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12<meta property="og:description" content="Acid-Base Equilibria Fundamentals    Description Equations     Autoionization of water $\color{blue} \ce{2H2O &lt;=&gt; H3O&#43; &#43; OH-}$  $K_{w} = \ce{[H3O&#43;][OH-]} = 10^{-14}$   pH function $\ce{pH} = -\log \ce{[H3O&#43;]}$   Acid dissociation constant $\color{blue} \ce{HA &#43; H2O &lt;=&gt; H3O&#43; &#43; A-}$  $K_{a} = \dfrac{\ce{[H3O&#43;][A-]}}{\ce{HA}}$   Base dissociation constant $\color{blue} \ce{B &#43; H2O &lt;=&gt; HB&#43; &#43; OH-}$  $K_{b} = \dfrac{\ce{[HB&#43;][OH-]}}{\ce{[B]}}$   Relationship between dissociation constants $K_{w} = K_{a}K_{b}$  $\ce{p}K_{a} &#43; \ce{p}K_{b} = \ce{p}K_{w} = 14$   Indicators $\color{blue} \ce{HIn &#43; H2O &lt;=&gt; H3O&#43; &#43; In-}$  $\dfrac{\ce{[H3O&#43;]}}{K_{a}} = \dfrac{\ce{[HIn]}}{\ce{[In^{-}]}}$    Buffer and titration    Description Equations     Henderson–Hasselbalch equation  pH of a buffer $\ce{pH} = \ce{p}K_{a} &#43; \log{\dfrac{\ce{[A^{-}]_{0}}}{\ce{[HA]_{0}}}}$   Titration curve at half equivalence point $\ce{pH} = \ce{p}K_{a}$ (where $\ce{[HA] = [A-]}$)   Titration curve until equivalence point  ($0 &lt; V &lt; V_{e}$) (1) Stoichiometric calculation of neutralization gives new acid/base concentration." />
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20<meta name="twitter:description" content="Acid-Base Equilibria Fundamentals    Description Equations     Autoionization of water $\color{blue} \ce{2H2O &lt;=&gt; H3O&#43; &#43; OH-}$  $K_{w} = \ce{[H3O&#43;][OH-]} = 10^{-14}$   pH function $\ce{pH} = -\log \ce{[H3O&#43;]}$   Acid dissociation constant $\color{blue} \ce{HA &#43; H2O &lt;=&gt; H3O&#43; &#43; A-}$  $K_{a} = \dfrac{\ce{[H3O&#43;][A-]}}{\ce{HA}}$   Base dissociation constant $\color{blue} \ce{B &#43; H2O &lt;=&gt; HB&#43; &#43; OH-}$  $K_{b} = \dfrac{\ce{[HB&#43;][OH-]}}{\ce{[B]}}$   Relationship between dissociation constants $K_{w} = K_{a}K_{b}$  $\ce{p}K_{a} &#43; \ce{p}K_{b} = \ce{p}K_{w} = 14$   Indicators $\color{blue} \ce{HIn &#43; H2O &lt;=&gt; H3O&#43; &#43; In-}$  $\dfrac{\ce{[H3O&#43;]}}{K_{a}} = \dfrac{\ce{[HIn]}}{\ce{[In^{-}]}}$    Buffer and titration    Description Equations     Henderson–Hasselbalch equation  pH of a buffer $\ce{pH} = \ce{p}K_{a} &#43; \log{\dfrac{\ce{[A^{-}]_{0}}}{\ce{[HA]_{0}}}}$   Titration curve at half equivalence point $\ce{pH} = \ce{p}K_{a}$ (where $\ce{[HA] = [A-]}$)   Titration curve until equivalence point  ($0 &lt; V &lt; V_{e}$) (1) Stoichiometric calculation of neutralization gives new acid/base concentration."/>
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84        <h1 class="single-title posts-title">CHEM 155 Honors General Chemistry II</h1>
85
86        <div class="post-meta">
87            <div class="post-meta-line"><i class="far fa-calendar-alt fa-fw"></i>&nbsp;<time datetime="2021-08-17">2021-08-17</time>&nbsp;</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&rsquo;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">$\color{blue} \ce{2H2O &lt;=&gt; H3O+ + OH-}$ <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">$\color{blue} \ce{HA + H2O &lt;=&gt; H3O+ + A-}$ <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">$\color{blue} \ce{B + H2O &lt;=&gt; HB+ + OH-}$ <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">$\color{blue} \ce{HIn + H2O &lt;=&gt; H3O+ + In-}$ <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>Henderson–Hasselbalch 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/> ($0 &lt; V &lt; V_{e}$)</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/> $V &gt; V_{e}$</td>
