> 672</tr> 673</thead> 674<tbody> 675<tr> 676<td style="text-align:right">Mass-energy equivalence</td> 677<td style="text-align:left">$E^{2} = m_{0}^{2}c^{4} + p^{2}c^{2}$ <br/> $\Delta E = c^{2} \Delta m \ \ (p = 0)$</td> 678</tr> 679<tr> 680<td style="text-align:right">Spontaneity of nuclear reactions</td> 681<td style="text-align:left">$\Delta E < 0 \implies \Delta m < 0$</td> 682</tr> 683<tr> 684<td style="text-align:right">Energy equivalent conversion</td> 685<td style="text-align:left">$\dfrac{1 \mathrm{u}}{931.494 \mathrm{MeV}} = 1$</td> 686</tr> 687</tbody> 688</table> 689<h3 id="kinetics-of-radioactive-decay">Kinetics of radioactive decay</h3> 690<table> 691<thead> 692<tr> 693<th style="text-align:right">Description</th> 694<th style="text-align:left">Equations</th> 695</tr> 696</thead> 697<tbody> 698<tr> 699<td style="text-align:right">Activity</td> 700<td style="text-align:left">$A = -\dfrac{dN}{dt} = kN$</td> 701</tr> 702<tr> 703<td style="text-align:right">Activity and number of nuclei over time</td> 704<td style="text-align:left">$A_{t} = A_{0}e^{-kt}$ <br/> $N_{t} = N_{0} e^{-kt}$</td> 705</tr> 706<tr> 707<td style="text-align:right">Decay constant and half life</td> 708<td style="text-align:left">$k = \dfrac{\ln 2}{t_{1/2}}$</td> 709</tr> 710</tbody> 711</table> 712<h2 id="introduction-to-quantum-mechanics">Introduction to Quantum Mechanics</h2> 713<h3 id="waves-and-energy-quantization">Waves and energy quantization</h3> 714<table> 715<thead> 716<tr> 717<th style="text-align:right">Description</th> 718<th style="text-align:left">Equations</th> 719</tr> 720</thead> 721<tbody> 722<tr> 723<td style="text-align:right">Wavelength and frequency of electromagnetic waves</td> 724<td style="text-align:left">$c = \lambda \nu$</td> 725</tr> 726<tr> 727<td style="text-align:right">Quantization of energy</td> 728<td style="text-align:left">$\varepsilon = nh\nu \newline n = 1, 2, 3, …$</td> 729</tr> 730<tr> 731<td style="text-align:right">Atomic spectra of H atom</td> 732<td style="text-align:left">$\nu = \left( \dfrac{1}{4} - \dfrac{1}{n^{2}} \right) \times 3.29 \times 10^{15} \ \mathrm{s^{-1}} \newline n = 3,4,5,…$</td> 733</tr> 734<tr> 735<td style="text-align:right">Energy quantization of photon</td> 736<td style="text-align:left">$\Delta E = h \nu$</td> 737</tr> 738<tr> 739<td style="text-align:right">Frank-Hertz experiment <br/> verifies Bohr’s model</td> 740<td style="text-align:left">$\nu = \dfrac{\Delta E}{h} = \dfrac{eV_{\text{thr}}}{h}$</td> 741</tr> 742</tbody> 743</table> 744<h3 id="bohrs-model">Bohr’s model</h3> 745<table> 746<thead> 747<tr> 748<th style="text-align:right">Description</th> 749<th style="text-align:left">Equations</th> 750</tr> 751</thead> 752<tbody> 753<tr> 754<td style="text-align:right">Total mechanical energy of H atom</td> 755<td style="text-align:left">$E = \dfrac{1}{2}m_{e}v^{2} - \dfrac{Ze^{2}}{4\pi\epsilon_{0}r}$</td> 756</tr> 757<tr> 758<td style="text-align:right">Uniform circular motion of electron <br/> a classical description</td> 759<td style="text-align:left">$\dfrac{Ze^{2}}{4\pi\epsilon_{0}r} = m_{e} \dfrac{v^{2}}{r}$</td> 760</tr> 761<tr> 762<td style="text-align:right">Quantized angular momentum of electron</td> 763<td style="text-align:left">$L = m_{e}vr = n\dfrac{h}{2\pi} \newline n = 1,2,3,…$</td> 764</tr> 765<tr> 766<td style="text-align:right">Allowed radius of H atom</td> 767<td