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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 $(P &#43; a\dfrac{n^{2}}{V^{2}})(V - bn) = nRT$   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&rsquo;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 $\mathrm{J}$ $H = U &#43;
12 PV$       Description Equations     First law of thermodynamics $\Delta U = q &#43; w$  $dU = \cancel{d}q &#43; \cancel{d}w$  $\Delta U_{\mathrm{univ}} = \Delta U_{\mathrm{sys}} &#43; \Delta U_{\mathrm{surr}} = 0$   Enthalpy change $q_{P} = \Delta (U &#43; PV) = \Delta H$   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 $c_{P} = c_{V} &#43; R = \dfrac{5}{2}R$   Internal energy change of any ideal gas $\Delta U = nc_{V}\Delta T$   Enthalpy change of any ideal gas $\begin{aligned}\Delta H &amp;= nc_{P}\Delta T \cr &amp;= \Delta U &#43; \Delta(PV) \cr &amp;= nc_{V}\Delta T &#43; nR\Delta T\end{aligned}$   Hess&rsquo;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&rsquo;s Hoff dissociation factor) $\Delta T_{\mathrm{boil}} = ibK_{\mathrm{boil}}$   Freezing point depression  ($i$ is vant&rsquo;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 $\Delta S_{\mathrm{total}} = \Delta S_{\mathrm{sys}} &#43; \Delta S_{\mathrm{surr}} &gt; 0$   Standard molar entropy $S^{\circ} = \displaystyle\int_{0K}^{298." />
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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 $(P &#43; a\dfrac{n^{2}}{V^{2}})(V - bn) = nRT$   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&rsquo;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 $\mathrm{J}$ $H = U &#43; PV$       Description Equations     First law of thermodynamics $\Delta U = q &#43; w$  $dU = \cancel{d}q &#43; \cancel{d}w$  $\Delta U_{\mathrm{univ}} = \Delta U_{\mathrm{sys}} &#43; \Delta U_{\mathrm{surr}} = 0$   Enthalpy change $q_{P} = \Delta (U &#43; PV) = \Delta H$   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 $c_{P} = c_{V} &#43; R = \dfrac{5}{2}R$   Internal energy change of any ideal gas $\Delta U = nc_{V}\Delta T$   Enthalpy change of any ideal gas $\begin{aligned}\Delta H &amp;= nc_{P}\Delta T \cr &amp;= \Delta U &#43; \Delta(PV) \cr &amp;= nc_{V}\Delta T &#43; nR\Delta T\end{aligned}$   Hess&rsquo;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&rsquo;s Hoff dissociation factor) $\Delta T_{\mathrm{boil}} = ibK_{\mathrm{boil}}$   Freezing point depression  ($i$ is vant&rsquo;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 $\Delta S_{\mathrm{total}} = \Delta S_{\mathrm{sys}} &#43; \Delta S_{\mathrm{surr}} &gt; 0$   Standard molar entropy $S^{\circ} = \displaystyle\int_{0K}^{298."/>
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84        <h1 class="single-title posts-title">CHEM 145 Honors General Chemistry I</h1>
85
86        <div class="post-meta">
87            <div class="post-meta-line"><i class="far fa-calendar-alt fa-fw"></i>&nbsp;<time datetime="2021-08-17">2021-08-17</time>&nbsp;</div>
88        </div>
89
90        <div class="details toc" id="toc-static"  kept="">
91                <div class="details-summary toc-title">
92                    <span>Contents</span>
93                    <span><i class="details-icon fas fa-angle-right"></i></span>
94                </div>
95                <div class="details-content toc-content" id="toc-content-static"><nav id="TableOfContents">
96  <ul>
97    <li><a href="#gas">Gas</a>
98      <ul>
99        <li><a href="#kinetic-theory-of-gas">Kinetic Theory of Gas</a></li>
100        <li><a href="#molecular-collisions-and-rate-processes">Molecular Collisions and Rate Processes</a></li>
101        <li><a href="#intermolecular-interactions">Intermolecular Interactions</a></li>
102      </ul>
103    </li>
104    <li><a href="#thermodynamics">Thermodynamics</a>
105      <ul>
106        <li><a href="#first-law-of-thermodynamics">First law of thermodynamics</a></li>
107        <li><a href="#second-law-of-thermodynamics">Second law of thermodynamics</a></li>
108      </ul>
109    </li>
110    <li><a href="#equilibrium">Equilibrium</a></li>
111  </ul>
112</nav></div>
113            </div><div class="content" id="content"><h2 id="gas">Gas</h2>
114<h3 id="kinetic-theory-of-gas">Kinetic Theory of Gas</h3>
115<table>
116<thead>
117<tr>
118<th style="text-align:right">Quantity</th>
119<th style="text-align:center">Unit</th>
120<th style="text-align:left">Definition</th>
121</tr>
122</thead>
123<tbody>
124<tr>
125<td style="text-align:right">Pressure</td>
126<td style="text-align:center">$\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&rsquo;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">$\begin{aligned}\Delta H &amp;= nc_{P}\Delta T \cr &amp;= \Delta U + \Delta(PV) \cr &amp;= nc_{V}\Delta T + nR\Delta T\end{aligned}$</td>
367</tr>
368<tr>
369<td style="text-align:right">Hess&rsquo;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&rsquo;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&rsquo;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">$\Delta S_{\mathrm{total}} = \Delta S_{\mathrm{sys}} + \Delta S_{\mathrm{surr}} &gt; 0$</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">$\begin{aligned}\varepsilon &amp;= \dfrac{\text{work on surrounding}}{\text{heat into system}} \cr &amp;= \dfrac{T_{\mathrm{high}} - T_{\mathrm{low}}}{T_{\mathrm{high}}} = 1-\dfrac{T_{\mathrm{low}}}{T_{\mathrm{high}}}\end{aligned}$</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&rsquo;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>
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575</body>
576</html>

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.