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 145 Honors General Chemistry I - Equation Sheets</title><meta name="Description" content=""><meta property="og:title" content="CHEM 145 Honors General Chemistry I" /> 12<meta property="og:description" content="Gas Kinetic Theory of Gas Quantity Unit Definition Pressure $\mathrm{N/m^{2}}$ $P = \dfrac{F}{A}$ Mole fraction of $a$ - $X_{a} = \dfrac{n_{a}}{n_{\mathrm{total}}}$ Partial pressure of $a$ $\mathrm{N/m^{2}}$ $P_{a} = X_{a}P_{\mathrm{total}}$ Pressure-volume work $\mathrm{J}$ $W = -P\Delta V$ Compressibility of gas - $z = \dfrac{PV}{nRT}$ Description Equations Ideal gas law $PV = nRT$ Ideal gas constant and Boltzmann constant $R = N_{A}k_{B}$ Average kinetic energy of one gas molecule $\varepsilon = \dfrac{3}{2}k_{B}T$ Root mean square speed gas $v_{\mathrm{RMS}} = \sqrt{\dfrac{3RT}{\mathcal{M}}} = \sqrt{\dfrac{3k_{B}T}{m}}$ Mean speed of gas $\bar{v} = \sqrt{\dfrac{8RT}{\pi\mathcal{M}}} = \sqrt{\dfrac{8k_{B}T}{\pi m}}$ Most probable speed of gas $v_{\mathrm{mp}} = \sqrt{\dfrac{2RT}{\mathcal{M}}} = \sqrt{\dfrac{2k_{B}T}{m}}$ Maxwell-Boltzmann speed distribution $f(v) = 4\pi \left(\dfrac{m}{2\pi k_{B}T}\right) v^{2} \exp\left(\dfrac{-mv^{2}}{2k_{B}T}\right)$ Van der Waals equation of state $(P + a\dfrac{n^{2}}{V^{2}})(V - bn) = nRT$ Lennard-Jones potential $V_{\mathrm{LJ}} = 4\varepsilon \left(\left(\dfrac{\sigma}{R}\right)^{12} - \left(\dfrac{\sigma}{R}\right)^{6} \right)$ Molecular Collisions and Rate Processes Description Equations Molecule collision rate with wall $Z_{w} \propto \dfrac{1}{4}\dfrac{N}{V}\bar{v}A = \dfrac{1}{4}\dfrac{N}{V}\sqrt{\dfrac{8RT}{\pi\mathcal{M}}}A$ Graham’s law of effusion $\dfrac{\text{rate of effusion of A}}{\text{rate of effusion of B}} = \dfrac{N_{\mathrm{A}}}{N_{\mathrm{B}}} \sqrt{\dfrac{\mathcal{M}_{\mathrm{B}}}{\mathcal{M}_{\mathrm{A}}}}$ Molecule-molecule collision rate $Z_{1} = 4\dfrac{N}{V}d^{2}\sqrt{\dfrac{\pi RT}{\mathcal{M}}}$ Mean free path $\lambda = \dfrac{\bar{v}}{Z_{1}} = \dfrac{V}{\sqrt{2}\pi d^{2}N}$ Mean square displacement of diffusion in 3D $\overline{\Delta r}^{2} = 6Dt$ Gas diffusion constant $D = \dfrac{3}{8} \sqrt{\dfrac{RT}{\pi\mathcal{M}}} \dfrac{V}{Nd^{2}}$ Intermolecular Interactions Quantity Unit Definition Electrostatic force $\mathrm{N}$ $F = \dfrac{q_{1}q_{2}}{4\pi\varepsilon r^{2}}$ Electrostatic potential energy $\mathrm{N}$ $V = \dfrac{q_{1}q_{2}}{4\pi\varepsilon r}$ Dipole moment $\mathrm{C\cdot m}$ $\mu = qd$ Polarizability $\mathrm{C\cdot m^{2}/V}$ $\alpha = \dfrac{\mu}{E}$ Induced dipole moment $\mathrm{C\cdot m}$ $\mu^{*} = \alpha E$ Description Equations Ion-ion interactions $E \propto \dfrac{q_{1}q_{2}}{r}$ Ion-dipole interactions $E \propto -\dfrac{q\mu}{r^{2}}$ Dipole-dipole interactions $E \propto -\dfrac{\mu_{1}\mu_{2}}{r^{3}}$ Induced-dipole-induced-dipole (London) $E \propto -\dfrac{\alpha_{1}\alpha_{2}}{r^{6}}$ Dipole-induced-dipole $E \propto -\dfrac{\mu_{1}^{2}\alpha_{2}}{r^{6}}$ Rotating fixed dipole (Keesom) $E \propto -\dfrac{\mu_{1}^{2}\mu_{2}^{2}}{r^{6}}$ Thermodynamics First law of thermodynamics Quantity Unit Definition Specific heat capacities $\mathrm{J \cdot kg^{-1} \cdot K^{-1}}$ $q = mc_{s}\Delta T = n\bar{c}\Delta T$ Heat capacity $\mathrm{J/K}$ $q = C\Delta T$ Molar heat capacities $\mathrm{J \cdot mol^{-1} \cdot K^{-1}}$ $q_{V} = nc_{V}\Delta T$ $q_{P} = nc_{P}\Delta T$ Enthalpy $\mathrm{J}$ $H = U +
12 PV$ Description Equations First law of thermodynamics $\Delta U = q + w$ $dU = \cancel{d}q + \cancel{d}w$ $\Delta U_{\mathrm{univ}} = \Delta U_{\mathrm{sys}} + \Delta U_{\mathrm{surr}} = 0$ Enthalpy change $q_{P} = \Delta (U + PV) = \Delta H$ Molar heat capacity of monoatomic ideal gas at constant volume $c_{V} = \dfrac{3}{2}R$ Molar heat capacity of any ideal gas at constant pressure $c_{P} = c_{V} + R = \dfrac{5}{2}R$ Internal energy change of any ideal gas $\Delta U = nc_{V}\Delta T$ Enthalpy change of any ideal gas $\begin{aligned}\Delta H &= nc_{P}\Delta T \cr &= \Delta U + \Delta(PV) \cr &= nc_{V}\Delta T + nR\Delta T\end{aligned}$ Hess’s law $\Delta H^{\circ} = \sum\limits_{i}^{\text{prod}} n_{i}\Delta H_{i}^{\circ} - \sum\limits_{j}^{\text{react}} n_{j}\Delta H_{j}^{\circ}$ Molality $b = \dfrac{n_{\mathrm{solute}}}{m_{\mathrm{solvent}}}$ Boiling point elevation ($i$ is vant’s Hoff dissociation factor) $\Delta T_{\mathrm{boil}} = ibK_{\mathrm{boil}}$ Freezing point depression ($i$ is vant’s Hoff dissociation factor) $\Delta T_{\mathrm{freeze}} = ibK_{\mathrm{freeze}}$ Reversible isothermal process of ideal gas $\Delta T = 0$ $\Delta U = 0$ $\Delta H = 0$ $w = -\displaystyle\int_{V_{1}}^{V_{2}} P \ dV = -nRT \ln\dfrac{V_{2}}{V_{1}}$ $q = -w$ Reversible adiabatic process of ideal gas $q = 0$ $\Delta U = nc_{V}\Delta T = w$ $\Delta H = nc_{P}\Delta T$ $\gamma = \dfrac{c_{P}}{c_{V}}$ $T_{1}V_{1}^{\gamma - 1} = T_{2}V_{2}^{\gamma - 1}$ $P_{1}V_{1}^{\gamma} = P_{2}V_{2}^{\gamma}$ Second law of thermodynamics Description Equations Entropy $S = k_{B} \ln\Omega$ Entropy change $\Delta S = \displaystyle\int_{i}^{f} \dfrac{dq_{\mathrm{rev}}}{T}$ $\Delta S_{\mathrm{sys}}$ for reversible isothermal process $\Delta S = \displaystyle\int_{i}^{f} \dfrac{dq_{\mathrm{rev}}}{T} = \dfrac{1}{T} \displaystyle\int_{i}^{f} dq_{\mathrm{rev}} = \dfrac{q_{\mathrm{rev}}}{T}$ $\Delta S_{\mathrm{sys}}$ for reversible isothermal process - compression/expansion of ideal gas $q_{\mathrm{rev}} = nRT \ln \left( \dfrac{V_{2}}{V_{1}} \right)$ $\Delta S = nR \ln \left( \dfrac{V_{2}}{V_{1}} \right)$ $\Delta S_{\mathrm{sys}}$ for reversible isothermal process - phase transitions $q_{\mathrm{rev}} = \Delta H_{\mathrm{fus}}$ $\Delta S_{\mathrm{fus}} = \dfrac{q_{\mathrm{rev}}}{T_{\mathrm{fus}}} = \dfrac{\Delta H_{\mathrm{fus}}}{T_{\mathrm{fus}}}$ $\Delta S_{\mathrm{sys}}$ for reversible adiabatic process $q = 0$ $\Delta S = 0$ $\Delta S_{\mathrm{sys}}$ for reversible isochoric process $\Delta V = 0$ $dq_{\mathrm{rev}} = nc_{V}dT$ $\Delta S = nc_{V} \displaystyle\int_{T_{1}}^{T_{2}} \dfrac{dT}{T} = nc_{V} \ln \left( \dfrac{T_{2}}{T_{1}} \right)$ $\Delta S_{\mathrm{sys}}$ for reversible isobaric process $\Delta P = 0$ $dq_{\mathrm{rev}} = nc_{P}dT$ $\Delta S = nc_{P} \displaystyle\int_{T_{1}}^{T_{2}} \dfrac{dT}{T} = nc_{P} \ln \left( \dfrac{T_{2}}{T_{1}} \right)$ Entropy change of surrounding $\Delta S_{\mathrm{surr}} = \dfrac{-\Delta H_{\mathrm{sys}}}{T_{\mathrm{surr}}}$ Second law of thermodynamics $\Delta S \geq \dfrac{q_{\mathrm{rev}}}{T}$ Enthalpy of spontaneous process $\Delta S_{\mathrm{total}} = \Delta S_{\mathrm{sys}} + \Delta S_{\mathrm{surr}} > 0$ Standard molar entropy $S^{\circ} = \displaystyle\int_{0K}^{298." /> 13<meta property="og:type" content="article" /> 14<meta property="og:url" content="https://equation-sheets.netlify.app/chem/chem145/" /><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 145 Honors General Chemistry I"/> 20<meta name="twitter:description" content="Gas Kinetic Theory of Gas Quantity Unit Definition Pressure $\mathrm{N/m^{2}}$ $P = \dfrac{F}{A}$ Mole fraction of $a$ - $X_{a} = \dfrac{n_{a}}{n_{\mathrm{total}}}$ Partial pressure of $a$ $\mathrm{N/m^{2}}$ $P_{a} = X_{a}P_{\mathrm{total}}$ Pressure-volume work $\mathrm{J}$ $W = -P\Delta V$ Compressibility of gas - $z = \dfrac{PV}{nRT}
20$ Description Equations Ideal gas law $PV = nRT$ Ideal gas constant and Boltzmann constant $R = N_{A}k_{B}$ Average kinetic energy of one gas molecule $\varepsilon = \dfrac{3}{2}k_{B}T$ Root mean square speed gas $v_{\mathrm{RMS}} = \sqrt{\dfrac{3RT}{\mathcal{M}}} = \sqrt{\dfrac{3k_{B}T}{m}}$ Mean speed of gas $\bar{v} = \sqrt{\dfrac{8RT}{\pi\mathcal{M}}} = \sqrt{\dfrac{8k_{B}T}{\pi m}}$ Most probable speed of gas $v_{\mathrm{mp}} = \sqrt{\dfrac{2RT}{\mathcal{M}}} = \sqrt{\dfrac{2k_{B}T}{m}}$ Maxwell-Boltzmann speed distribution $f(v) = 4\pi \left(\dfrac{m}{2\pi k_{B}T}\right) v^{2} \exp\left(\dfrac{-mv^{2}}{2k_{B}T}\right)$ Van der Waals equation of state $(P + a\dfrac{n^{2}}{V^{2}})(V - bn) = nRT$ Lennard-Jones potential $V_{\mathrm{LJ}} = 4\varepsilon \left(\left(\dfrac{\sigma}{R}\right)^{12} - \left(\dfrac{\sigma}{R}\right)^{6} \right)$ Molecular Collisions and Rate Processes Description Equations Molecule collision rate with wall $Z_{w} \propto \dfrac{1}{4}\dfrac{N}{V}\bar{v}A = \dfrac{1}{4}\dfrac{N}{V}\sqrt{\dfrac{8RT}{\pi\mathcal{M}}}A$ Graham’s law of effusion $\dfrac{\text{rate of effusion of A}}{\text{rate of effusion of B}} = \dfrac{N_{\mathrm{A}}}{N_{\mathrm{B}}} \sqrt{\dfrac{\mathcal{M}_{\mathrm{B}}}{\mathcal{M}_{\mathrm{A}}}}$ Molecule-molecule collision rate $Z_{1} = 4\dfrac{N}{V}d^{2}\sqrt{\dfrac{\pi RT}{\mathcal{M}}}$ Mean free path $\lambda = \dfrac{\bar{v}}{Z_{1}} = \dfrac{V}{\sqrt{2}\pi d^{2}N}$ Mean square displacement of diffusion in 3D $\overline{\Delta r}^{2} = 6Dt$ Gas diffusion constant $D = \dfrac{3}{8} \sqrt{\dfrac{RT}{\pi\mathcal{M}}} \dfrac{V}{Nd^{2}}$ Intermolecular Interactions Quantity Unit Definition Electrostatic force $\mathrm{N}$ $F = \dfrac{q_{1}q_{2}}{4\pi\varepsilon r^{2}}$ Electrostatic potential energy $\mathrm{N}$ $V = \dfrac{q_{1}q_{2}}{4\pi\varepsilon r}$ Dipole moment $\mathrm{C\cdot m}$ $\mu = qd$ Polarizability $\mathrm{C\cdot m^{2}/V}$ $\alpha = \dfrac{\mu}{E}$ Induced dipole moment $\mathrm{C\cdot m}$ $\mu^{*} = \alpha E$ Description Equations Ion-ion interactions $E \propto \dfrac{q_{1}q_{2}}{r}$ Ion-dipole interactions $E \propto -\dfrac{q\mu}{r^{2}}$ Dipole-dipole interactions $E \propto -\dfrac{\mu_{1}\mu_{2}}{r^{3}}$ Induced-dipole-induced-dipole (London) $E \propto -\dfrac{\alpha_{1}\alpha_{2}}{r^{6}}$ Dipole-induced-dipole $E \propto -\dfrac{\mu_{1}^{2}\alpha_{2}}{r^{6}}$ Rotating fixed dipole (Keesom) $E \propto -\dfrac{\mu_{1}^{2}\mu_{2}^{2}}{r^{6}}$ Thermodynamics First law of thermodynamics Quantity Unit Definition Specific heat capacities $\mathrm{J \cdot kg^{-1} \cdot K^{-1}}$ $q = mc_{s}\Delta T = n\bar{c}\Delta T$ Heat capacity $\mathrm{J/K}$ $q = C\Delta T$ Molar heat capacities $\mathrm{J \cdot mol^{-1} \cdot K^{-1}}$ $q_{V} = nc_{V}\Delta T$ $q_{P} = nc_{P}\Delta T$ Enthalpy $\mathrm{J}$ $H = U + PV$ Description Equations First law of thermodynamics $\Delta U = q + w$ $dU = \cancel{d}q + \cancel{d}w$ $\Delta U_{\mathrm{univ}} = \Delta U_{\mathrm{sys}} + \Delta U_{\mathrm{surr}} = 0$ Enthalpy change $q_{P} = \Delta (U + PV) = \Delta H$ Molar heat capacity of monoatomic ideal gas at constant volume $c_{V} = \dfrac{3}{2}R$ Molar heat capacity of any ideal gas at constant pressure $c_{P} = c_{V} + R = \dfrac{5}{2}R$ Internal energy change of any ideal gas $\Delta U = nc_{V}\Delta T$ Enthalpy change of any ideal gas $\begin{aligned}\Delta H &= nc_{P}\Delta T \cr &= \Delta U + \Delta(PV) \cr &= nc_{V}\Delta T + nR\Delta T\end{aligned}$ Hess’s law $\Delta H^{\circ} = \sum\limits_{i}^{\text{prod}} n_{i}\Delta H_{i}^{\circ} - \sum\limits_{j}^{\text{react}} n_{j}\Delta H_{j}^{\circ}$ Molality $b = \dfrac{n_{\mathrm{solute}}}{m_{\mathrm{solvent}}}$ Boiling point elevation ($i$ is vant’s Hoff dissociation factor) $\Delta T_{\mathrm{boil}} = ibK_{\mathrm{boil}}$ Freezing point depression ($i$ is vant’s Hoff dissociation factor) $\Delta T_{\mathrm{freeze}} = ibK_{\mathrm{freeze}}$ Reversible isothermal process of ideal gas $\Delta T = 0$ $\Delta U = 0$ $\Delta H = 0$ $w = -\displaystyle\int_{V_{1}}^{V_{2}} P \ dV = -nRT \ln\dfrac{V_{2}}{V_{1}}$ $q = -w$ Reversible adiabatic process of ideal gas $q = 0$ $\Delta U = nc_{V}\Delta T = w$ $\Delta H = nc_{P}\Delta T$ $\gamma = \dfrac{c_{P}}{c_{V}}$ $T_{1}V_{1}^{\gamma - 1} = T_{2}V_{2}^{\gamma - 1}$ $P_{1}V_{1}^{\gamma} = P_{2}V_{2}^{\gamma}
20$ Second law of thermodynamics Description Equations Entropy $S = k_{B} \ln\Omega$ Entropy change $\Delta S = \displaystyle\int_{i}^{f} \dfrac{dq_{\mathrm{rev}}}{T}$ $\Delta S_{\mathrm{sys}}$ for reversible isothermal process $\Delta S = \displaystyle\int_{i}^{f} \dfrac{dq_{\mathrm{rev}}}{T} = \dfrac{1}{T} \displaystyle\int_{i}^{f} dq_{\mathrm{rev}} = \dfrac{q_{\mathrm{rev}}}{T}$ $\Delta S_{\mathrm{sys}}$ for reversible isothermal process - compression/expansion of ideal gas $q_{\mathrm{rev}} = nRT \ln \left( \dfrac{V_{2}}{V_{1}} \right)$ $\Delta S = nR \ln \left( \dfrac{V_{2}}{V_{1}} \right)$ $\Delta S_{\mathrm{sys}}$ for reversible isothermal process - phase transitions $q_{\mathrm{rev}} = \Delta H_{\mathrm{fus}}$ $\Delta S_{\mathrm{fus}} = \dfrac{q_{\mathrm{rev}}}{T_{\mathrm{fus}}} = \dfrac{\Delta H_{\mathrm{fus}}}{T_{\mathrm{fus}}}$ $\Delta S_{\mathrm{sys}}$ for reversible adiabatic process $q = 0$ $\Delta S = 0$ $\Delta S_{\mathrm{sys}}$ for reversible isochoric process $\Delta V = 0$ $dq_{\mathrm{rev}} = nc_{V}dT$ $\Delta S = nc_{V} \displaystyle\int_{T_{1}}^{T_{2}} \dfrac{dT}{T} = nc_{V} \ln \left( \dfrac{T_{2}}{T_{1}} \right)$ $\Delta S_{\mathrm{sys}}$ for reversible isobaric process $\Delta P = 0$ $dq_{\mathrm{rev}} = nc_{P}dT$ $\Delta S = nc_{P} \displaystyle\int_{T_{1}}^{T_{2}} \dfrac{dT}{T} = nc_{P} \ln \left( \dfrac{T_{2}}{T_{1}} \right)$ Entropy change of surrounding $\Delta S_{\mathrm{surr}} = \dfrac{-\Delta H_{\mathrm{sys}}}{T_{\mathrm{surr}}}$ Second law of thermodynamics $\Delta S \geq \dfrac{q_{\mathrm{rev}}}{T}$ Enthalpy of spontaneous process $\Delta S_{\mathrm{total}} = \Delta S_{\mathrm{sys}} + \Delta S_{\mathrm{surr}} > 0$ Standard molar entropy $S^{\circ} = \displaystyle\int_{0K}^{298."