style="text-align:left">$r_{e} = \dfrac{\epsilon_{0}n^{2}h^{2}}{\pi Z e^{2}m_{e}} = \dfrac{n^{2}}{Z} a_{0}$</td> 768</tr> 769<tr> 770<td style="text-align:right">Allowed velocity of H atom</td> 771<td style="text-align:left">$v_{n} = \dfrac{nh}{n\pi m_{e}r_{n}} = \dfrac{Ze^{2}}{2\epsilon_{0}nh}$</td> 772</tr> 773<tr> 774<td style="text-align:right">Allowed energy of H atom</td> 775<td style="text-align:left">$\begin{aligned}E_{n} &= -\dfrac{Z^{2}e^{4}m_{e}}{8\epsilon_{0}^{2}n^{2}h^{2}} \cr &= -(2.18 \times 10^{-18}\mathrm{J}) \dfrac{Z^{2}}
775{n^{2}} \cr &= -(13.60 \mathrm{eV}) \dfrac{Z^{2}}{n^{2}}\end{aligned} \newline n = 1,2,3,…$</td> 776</tr> 777<tr> 778<td style="text-align:right">Emission atomic spectra of H atom</td> 779<td style="text-align:left">$\nu = (3.29 \times 10^{15} \mathrm{s^{-1}})Z^{2} \left( \dfrac{1}{n_{f}^{2}} - \dfrac{1}{n_{i}^{2}} \right) \newline n_{i} > n_{f} = 1,2,3,…$</td> 780</tr> 781<tr> 782<td style="text-align:right">Absorption atomic spectra of H atom</td> 783<td style="text-align:left">$\nu = (3.29 \times 10^{15} \mathrm{s^{-1}})Z^{2} \left( \dfrac{1}{n_{i}^{2}} - \dfrac{1}{n_{f}^{2}} \right) \newline n_{f} > n_{i} = 1,2,3,…$</td> 784</tr> 785</tbody> 786</table> 787<h3 id="wave-particle-duality">Wave-particle duality</h3> 788<table> 789<thead> 790<tr> 791<th style="text-align:right">Description</th> 792<th style="text-align:left">Equations</th> 793</tr> 794</thead> 795<tbody> 796<tr> 797<td style="text-align:right">Planck’s constant <br/> lazy physicist/chemist</td> 798<td style="text-align:left">$\hbar = \dfrac{h}{2\pi}$</td> 799</tr> 800<tr> 801<td style="text-align:right">Conservation of energy in photoelectric effect</td> 802<td style="text-align:left">$h\nu = E_{lost} + K + \Phi$</td> 803</tr> 804<tr> 805<td style="text-align:right">Work function of metal</td> 806<td style="text-align:left">$\Phi = h\nu_{0}$</td> 807</tr> 808<tr> 809<td style="text-align:right">Maximum kinetic energy of photoelectrons <br/> $(E_{lost} = 0)$</td> 810<td style="text-align:left">$E_{\text{max}} = K = \dfrac{1}{2}mv_{e}^{2} = \dfrac{p^{2}}{2m} = h\nu - \Phi$</td> 811</tr> 812<tr> 813<td style="text-align:right">de Broglie wavelength</td> 814<td style="text-align:left">$\lambda = \dfrac{h}{p} = \dfrac{h}{mv}$</td> 815</tr> 816<tr> 817<td style="text-align:right">Heisenberg uncertainty principle</td> 818<td style="text-align:left">$(\Delta x)(\Delta p) \ge \dfrac{h}{4\pi} = \dfrac{\hbar}{2}$</td> 819</tr> 820<tr> 821<td style="text-align:right">1D standing wave</td> 822<td style="text-align:left">$f(x, t) = A\sin(kt)\sin(\omega t)$</td> 823</tr> 824<tr> 825<td style="text-align:right">Wave traveling to the right</td> 826<td style="text-align:left">$f(x, t) = A\sin(\omega t - kx)$</td> 827</tr> 828</tbody> 829</table> 830<h3 id="the-schrodinger-equation">The Schrodinger equation</h3> 831<table> 832<thead> 833<tr> 834<th style="text-align:right">Description</th> 835<th style="text-align:left">Equations</th> 836</tr> 837</thead> 838<tbody> 839<tr> 840<td style="text-align:right">Time independent Schrodinger equation</td> 841<td style="text-align:left">$-\dfrac{h^{2}}{8\pi^{2}m} \dfrac{d^{2}\psi(x)}{dx^{2}} + V(x)\psi(x) = E\psi(x)$ <br/> $\hat{H}\Psi = E\Psi \newline \hat{H} = -\dfrac{h^{2}}{8\pi^{2}m} \dfrac{d^{2}}{dx^{2}} + V(x)$</td> 842</tr> 843<tr> 844<td style="text-align:right">Time dependent Schrodinger