CHEM E 457 Principles of Molecular Engineering
Contents
Counting and Probability
Counting
| Description | Equations |
|---|---|
| Total possible events if each event $E_i$ can occur in $n_i$ ways | $\prod n_i$ |
| Permutation of $n$ elements taken $r$ at a time | $P(n, r) = \dfrac{n!}{(n-r)!}$ |
| Distinguishable permutations of $n$ objects, with $n_i$ are alike of one kind | $\dfrac{n!}{n_1! n_2! \cdots n_k!}$ |
| Combination of $n$ elements taken $r$ at a time | $\displaystyle C(n, r) = \binom{n}{r} = \dfrac{n!}{r!(n-r)!}$ |
| Stirling’s approximation | $x! \approx \left(\dfrac{x}{e}\right)^x$ |
| Stirling’s approximation | $\ln(x!) = x \ln(x) - x$ |
Probability
| Description | Equations |
|---|---|
| Probability | $\mathbf{P}(A) = \dfrac{n_A}{N}$ |
| Addition rule of mutually exclusive outcomes | $\mathbf{P}\left(\bigcup A_i\right) = \sum \mathbf{P}(A_i)$ |
| Multiplication rule of independent outcomes | $\mathbf{P}\left(\bigcap A_i\right) = \prod \mathbf{P}(A_i)$ |
| General addition rule | $\mathbf{P}(A\cup B) = \mathbf{P}(A) + \mathbf{P}(B) - \mathbf{P}(A\cap B)$ |
| Conditional probability | $\mathbf{P}(A\vert B) \equiv \dfrac{\mathbf{P}(A\cap B)}{\mathbf{P}(B)}$ |
| Bayes' rule | $\mathbf{P}(A\vert B) = \dfrac{\mathbf{P}(A) \mathbf{P}(B\vert A)}{\mathbf{P}(B)}$ |
| Total probability theorem | $\mathbf{P}(B) = \mathbf{P}(A)\mathbf{P}(B\vert A) + \mathbf{P}(A^c)\mathbf{P}(B\vert A^c)$ |
| Degree of correlation | $g = \dfrac{\mathbf{P}(B\vert A)}{\mathbf{P}(B)} = \dfrac{\mathbf{P}(A \cap B)}{\mathbf{P}(A)\mathbf{P}(B)}$ |
Continuous probability distribution
| Description | Equations |
|---|---|
| Normalization condition of probability distribution function | $\displaystyle\int_a^b P(x) dx = 1$ |
| Binomial distribution | $P(n, N) = \dfrac{N!}{n!(N-n)!}p^n (1-p)^{N-n}$ |
| Multinomial distribution | $P(n_1, n_2, …, n_t, N) = \dfrac{N!}{\prod n_i!}\prod p_i^{n_i}$ |
| Average | $\langle x \rangle = \displaystyle\int_a^b xP(x) dx$ |
| Average of a function | $\langle f(x) \rangle = \displaystyle\int_a^b f(x)P(x) dx$ |
| $n$th moment | $\langle x^n \rangle = \displaystyle\int_a^b x^nP(x) dx$ |
| Variance | $\sigma^2 = \langle x^2 \rangle - \langle x \rangle^2$ |
Extremum Principles Predicts Equilibria
| Physical Description | Math Description | Equations |
|---|---|---|
| Equilibrium | Critical point | $f'(x) = 0$ |
| Stable equilibrium | Minimum | $f'(x) = 0, f''(x) > 0$ |
| Unstable equilibrium | Maximum | $f'(x) = 0, f''(x) < 0$ |
| Metastable equilibrium | Local minimum | $f'(x) = 0, f''(x) > 0$ in some $dx$ |
| Neutral equilibrium | Constant | $f'(x) = 0$ for all $x$ |
- Extremum principles
- Minimization of energy
- Maximization of entropy (multiplicity)
| Description | Equations |
|---|---|
| Method of Lagrange multiplier Finding extremum of objective function $f(\mathbf{x})$ subjected to constraint $g(\mathbf{x})$ |
$\nabla f(\mathbf{x}) = \lambda \nabla g(\mathbf{x})$ |
Entropy and Boltzmann Law
- Ground state - state of lowest energy
- Excited state - states of higher energy
- Microstate - microscopic configuration
