CHEM 145 Honors General Chemistry I
Contents
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 | $\mathrm{J}$ | $H = U + PV$ |
| 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 | $S^{\circ} = \displaystyle\int_{0K}^{298.15\mathrm{K}} \dfrac{c_{P}}{T} dT + \Delta S \text{(phase changes between 0K and 298.15K)}$ |
| Gibbs free energy for reaction at constant temperature | $\Delta G = \Delta H - T\Delta S$ |
| Efficiency of Carnot engines | $\begin{aligned}\varepsilon &= \dfrac{\text{work on surrounding}}{\text{heat into system}} \cr &= \dfrac{T_{\mathrm{high}} - T_{\mathrm{low}}}{T_{\mathrm{high}}} = 1-\dfrac{T_{\mathrm{low}}}{T_{\mathrm{high}}}\end{aligned}$ |
| Relationship between heat and temperature in Carnot cycle | $\dfrac{q_{\mathrm{high}}}{T_{\mathrm{high}}} + \dfrac{q_{\mathrm{low}}}{T_{\mathrm{low}}} = 0$ |
| Work done by Carnot cycle in one cycle | $w_{\mathrm{cycle}} = -nR(T_{\mathrm{hot}} - T_{\mathrm{cold}}) \ln\dfrac{V_{B}}{V_{A}}$ |
Equilibrium
| Description | Equations |
|---|---|
| Law of mass action - partial pressure | $K = \dfrac{\prod\limits_{j}(P_{\text{product }j} / P_{\mathrm{ref}})_{eq}^{b_{j}}}{\prod\limits_{i}(P_{\text{reactant }i} / P_{\mathrm{ref}})_{eq}^{a_{i}}}$ |
| Law of mass action - concentration | $K = \dfrac{\prod\limits_{j}(c_{\text{product }j} / c_{\mathrm{ref}})_{eq}^{b_{j}}}{\prod\limits_{i}(c_{\text{reactant }i} / c_{\mathrm{ref}})_{eq}^{a_{i}}}$ |
| Gibbs free energy of isothermal reactions | $\Delta G = -T\Delta S = nRT\ln\dfrac{P_{2}}{P_{1}}$ |
| Equilibrium expression: relationship between Gibbs free energy and equilibrium constant (gas phase reaction) | $\Delta G^{\circ} = -RT\ln K$ |
| Equilibrium expression: alternative form | $\ln K = -\dfrac{\Delta G^{\circ}}{RT} = - \dfrac{\Delta H^{\circ}}{RT} + \dfrac{\Delta S}{R}$ |
| Change in Gibbs free energy at non-standard conditions | $\Delta G = \Delta G^{\circ} + RT\ln Q = RT\ln\dfrac{Q}{K}$ |
| Reaction quotient - partial pressure | $Q = \dfrac{\prod\limits_{j}(P_{\text{product }j} / P_{\mathrm{ref}})^{b_{j}}}{\prod\limits_{i}(P_{\text{reactant }i} / P_{\mathrm{ref}})^{a_{i}}}$ |
| Reaction quotient - concentration | $Q = \dfrac{\prod\limits_{j}(c_{\text{product }j} / c_{\mathrm{ref}})^{b_{j}}}{\prod\limits_{i}(c_{\text{reactant }i} / c_{\mathrm{ref}})^{a_{i}}}$ |
| vant’s Hoff equation temperature dependence of equilibrium constant of a reaction |
$\ln\dfrac{K_{2}}{K_{1}} = -\dfrac{\Delta H^{\circ}}{R} \left( \dfrac{1}{T_{2}} - \dfrac{1}{T_{1}} \right)$ |
| Clapeyron equation for two phase in equilibrium, construct phase diagram by finding change of pressure as a function of temperature |
$\left( \dfrac{dP}{dT} \right)_{eq} = \dfrac{\Delta S}{\Delta V} = \dfrac{\Delta H}{T\Delta V}$ |
| Clausius-Clapeyron equation temperature dependence of vapor pressure for condensed phase and gas phase in equilibrium |
$\ln\dfrac{P_{2}}{P_{1}} = -\dfrac{\Delta H_{\mathrm{vap}}}{nR} \left( \dfrac{1}{T_{2}} - \dfrac{1}{T_{1}} \right)$ |
| Clausius-Clapeyron equation for $P_{1} = P^{\circ}; \Delta S^{\circ} = \dfrac{\Delta H^{\circ}}{T^{\circ}}$ |
$\ln\dfrac{P_{2}}{P^{\circ}} = -\dfrac{\Delta H_{\mathrm{vap}}}{nRT_{2}} + \dfrac{\Delta S^{\circ}_{\mathrm{vap}}}{nR}$ |