equation</td> 845<td style="text-align:left">$\hat{H}\Psi = i\hbar \dfrac{\partial\Psi}{\partial t}$</td> 846</tr> 847<tr> 848<td style="text-align:right">Normalization of wave function</td> 849<td style="text-align:left">$\displaystyle\int_{-\infty}^{\infty} \psi^{*} \psi \ dx = 1$</td> 850</tr> 851<tr> 852<td style="text-align:right">Boundary conditions of wave functions</td> 853<td style="text-align:left">$\lim\limits_{x \to \pm\infty} \psi(x) = 0$</td> 854</tr> 855</tbody> 856</table> 857<h3 id="particle-in-a-box">Particle in a box</h3> 858<table> 859<thead> 860<tr> 861<th style="text-align:right">Description</th> 862<th style="text-align:left">Equations</th> 863</tr> 864</thead> 865<tbody> 866<tr> 867<td style="text-align:right">Wave function of 1D particles in a box</td> 868<td style="text-align:left">$\psi_{n}(x) = \sqrt{\dfrac{2}{L}} \sin\left( \dfrac{n\pi x}{L} \right)n \newline n = 1,2,3,…$</td> 869</tr> 870<tr> 871<td style="text-align:right">Allowed energy of 1D particles in a box</td> 872<td style="text-align:left">$E_{n} = \dfrac{n^{2}h^{2}}{8mL^{2}} \newline n = 1,2,3,…$</td> 873</tr> 874<tr> 875<td style="text-align:right">Allowed energy of 3D particles in cubic boxes</td> 876<td style="text-align:left">$E_{n_{x}n_{y}n_{z}} = \dfrac{h^{2}}{8mL^{2}}[n_{x}^{2} + n_{y}^{2} + n_{z}^{2}] \newline n_{x} = 1,2,3,… \newline n_{y} = 1,2,3,… \newline n_{z} = 1,2,3,…$</td> 877</tr> 878<tr> 879<td style="text-align:right">Wave function of 2D particles in square boxes</td> 880<td style="text-align:left">$\begin{aligned}&\Psi_{n_{x}n_{y}}(x, y) \cr = &\psi_{n_{x}}(x) \psi_{n_{y}}(y) \cr = &\dfrac{2}{L} \sin \left( \dfrac{n_{x}\pi x}{L} \right) \sin \left( \dfrac{n_{y}\pi y}{L} \right)\end{aligned}$</td> 881</tr> 882<tr> 883<td style="text-align:right">Wave function of 3D particles in cubic boxes</td> 884<td style="text-align:left">$\begin{aligned}&\Psi_{n_{x}n_{y}n_{z}}(x, y, z) \cr = &\psi_{n_{x}}(x) \psi_{n_{y}}(y) \psi_{n_{z}}(z) \cr = &\left( \dfrac{2}{L} \right)^{3/2} \sin \left( \dfrac{n_{x}\pi x}{L} \right) \sin \left( \dfrac{n_{y}\pi y}{L} \right) \sin \left( \dfrac{n_{z}\pi z}{L} \right)\end{aligned}$</td> 885</tr> 886</tbody> 887</table> 888<h2 id="quantum-mechanics-and-atomic-structure">Quantum Mechanics and Atomic Structure</h2> 889<h3 id="the-hydrogen-atom">The hydrogen atom</h3> 890<table> 891<thead> 892<tr> 893<th style="text-align:right">Description</th> 894<th style="text-align:left">Equations</th> 895</tr> 896</thead> 897<tbody> 898<tr> 899<td style="text-align:right"><strong>Principle quantum number</strong> $n$ <br/> determines energy of the electron</td> 900<td style="text-align:left">$E_{n} = -\dfrac{Z^{2}e^{4}m_{e}}{8\epsilon_{0}^{2}n^{2}h^{2}} \newline = -(2.18 \times 10^{-18}\mathrm{J}) \dfrac{Z^{2}}
900{n^{2}} \newline = -(13.60 \mathrm{eV}) \dfrac{Z^{2}}{n^{2}} \newline n = 1,2,3,…$</td> 901</tr> 902<tr> 903<td style="text-align:right"><strong>Angular momentum quantum number</strong> $l$ <br/> determines angular momentum of electron</td> 904<td style="text-align:left">$L^{2} = l(l+1)\dfrac{h^{2}}{4\pi^{2}} \newline l = 0,1,…,n-1$</td> 905</tr> 906<tr> 907<td style="text-align:right"><strong>Magnetic quantum number</strong> $m$ <br/> determines z-component of angular momentum of electron</td> 908<td style="text-align:left">$L_{z} = m\dfrac{h}{2\pi} \newline m = -l, -l+1, …, 0, …, l-1, l$</td> 909</tr> 910<tr> 