/> 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/chem145/" /><link rel="prev" href="https://equation-sheets.netlify.app/chem/chem155/" /><link rel="next" href="https://equation-sheets.netlify.app/chem/chem455/" /><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="#gas">Gas</a> 98 <ul> 99 <li><a href="#kinetic-theory-of-gas">Kinetic Theory of Gas</a></li> 100 <li><a href="#molecular-collisions-and-rate-processes">Molecular Collisions and Rate Processes</a></li> 101 <li><a href="#intermolecular-interactions">Intermolecular Interactions</a></li> 102 </ul> 103 </li> 104 <li><a href="#thermodynamics">Thermodynamics</a> 105 <ul> 106 <li><a href="#first-law-of-thermodynamics">First law of thermodynamics</a></li> 107 <li><a href="#second-law-of-thermodynamics">Second law of thermodynamics</a></li> 108 </ul> 109 </li> 110 <li><a href="#equilibrium">Equilibrium</a></li> 111 </ul> 112</nav></div> 113 </div><div class="content" id="content"><h2 id="gas">Gas</h2> 114<h3 id="kinetic-theory-of-gas">Kinetic Theory of Gas</h3> 115<table> 116<thead> 117<tr> 118<th style="text-align:right">Quantity</th> 119<th style="text-align:center">Unit</th> 120<th style="text-align:left">Definition</th> 121</tr> 122</thead> 123<tbody> 124<tr> 125<td style="text-align:right">Pressure</td> 126<td style="text-align:center">$\mathrm{N/m^{2}}$</td> 127<td style="text-align:left">$P = \dfrac{F}{A}$</td> 128</tr> 129<tr> 130<td style="text-align:right">Mole fraction of $a$</td> 131<td style="text-align:center">-</td> 132<td style="text-align:left">$X_{a} = \dfrac{n_{a}}{n_{\mathrm{total}}}$</td> 133</tr> 134<tr> 135<td style="text-align:right">Partial pressure of $a$</td> 136<td style="text-align:center">$\mathrm{N/m^{2}}$</td> 137<td style="text-align:left">$P_{a} = X_{a}P_{\mathrm{total}}$</td> 138</tr> 139<tr> 140<td style="text-align:right">Pressure-volume work</td> 141<td style="text-align:center">$\mathrm{J}$</td> 142<td style="text-align:left">$W = -P\Delta V$</td> 143</tr> 144<tr> 145<td style="text-align:right">Compressibility of gas</td> 146<td style="text-align:center">-</td> 147<td style="text-align:left">$z = \dfrac{PV}{nRT}$</td> 148</tr> 149</tbody> 150</table> 151<table> 152<thead> 153<tr> 154<th style="text-align:right">Description</th> 155<th style="text-align:left">Equations</th> 156</tr> 157</thead> 158<tbody> 159<tr> 160<td style="text-align:right">Ideal gas law</td> 161<td style="text-align:left">$PV = nRT$</td> 162</tr> 163<tr> 164<td style="text-align:right">Ideal gas constant and Boltzmann constant</td> 165<td style="text-align:left">$R = N_{A}k_{B}$</td> 166</tr> 167<tr> 168<td style="text-align:right">Average kinetic energy of one gas molecule</td> 169<td style="text-align:left">$\varepsilon = \dfrac{3}{2}k_{B}T$</td> 170</tr> 171<tr> 172<td style="text-align:right">Root mean square speed gas</td> 173<td style="text-align:left">$v_{\mathrm{RMS}} = \sqrt{\dfrac{3RT}{\mathcal{M}}} = \sqrt{\dfrac{3k_{B}T}{m}}$</td> 174</tr> 175<tr> 176<td style="text-align:right">Mean speed of gas</td> 177<td style="text-align:left">$\bar{v} = \sqrt{\dfrac{8RT}{\pi\mathcal{M}}} = \sqrt{\dfrac{8k_{B}T}{\pi m}}$</td> 178</tr> 179<tr> 180<td style="text-align:right">Most probable speed of gas</td> 181<td style="text-align:left">$v_{\mathrm{mp}} = \sqrt{\dfrac{2RT}{\mathcal{M}}} = \sqrt{\dfrac{2k_{B}T}{m}}$</td> 182</tr> 183<tr> 184<td style="text-align:right">Maxwell-Boltzmann speed distribution</td> 185<td style="text-align:left">$f(v) = 4\pi \left(\dfrac{m}{2\pi k_{B}T}\right) v^{2} \exp\left(\dfrac{-mv^{2}}{2k_{B}T}\right)$</td> 186</tr> 187<tr> 188<td style="text-align:right">Van der Waals equation of state</td> 189<td style="text-align:left">$(P + a\dfrac{n^{2}}{V^{2}}