221<td style="text-align:left">Calculate pH using excess base that wasn&rsquo;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">$\color{blue} \ce{H2A + H2O &lt;=&gt; HA- + H3O+} (K_{a1}) \newline \ce{HA- + H2O &lt;=&gt; A^2- + H3O+} (K_{a2}) \newline \ce{A^2- + H2O &lt;=&gt; HA- + OH-} (K_{b1}) \newline \ce{HA- + H2O &lt;=&gt; H2A + OH-} (K_{b2})$</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">$\color{blue} \ce{M_aX_b &lt;=&gt; aM^b+ + bX^a-}$ <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">$\color{blue} \ce{M^a+ + X &lt;=&gt; MX^a+} (K_{1}) \newline \ce{MX^a+ + X &lt;=&gt; MX_2^a+} (K_{2}) \newline \ce{M^a+ + 2X &lt;=&gt; MX_2^a+ (K_{f})}$ <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">$\color{blue} \ce{M_aX_b &lt;=&gt; aM^b+ + bX^a-}$ <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">$\color{blue} \ce{H2S + H2O &lt;=&gt; H3O+ + HS-} \newline \ce{MS + H2O &lt;=&gt; M^2+ + HS- + OH-}$</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">$\color{blue} \ce{2H3O+ + 2e- -&gt; H2 + 2H2O}$</td>
339</tr>
340<tr>
341<td style="text-align:right">pH meter reaction at anode</td>
342<td style="text-align:left">$\color{blue} \ce{H2 + 2H2O -&gt; 2H3O+ + 2e-}$</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">$\color{blue}\ce{aA + b B -&gt; cC + dD}$ <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">$\ce{aA + bB &lt;=&gt;[\mathit{k_{1}}][\mathit{k_{-1}}] cC + dD}$ <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">$\color{blue} \ce{A -&gt; C} \ (\text{overall}) \newline \ce{A + B &lt;-&gt;[\mathit{k_{1}}][\mathit{k_{-1}}] I + B} \newline \ce{I -&gt;[\mathit{k_{2}}] C}$ <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">$\color{blue} \ce{E + S -&gt; P} \ (\text{overall}) \newline \ce{E + S &lt;=&gt;[\mathit{k_{1}}][\mathit{k_{-1}}] ES} \newline \ce{ES -&gt;[\mathit{k_{2}}] E + P}$ <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">$\ce{^238U -&gt; ^234Th + ^4_2He}$</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">$\ce{^14C -&gt; ^14N + \beta-} + \bar{\nu}_{e}$</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">$\ce{^22Na -&gt; ^22Ne + \beta+ + \nu_{e}}$</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">$\ce{^207Bi + e- -&gt; ^207Pb + \nu_{e}}$</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">$\ce{^60Ni^* -&gt; ^60Ni + \gamma}$</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">$\ce{^125Sb^m -&gt; ^125Sb + e-}$</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">$\ce{^1_1p+ -&gt; ^1_0n + ^0_1e+ + \nu_e}$ <br/> $\ce{^1_0n -&gt; ^1_1p+ + ^0_-1e- + \bar{\nu_{e}}}$</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">$\Delta E &lt; 0 \implies \Delta m &lt; 0$</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">$\varepsilon = nh\nu \newline n = 1, 2, 3, &hellip;$</td>
729</tr>
730<tr>
731<td style="text-align:right">Atomic spectra of H atom</td>