911<td style="text-align:right">Spin</td> 912<td style="text-align:left">$m_{s} = -\dfrac{1}{2}, \dfrac{1}{2}$</td> 913</tr> 914<tr> 915<td style="text-align:right">Wave function of electron in quantum state $(n, l, m)$ have radial part and angular part</td> 916<td style="text-align:left">$\psi_{nlm}(r, \theta, \phi) = R_{nl}(r) Y_{lm}(\theta, \phi)$</td> 917</tr> 918<tr> 919<td style="text-align:right">Wave function as probability density</td> 920<td style="text-align:left">$(\psi_{nlm})^{2} dV = [R_{nl}(r)]^{2} [Y_{lm}]^{2} dV \newline dV = r^{2}\sin\theta \ dr \ d\theta \ d\phi$</td> 921</tr> 922<tr> 923<td style="text-align:right">Radial probability density (s orbital)</td> 924<td style="text-align:left">$r^{2} [R_{nl}(r)]^{2} \ dr$</td> 925</tr> 926<tr> 927<td style="text-align:right">Average value of distance of electron from nucleus in an orbital</td> 928<td style="text-align:left">$\bar{r}_{nl} = \dfrac{n^{2}a_{0}}{Z} \left[ 1 + \dfrac{1}{2} \left[ 1 - \dfrac{l(l+1)}{n^{2}} \right] \right]$</td> 929</tr> 930</tbody> 931</table> 932<h3 id="hartree-orbital-model-for-many-electron-atoms">Hartree orbital model for many-electron atoms</h3> 933<table> 934<thead> 935<tr> 936<th style="text-align:right">Description</th> 937<th style="text-align:left">Equations</th> 938</tr> 939</thead> 940<tbody> 941<tr> 942<td style="text-align:right">Orbital approximation for atoms</td> 943<td style="text-align:left">$\psi_{\text{atom}} = \prod\limits_{i} \varphi_{i}(r_{i})$</td> 944</tr> 945<tr> 946<td style="text-align:right">Coulomb potential of electron moving in shell n and effective nuclear charge</td> 947<td style="text-align:left">$V_{n}^{\text{eff}}(r) \approx -\dfrac{Z_{\text{eff}}(n)e^{2}}{r}$</td> 948</tr> 949</tbody> 950</table> 951<h2 id="quantum-mechanics-and-molecular-structure">Quantum Mechanics and Molecular Structure</h2> 952<h3 id="exact-molecular-model-for-ceh2">Exact molecular model for $\ce{H2+}$</h3> 953<table> 954<thead> 955<tr> 956<th style="text-align:right">Description</th> 957<th style="text-align:left">Equations</th> 958</tr> 959</thead> 960<tbody> 961<tr> 962<td style="text-align:right"><strong>Born-Oppenheimer approximation</strong> <br/> light slow nuclei; heavy fast electron</td> 963<td style="text-align:left">$\psi \approx \psi_{e^{-}} + \psi_{\text{nuclei}}$ <br/> $E_{\text{total}} = E_{e^{-}} + E_{\text{nuclei}}$</td> 964</tr> 965</tbody> 966</table> 967<h3 id="molecular-orbital-mo-and-linear-combination-of-atomic-orbitals-lcao-approximation">Molecular Orbital (MO) and Linear Combination of Atomic Orbitals (LCAO) Approximation</h3> 968<table> 969<thead> 970<tr> 971<th style="text-align:right">Description</th> 972<th style="text-align:left">Equations</th> 973</tr> 974</thead> 975<tbody> 976<tr> 977<td style="text-align:right">MO-LCAO approximation for bonding orbital of $\ce{H2+}$</td> 978<td style="text-align:left">$1\sigma_{g} \approx \sigma_{g1s} = C_{g}[\varphi_{1s}^{A} + \varphi_{1s}^{B}]$</td> 979</tr> 980<tr> 981<td style="text-align:right">MO-LCAO approximation for antibonding orbital of $\ce{H2+}$</td> 982<td style="text-align:left">$1\sigma_{u}^{*} \approx \sigma_{u1s}^{*} = C_{u}[\varphi_{1s}^{A} - \varphi_{1s}^{B}]$</td> 983</tr> 984<tr> 985<td style="text-align:right">Bond order</td> 986<td style="text-align:left">$\mathrm{B.O.