189)(V - bn) = nRT$</td> 190</tr> 191<tr> 192<td style="text-align:right">Lennard-Jones potential</td> 193<td style="text-align:left">$V_{\mathrm{LJ}} = 4\varepsilon \left(\left(\dfrac{\sigma}{R}\right)^{12} - \left(\dfrac{\sigma}{R}\right)^{6} \right)$</td> 194</tr> 195</tbody> 196</table> 197<h3 id="molecular-collisions-and-rate-processes">Molecular Collisions and Rate Processes</h3> 198<table> 199<thead> 200<tr> 201<th style="text-align:right">Description</th> 202<th style="text-align:left">Equations</th> 203</tr> 204</thead> 205<tbody> 206<tr> 207<td style="text-align:right">Molecule collision rate with wall</td> 208<td style="text-align:left">$Z_{w} \propto \dfrac{1}{4}\dfrac{N}{V}\bar{v}A = \dfrac{1}{4}\dfrac{N}{V}\sqrt{\dfrac{8RT}{\pi\mathcal{M}}}A$</td> 209</tr> 210<tr> 211<td style="text-align:right">Graham’s law of effusion</td> 212<td style="text-align:left">$\dfrac{\text{rate of effusion of A}}{\text{rate of effusion of B}} = \dfrac{N_{\mathrm{A}}}{N_{\mathrm{B}}} \sqrt{\dfrac{\mathcal{M}_{\mathrm{B}}}{\mathcal{M}_{\mathrm{A}}}}$</td> 213</tr> 214<tr> 215<td style="text-align:right">Molecule-molecule collision rate</td> 216<td style="text-align:left">$Z_{1} = 4\dfrac{N}{V}d^{2}\sqrt{\dfrac{\pi RT}{\mathcal{M}}}$</td> 217</tr> 218<tr> 219<td style="text-align:right">Mean free path</td> 220<td style="text-align:left">$\lambda = \dfrac{\bar{v}}{Z_{1}} = \dfrac{V}{\sqrt{2}\pi d^{2}N}$</td> 221</tr> 222<tr> 223<td style="text-align:right">Mean square displacement of diffusion in 3D</td> 224<td style="text-align:left">$\overline{\Delta r}^{2} = 6Dt$</td> 225</tr> 226<tr> 227<td style="text-align:right">Gas diffusion constant</td> 228<td style="text-align:left">$D = \dfrac{3}{8} \sqrt{\dfrac{RT}{\pi\mathcal{M}}} \dfrac{V}{Nd^{2}}$</td> 229</tr> 230</tbody> 231</table> 232<h3 id="intermolecular-interactions">Intermolecular Interactions</h3> 233<table> 234<thead> 235<tr> 236<th style="text-align:right">Quantity</th> 237<th style="text-align:center">Unit</th> 238<th style="text-align:left">Definition</th> 239</tr> 240</thead> 241<tbody> 242<tr> 243<td style="text-align:right">Electrostatic force</td> 244<td style="text-align:center">$\mathrm{N}$</td> 245<td style="text-align:left">$F = \dfrac{q_{1}q_{2}}{4\pi\varepsilon r^{2}}$</td> 246</tr> 247<tr> 248<td style="text-align:right">Electrostatic potential energy</td> 249<td style="text-align:center">$\mathrm{N}$</td> 250<td style="text-align:left">$V = \dfrac{q_{1}q_{2}}{4\pi\varepsilon r}$</td> 251</tr> 252<tr> 253<td style="text-align:right">Dipole moment</td> 254<td style="text-align:center">$\mathrm{C\cdot m}$</td> 255<td style="text-align:left">$\mu = qd$</td> 256</tr> 257<tr> 258<td style="text-align:right">Polarizability</td> 259<td style="text-align:center">$\mathrm{C\cdot m^{2}/V}$</td> 260<td style="text-align:left">$\alpha = \dfrac{\mu}{E}$</td> 261</tr> 262<tr> 263<td style="text-align:right">Induced dipole moment</td> 264<td style="text-align:center">$\mathrm{C\cdot m}$</td> 265<td style="text-align:left">$\mu^{*} = \alpha E$</td> 266</tr> 267</tbody> 268</table> 269<table> 270<thead> 271<tr> 272<th style="text-align:right">Description</th> 273<th style="text-align:left">Equations</th> 274</tr> 275</thead> 276<tbody> 277<tr> 278<td style="text-align:right">Ion-ion interactions</td> 279<td style="text-align:left">$E \propto \dfrac{q_{1}q_{2}}{r}$</td> 280</tr> 281<tr> 282<td style="text-align:right">Ion-dipole interactions</td> 283<td style="text-align:left">$E \propto -\dfrac{q\mu}{r^{2}}$</td> 284</tr> 285<tr> 286<td style="text-align:right">Dipole-dipole interactions</td> 287<td style="text-align:left">$E \propto -\dfrac{\mu_{1}\mu_{2}}{r^{3}}$</td> 288</tr> 289<tr> 290<td style="text-align:right">Induced-dipole-induced-dipole (London)</td> 291<td style="text-align:left">$E \propto -\dfrac{\alpha_{1}\alpha_{2}}{r^{6}}$</td> 292</tr> 293<tr> 294<td style="text-align:right">Dipole-induced-dipole</td> 295<td style="text-align:left">$E \propto -\dfrac{\mu_{1}^{2}\alpha_{2}}{r^{6}}$</td> 296</tr> 297<tr> 298<td style="text-align:right">Rotating fixed dipole (Keesom)</td> 299<td style="text-align:left">$E \propto -\dfrac{\mu_{1}^{2}\mu_{2}^{2}}{r^{6}}$</td> 300</tr> 301</tbody> 