732<td style="text-align:left">$\nu = \left( \dfrac{1}{4} - \dfrac{1}{n^{2}} \right) \times 3.29 \times 10^{15} \ \mathrm{s^{-1}} \newline n = 3,4,5,&hellip;$</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&rsquo;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&rsquo;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">$L = m_{e}vr = n\dfrac{h}{2\pi} \newline n = 1,2,3,&hellip;$</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">$\begin{aligned}E_{n} &amp;= -\dfrac{Z^{2}e^{4}m_{e}}{8\epsilon_{0}^{2}n^{2}h^{2}} \cr &amp;= -(2.18 \times 10^{-18}\mathrm{J}) \dfrac{Z^{2}}
775{n^{2}} \cr &amp;= -(13.60 \mathrm{eV}) \dfrac{Z^{2}}{n^{2}}\end{aligned} \newline n = 1,2,3,&hellip;$</td>
776</tr>
777<tr>
778<td style="text-align:right">Emission atomic spectra of H atom</td>
779<td style="text-align:left">$\nu = (3.29 \times 10^{15} \mathrm{s^{-1}})Z^{2} \left( \dfrac{1}{n_{f}^{2}} - \dfrac{1}{n_{i}^{2}} \right) \newline n_{i} &gt; n_{f} = 1,2,3,&hellip;$</td>
780</tr>
781<tr>
782<td style="text-align:right">Absorption atomic spectra of H atom</td>
783<td style="text-align:left">$\nu = (3.29 \times 10^{15} \mathrm{s^{-1}})Z^{2} \left( \dfrac{1}{n_{i}^{2}} - \dfrac{1}{n_{f}^{2}} \right) \newline n_{f} &gt; n_{i} = 1,2,3,&hellip;$</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&rsquo;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">$\psi_{n}(x) = \sqrt{\dfrac{2}{L}} \sin\left( \dfrac{n\pi x}{L} \right)n \newline n = 1,2,3,&hellip;$</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">$E_{n} = \dfrac{n^{2}h^{2}}{8mL^{2}} \newline n = 1,2,3,&hellip;$</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">$E_{n_{x}n_{y}n_{z}} = \dfrac{h^{2}}{8mL^{2}}[n_{x}^{2} + n_{y}^{2} + n_{z}^{2}] \newline n_{x} = 1,2,3,&hellip; \newline n_{y} = 1,2,3,&hellip; \newline n_{z} = 1,2,3,&hellip;$</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">$\begin{aligned}&amp;\Psi_{n_{x}n_{y}}(x, y) \cr = &amp;\psi_{n_{x}}(x) \psi_{n_{y}}(y) \cr = &amp;\dfrac{2}{L} \sin \left( \dfrac{n_{x}\pi x}{L} \right) \sin \left( \dfrac{n_{y}\pi y}{L} \right)\end{aligned}$</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">$\begin{aligned}&amp;\Psi_{n_{x}n_{y}n_{z}}(x, y, z) \cr = &amp;\psi_{n_{x}}(x) \psi_{n_{y}}(y) \psi_{n_{z}}(z) \cr = &amp;\left( \dfrac{2}{L} \right)^{3/2} \sin \left( \dfrac{n_{x}\pi x}{L} \right) \sin \left( \dfrac{n_{y}\pi y}{L} \right) \sin \left( \dfrac{n_{z}\pi z}{L} \right)\end{aligned}$</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">$E_{n} = -\dfrac{Z^{2}e^{4}m_{e}}{8\epsilon_{0}^{2}n^{2}h^{2}} \newline = -(2.18 \times 10^{-18}\mathrm{J}) \dfrac{Z^{2}}
900{n^{2}} \newline = -(13.60 \mathrm{eV}) \dfrac{Z^{2}}{n^{2}} \newline n = 1,2,3,&hellip;$</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">$L^{2} = l(l+1)\dfrac{h^{2}}{4\pi^{2}} \newline l = 0,1,&hellip;,n-1$</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">$L_{z} = m\dfrac{h}{2\pi} \newline m = -l, -l+1, &hellip;, 0, &hellip;, l-1, l$</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&rsquo;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">$E_{n} = (n + \dfrac{1}{2})\hbar\omega = (n + \dfrac{1}{2})h\nu \newline n = 1,2,3,&hellip;$</td>
1040</tr>
1041</tbody>
1042</table>
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