} = \frac{1}{2} (\text{bonding } e^{-} - \text{antibonding } e^{-})$</td> 987</tr> 988</tbody> 989</table> 990<h2 id="spectroscopy">Spectroscopy</h2> 991<h3 id="electronic-spectroscopy">Electronic spectroscopy</h3> 992<p>Ultraviolet-Visible (UV-Vis) Spectroscopy</p> 993<table> 994<thead> 995<tr> 996<th style="text-align:right">Description</th> 997<th style="text-align:left">Equations</th> 998</tr> 999</thead> 1000<tbody> 1001<tr> 1002<td style="text-align:right">Transmittance <br/> ($I_{0}$ is incident light; $I$ is transmitted light)</td> 1003<td style="text-align:left">$T = \dfrac{I}{I_{0}}$</td> 1004</tr> 1005<tr> 1006<td style="text-align:right">Absorbance</td> 1007<td style="text-align:left">$A = \log \dfrac{I_{0}}{I} \newline A = -\log T$</td> 1008</tr> 1009<tr> 1010<td style="text-align:right"><strong>Beer’s Law</strong> <br/> absorbance depends on molar absorption coefficient, concentration, and path legth</td> 1011<td style="text-align:left">$I = I_{0}10^{-\varepsilon c l}$ <br/> $A = \varepsilon c l$</td> 1012</tr> 1013</tbody> 1014</table> 1015<h3 id="vibrational-spectroscopy---harmonic-oscillator-model">Vibrational spectroscopy - harmonic oscillator model</h3> 1016<p>Infrared (IR) Spectroscopy</p> 1017<table> 1018<thead> 1019<tr> 1020<th style="text-align:right">Description</th> 1021<th style="text-align:left">Equations</th> 1022</tr> 1023</thead> 1024<tbody> 1025<tr> 1026<td style="text-align:right">Reduced mass of system</td> 1027<td style="text-align:left">$\mu = \dfrac{m_{1}m_{2}}{m_{1}+m_{2}}
1027$</td> 1028</tr> 1029<tr> 1030<td style="text-align:right">Frequency</td> 1031<td style="text-align:left">$\omega = \sqrt{\dfrac{k}{\mu}}$</td> 1032</tr> 1033<tr> 1034<td style="text-align:right">Characteristic frequency</td> 1035<td style="text-align:left">$\nu = \dfrac{\omega}{2\pi} = \dfrac{1}{2\pi} \sqrt{\dfrac{k}{\mu}}$</td> 1036</tr> 1037<tr> 1038<td style="text-align:right">Vibrational energy</td> 1039<td style="text-align:left">$E_{n} = (n + \dfrac{1}{2})\hbar\omega = (n + \dfrac{1}{2})h\nu \newline n = 1,2,3,…$</td> 1040</tr> 1041</tbody> 1042</table> 1043</div><div class="post-footer" id="post-footer"> 1044 <div class="post-info"> 1045 <div class="post-info-line"> 1046 <div class="post-info-mod"> 1047 <span>Updated on 2021-08-17</span> 1048 </div> 1049 <div class="post-info-license"></div> 1050 </div> 1051 <div class="post-info-line"> 1052 <div class="post-info-md"></div> 1053 <div class="post-info-share"> 1054 <span></span> 1055 </div> 1056 </div> 1057 </div> 1058 1059 <div class="post-info-more"> 1060 <section class="post-tags"></section> 1061 <section> 1062 <span><a href="javascript:void(0);" onclick="window.history.back();">Back</a></span> | <span><a href="/">Home</a></span> 1063 </section> 1064 </div> 1065 1066 <div class="post-nav"><a href="/chem/chem238/" class="prev" rel="prev" title="CHEM 238 Organic Chemistry"><i class="fas fa-angle-left fa-fw"></i>CHEM 238 Organic Chemistry</a> 1067 <a href="/chem/chem145/" class="next" rel="next" title="CHEM 145 Honors General Chemistry I">CHEM 145 Honors General Chemistry I<i class="fas fa-angle-right fa-fw"></i></a></div> 1068</div> 1069</article> 1070 1071 1072
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Line numbers count LF bytes from the start of the resource, as the search results do. Vendor segments are library code the classifier recognised; they are stored but not indexed. Bytes are shown as Latin1 characters, one per byte.