302</table> 303<h2 id="thermodynamics">Thermodynamics</h2> 304<h3 id="first-law-of-thermodynamics">First law of thermodynamics</h3> 305<table> 306<thead> 307<tr> 308<th style="text-align:right">Quantity</th> 309<th style="text-align:center">Unit</th> 310<th style="text-align:left">Definition</th> 311</tr> 312</thead> 313<tbody> 314<tr> 315<td style="text-align:right">Specific heat capacities</td> 316<td style="text-align:center">$\mathrm{J \cdot kg^{-1} \cdot K^{-1}}$</td> 317<td style="text-align:left">$q = mc_{s}\Delta T = n\bar{c}\Delta T$</td> 318</tr> 319<tr> 320<td style="text-align:right">Heat capacity</td> 321<td style="text-align:center">$\mathrm{J/K}$</td> 322<td style="text-align:left">$q = C\Delta T$</td> 323</tr> 324<tr> 325<td style="text-align:right">Molar heat capacities</td> 326<td style="text-align:center">$\mathrm{J \cdot mol^{-1} \cdot K^{-1}}$</td> 327<td style="text-align:left">$q_{V} = nc_{V}\Delta T$ <br> $q_{P} = nc_{P}\Delta T$</td> 328</tr> 329<tr> 330<td style="text-align:right">Enthalpy</td> 331<td style="text-align:center">$\mathrm{J}$</td> 332<td style="text-align:left">$H = U + PV$</td> 333</tr> 334</tbody> 335</table> 336<table> 337<thead> 338<tr> 339<th style="text-align:right">Description</th> 340<th style="text-align:left">Equations</th> 341</tr> 342</thead> 343<tbody> 344<tr> 345<td style="text-align:right">First law of thermodynamics</td> 346<td style="text-align:left">$\Delta U = q + w$ <br/> $dU = \cancel{d}q + \cancel{d}w$ <br/> $\Delta U_{\mathrm{univ}} = \Delta U_{\mathrm{sys}} + \Delta U_{\mathrm{surr}} = 0$</td> 347</tr> 348<tr> 349<td style="text-align:right">Enthalpy change</td> 350<td style="text-align:left">$q_{P} = \Delta (U + PV) = \Delta H$</td> 351</tr> 352<tr> 353<td style="text-align:right">Molar heat capacity of monoatomic ideal gas at constant volume</td> 354<td style="text-align:left">$c_{V} = \dfrac{3}{2}R$</td> 355</tr> 356<tr> 357<td style="text-align:right">Molar heat capacity of any ideal gas at constant pressure</td> 358<td style="text-align:left">$c_{P} = c_{V} + R = \dfrac{5}{2}R$</td> 359</tr> 360<tr> 361<td style="text-align:right">Internal energy change of any ideal gas</td> 362<td style="text-align:left">$\Delta U = nc_{V}\Delta T$</td> 363</tr> 364<tr> 365<td style="text-align:right">Enthalpy change of any ideal gas</td> 366<td style="text-align:left">$\begin{aligned}\Delta H &= nc_{P}\Delta T \cr &= \Delta U + \Delta(PV) \cr &= nc_{V}\Delta T + nR\Delta T\end{aligned}$</td> 367</tr> 368<tr> 369<td style="text-align:right">Hess’s law</td> 370<td style="text-align:left">$\Delta H^{\circ} = \sum\limits_{i}^{\text{prod}} n_{i}\Delta H_{i}^{\circ} - \sum\limits_{j}^{\text{react}} n_{j}\Delta H_{j}^{\circ}$</td> 371</tr> 372<tr> 373<td style="text-align:right">Molality</td> 374<td style="text-align:left">$b = \dfrac{n_{\mathrm{solute}}}{m_{\mathrm{solvent}}}$</td> 375</tr> 376<tr> 377<td style="text-align:right">Boiling point elevation <br/> ($i$ is vant’s Hoff dissociation factor)</td> 378<td style="text-align:left">$\Delta T_{\mathrm{boil}} = ibK_{\mathrm{boil}}$</td> 379</tr> 380<tr> 381<td style="text-align:right">Freezing point depression <br/> ($i$ is vant’s Hoff dissociation factor)</td> 382<td style="text-align:left">$\Delta T_{\mathrm{freeze}} = ibK_{\mathrm{freeze}}$</td> 383</tr> 384<tr> 385<td style="text-align:right">Reversible isothermal process of ideal gas</td> 386<td style="text-align:left">$\Delta T = 0$ <br/> $\Delta U = 0$ <br/> $\Delta H = 0$ <br/> $w = -\displaystyle\int_{V_{1}}^{V_{2}} P \ dV = -nRT \ln\dfrac{V_{2}}{V_{1}}$ <br/> $q = -w$</td> 387</tr> 388<tr> 389<td style="text-align:right">Reversible adiabatic process of ideal gas</td> 390<td style="text-align:left">$q = 0$ <br/> $\Delta U = nc_{V}\Delta T = w$ <br/> $\Delta H = nc_{P}\Delta T$ <br/> $\gamma = \dfrac{c_{P}}{c_{V}}$ <br/> $T_{1}V_{1}^{\gamma - 1} = T_{2}V_{2}^{\gamma - 1}$ <br/> $P_{1}V_{1}^{\gamma} = P_{2}V_{2}^{\gamma}$</td> 391</tr> 392</tbody> 393</table> 394<h3 id="second-law-of-thermodynamics">Second law of thermodynamics</h3> 395<table> 396<thead> 397<tr> 398<th style="text-align:right">Description</th> 399<th style="text-align:left">Equations</th> 400</tr> 401</thead> 402<tbody> 403<tr> 404<td style="text-align:right">Entropy</td> 405<td style="text-align:left">$S = k_{B} \ln\Omega$</td> 406</tr> 407<tr> 408<td style="text-align:right">Entropy change</td> 409<td style="text-align:left">$\Delta S = \displaystyle\int_{i}^{f} \dfrac{dq_{\mathrm{rev}}}{T}$</td> 410</tr> 411<tr> 412<td style="text-align:right">$\Delta S_{\mathrm{sys}}$ for reversible isothermal process</td> 413<td style="text-align:left">$\Delta S = \displaystyle\int_{i}^{f} \dfrac{dq_{\mathrm{rev}}}{T} = \dfrac{1}{T} \displaystyle\int_{i}^{f} dq_{\mathrm{rev}} = \dfrac{q_{\mathrm{rev}}}{T}$</td> 414</tr> 415<tr> 416<td style="text-align:right">$\Delta S_{\mathrm{sys}}$ for reversible isothermal process - compression/expansion of ideal gas</td> 417<td style="text-align:left">$q_{\mathrm{rev}} = nRT \ln \left( \dfrac{V_{2}}{V_{1}} \right)$ <br/> $\Delta S = nR \ln \left( \dfrac{V_{2}}{V_{1}} \right)$</td> 418</tr> 419<tr> 420<td style="text-align:right">$\Delta S_{\mathrm{sys}}$ for reversible isothermal process - phase transitions</td> 421<td style="text-align:left">$q_{\mathrm{rev}} = \Delta H_{\mathrm{fus}}$ <br/> $\Delta S_{\mathrm{fus}} = \dfrac{q_{\mathrm{rev}}}{T_{\mathrm{fus}}} = \dfrac{\Delta H_{\mathrm{fus}}}{T_{\mathrm{fus}}}$</td> 422</tr> 423<tr> 424<td style="text-align:right">$\Delta S_{\mathrm{sys}}$ for reversible adiabatic process</td> 425<td style="text-align:left">$q = 0$ <br/> $\Delta S = 0$</td> 426</tr> 427<tr> 428<td style="text-align:right">$\Delta S_{\mathrm{sys}}$ for reversible isochoric process</td> 429<td style="text-align:left">$\Delta V = 0$ <br/> $dq_{\mathrm{rev}} = nc_{V}dT$ <br/> $\Delta S = nc_{V} \displaystyle\int_{T_{1}}^{T_{2}} \dfrac{dT}{T} = nc_{V} \ln \left( \dfrac{T_{2}}{T_{1}} \right)$</td> 430</tr> 431<tr> 432<td style="text-align:right">$\Delta S_{\mathrm{sys}}$ for reversible isobaric process</td> 433<td style="text-align:left">$\Delta P = 0$ <br/> $dq_{\mathrm{rev}} = nc_{P}dT$ <br/> $\Delta S = nc_{P}
433 \displaystyle\int_{T_{1}}^{T_{2}} \dfrac{dT}{T} = nc_{P} \ln \left( \dfrac{T_{2}}{T_{1}} \right)$</td> 434</tr> 435<tr> 436<td style="text-align:right">Entropy change of surrounding</td> 437<td style="text-align:left">$\Delta S_{\mathrm{surr}} = \dfrac{-\Delta H_{\mathrm{sys}}}{T_{\mathrm{surr}}}$</td> 438</tr> 439<tr> 440<td style="text-align:right">Second law of thermodynamics</td> 441<td style="text-align:left">$\Delta S \geq \dfrac{q_{\mathrm{rev}}}{T}$</td> 442</tr> 443<tr> 444<td style="text-align:right">Enthalpy of spontaneous process</td> 445<td style="text-align:left">$\Delta S_{\mathrm{total}} = \Delta S_{\mathrm{sys}} + \Delta S_{\mathrm{surr}} > 0$</td> 446</tr> 447<tr> 448<td style="text-align:right">Standard molar entropy</td> 449<td style="text-align:left">$S^{\circ} = \displaystyle\int_{0K}^{298.15\mathrm{K}} \dfrac{c_{P}}{T} dT + \Delta S \text{(phase changes between 0K and 298.15K)}$</td> 450</tr> 451<tr> 452<td style="text-align:right">Gibbs free energy for reaction at constant temperature</td> 453<td style="text-align:left">$\Delta G = \Delta H - T\Delta S$</td> 454</tr> 455<tr> 456<td style="text-align:right">Efficiency of Carnot engines</td> 457<td style="text-align:left">$\begin{aligned}\varepsilon &= \dfrac{\text{work on surrounding}}{\text{heat into system}} \cr &= \dfrac{T_{\mathrm{high}} - T_{\mathrm{low}}}{T_{\mathrm{high}}} = 1-\dfrac{T_{\mathrm{low}}}{T_{\mathrm{high}}}\end{aligned}$</td> 458</tr> 459<tr> 460<td style="text-align:right">Relationship between heat and temperature in Carnot cycle</td> 461<td style="text-align:left">$\dfrac{q_{\mathrm{high}}}{T_{\mathrm{high}}} + \dfrac{q_{\mathrm{low}}}{T_{\mathrm{low}}} = 0$</td> 462</tr> 463<tr> 464<td style="text-align:right">Work done by Carnot cycle in one cycle</td> 465<td style="text-align:left">$w_{\mathrm{cycle}} = -nR(T_{\mathrm{hot}} - T_{\mathrm{cold}}) \ln\dfrac{V_{B}}{V_{A}}$</td> 466</tr> 467</tbody> 468</table> 469<h2 id="equilibrium">Equilibrium</h2> 470<table> 471<thead> 472<tr> 473<th style="text-align:right">Description</th> 474<th style="text-align:left">Equations</th> 475</tr> 476</thead> 477<tbody> 478<tr> 479<td style="text-align:right">Law of mass action - partial pressure</td> 480<td style="text-align:left">$K = \dfrac{\prod\limits_{j}(P_{\text{product }j} / P_{\mathrm{ref}})_{eq}^{b_{j}}}{\prod\limits_{i}(P_{\text{reactant }i} / P_{\mathrm{ref}})_{eq}^{a_{i}}}$</td> 481</tr> 482<tr> 483<td style="text-align:right">Law of mass action - concentration</td> 484<td style="text-align:left">$K = \dfrac{\prod\limits_{j}(c_{\text{product }j} / c_{\mathrm{ref}})_{eq}^{b_{j}}}{\prod\limits_{i}(c_{\text{reactant }i} / c_{\mathrm{ref}})_{eq}^{a_{i}}}$</td> 485</tr> 486<tr> 487<td style="text-align:right">Gibbs free energy of isothermal reactions</td> 488<td style="text-align:left">$\Delta G = -T\Delta S = nRT\ln\dfrac{P_{2}}{P_{1}}$</td> 489</tr> 490<tr> 491<td style="text-align:right">Equilibrium expression: relationship between Gibbs free energy and equilibrium constant (gas phase reaction)</td> 492<td style="text-align:left">$\Delta G^{\circ} = -RT\ln K$</td> 493</tr> 494<tr> 495<td style="text-align:right">Equilibrium expression: alternative form</td> 496<td style="text-align:left">$\ln K = -\dfrac{\Delta G^{\circ}}{RT} = - \dfrac{\Delta H^{\circ}}{RT} + \dfrac{\Delta S}{R}$</td> 497</tr> 498<tr> 499<td style="text-align:right">Change in Gibbs free energy at non-standard conditions</td> 500<td style="text-align:left">$\Delta G = \Delta G^{\circ} + RT\ln Q = RT\ln\dfrac{Q}{K}$</td> 501</tr> 502<tr> 503<td style="text-align:right">Reaction quotient - partial pressure</td> 504<td style="text-align:left">$Q = \dfrac{\prod\limits_{j}(P_{\text{product }j} / P_{\mathrm{ref}})^{b_{j}}}{\prod\limits_{i}(P_{\text{reactant }i} / P_{\mathrm{ref}})^{a_{i}}}$</td> 505</tr> 506<tr> 507<td style="text-align:right">Reaction quotient - concentration</td> 508<td style="text-align:left">$Q = \dfrac{\prod\limits_{j}(c_{\text{product }j} / c_{\mathrm{ref}})^{b_{j}}}{\prod\limits_{i}(c_{\text{reactant }i} / c_{\mathrm{ref}})^{a_{i}}}$</td> 509</tr> 510<tr> 511<td style="text-align:right"><strong>vant’s Hoff equation</strong> <br/> temperature dependence of equilibrium constant of a reaction</td> 512<td style="text-align:left">$\ln\dfrac{K_{2}}{K_{1}} = -\dfrac{\Delta H^{\circ}}{R} \left( \dfrac{1}{T_{2}} - \dfrac{1}{T_{1}} \right)$</td> 513</tr> 514<tr> 515<td style="text-align:right"><strong>Clapeyron equation</strong> <br/> for two phase in equilibrium, construct phase diagram by finding change of pressure as a function of temperature</td> 516<td style="text-align:left">$\left( \dfrac{dP}{dT} \right)_{eq}
516 = \dfrac{\Delta S}{\Delta V} = \dfrac{\Delta H}{T\Delta V}$</td> 517</tr> 518<tr> 519<td style="text-align:right"><strong>Clausius-Clapeyron equation</strong> <br/> temperature dependence of vapor pressure for condensed phase and gas phase in equilibrium</td> 520<td style="text-align:left">$\ln\dfrac{P_{2}}{P_{1}} = -\dfrac{\Delta H_{\mathrm{vap}}}{nR} \left( \dfrac{1}{T_{2}} - \dfrac{1}{T_{1}} \right)$</td> 521</tr> 522<tr> 523<td style="text-align:right"><strong>Clausius-Clapeyron equation</strong> <br/> for $P_{1} = P^{\circ}; \Delta S^{\circ} = \dfrac{\Delta H^{\circ}}{T^{\circ}}$</td> 524<td style="text-align:left">$\ln\dfrac{P_{2}}{P^{\circ}} = -\dfrac{\Delta H_{\mathrm{vap}}}{nRT_{2}} + \dfrac{\Delta S^{\circ}_{\mathrm{vap}}}{nR}$</td> 525</tr> 526</tbody> 527</table> 528</div><div class="post-footer" id="post-footer"> 529 <div class="post-info"> 530 <div class="post-info-line"> 531 <div class="post-info-mod"> 532 <span>Updated on 2021-08-17</span> 533 </div> 534 <div class="post-info-license"></div> 535 </div> 536 <div class="post-info-line"> 537 <div class="post-info-md"></div> 538 <div class="post-info-share"> 539 <span></span> 540 </div> 541 </div> 542 </div> 543 544 <div class="post-info-more"> 545 <section class="post-tags"></section> 546 <section> 547 <span><a href="javascript:void(0);" onclick="window.history.back();">Back</a></span> | <span><a href="/">Home</a></span> 548 </section> 549 </div> 550 551 <div class="post-nav"><a href="/chem/chem155/" class="prev" rel="prev" title="CHEM 155 Honors General Chemistry II"><i class="fas fa-angle-left fa-fw"></i>CHEM 155 Honors General Chemistry II</a> 552 <a href="/chem/chem455/" class="next" rel="next" title="CHEM 455 Physical Chemistry">CHEM 455 Physical Chemistry<i class="fas fa-angle-right fa-fw"></i></a></div> 553</div> 554</article> 555 556 557
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