- Macrostate - collection of microstate
General applications
| Description | Equations |
|---|---|
| Entropy in terms of multiplicity | $S = k\ln(W)$ |
| Entropy in terms of probability | $S = -k \sum p_i \ln (p_i)$ |
| Probability of a microstate ★ No constraint on observation ★ Maximized entropy |
$p_i = \dfrac{1}{t}$ |
| Boltzmann distribution law Probability of a microstate ★ With constraint on observation ★ Maximized entropy |
$p_i = \dfrac{e^{-\beta \varepsilon_i}}{\sum e^{-\beta \varepsilon_i}} = \dfrac{e^{-\beta \varepsilon_i}}{q}$ |
| Partition function | $q = \sum e^{-\beta \varepsilon_i}$ |
| Average observation | $\langle \varepsilon \rangle = \sum \varepsilon_i p_i$ |
Molecular distributions
| Description | Boltzmann Distribution Law | Partition Function |
|---|---|---|
| System with energy levels | $p_i = \dfrac{\exp(-E_i/kT)}{Q}$ | $Q = \sum \exp\left(-E_i/kT\right)$ |
| System with energy differences | $p_i = \dfrac{\exp(-(E_i-E_j)/kT)}{Q}$ | $Q = \sum\exp\left(-(E_i-E_j)/kT\right)$ |
| System with degenerate energy levels | $p_i = \dfrac{W(E_l) \exp(-E_l/kT)}{Q}$ | $Q = \sum W(E_l) \exp(-E_l / kT)$ |
| Description | Equations |
|---|---|
| Thermodynamic beta | $\beta = \dfrac{1}{kT}$ |
| Relative populations of particles in energy level $i$ and $j$ at equilibrium | $\dfrac{p_i}{p_j} = \exp\left(-\dfrac{E_i - E_j}{kT}\right)$ |
| Partition function of subsystem of independent distinguishable particles (solid) | $Q = \prod_i^N q_i = q^N$ |
| Partition function of subsystem of independent indistinguishable particles (gas) | $Q = \dfrac{q^N}{N!}$ |
| Internal energy | $U = kT^2 \left(\dfrac{\partial\ln Q}{\partial T}\right)_{V, N}$ |
| Average particle energy | $\langle \varepsilon \rangle = \dfrac{U}{N} = \dfrac{kT^2}{N} \left(\dfrac{\partial\ln Q}{\partial T}\right)_{V, N}$ |
| Entropy | $S = k \ln Q + \dfrac{U}{T}$ |
| Helmholtz free energy | $F = U - TS = -kT \ln Q$ |
| Chemical potential | $\mu = \left(\dfrac{\partial F}{\partial N}\right)_{T, V} = -kT\left(\dfrac{\partial\ln Q}{\partial N}\right)_{T, V}$ |
| Pressure | $p = -\left(\dfrac{\partial F}{\partial V}\right)_{T, N} = kT \left(\dfrac{\partial\ln Q}{\partial N}\right)_{T, N}$ |
Ensembles
- controlled set of variables
- collection of all the possible microstates
| Description | Equations |
|---|---|
| Canonical ensemble | $(T, V, N)$ |
| Microcanonical ensemble | $(U, V, N)$ |
| Isobaric-isothermal ensemble | $(T, P, N)$ |
| Grand canonical ensemble | $(T, V, \mu)$ |
Ch 13 Chemical Equilibrium
Multicomponent reactions
| Description | Equations |
|---|---|
| Multicomponent gas phase reaction ★ No intermolecular interactions |
$\ce{aA + bB -> cC + dD}$ |
| Difference in ground state energy | $\Delta \varepsilon_0 = d \varepsilon_{0D} + c \varepsilon_{0C} - b \varepsilon_{0B} - a \varepsilon_{0A}$ |
| Difference in Dissociation energy | $\Delta D = d D_D + c D_C - b D_B - a D_A$ |
| Dissociation energy | $D = - \varepsilon_0 \\ \Delta D = -\Delta \varepsilon_0$ |
| Equilibrium constant | $K = \dfrac{N_C^c N_D^d}{N_A^a N_B^b} = \dfrac{q_C^c q_D^d}{q_A^a q_B^b} \exp\left(\dfrac{\Delta D}{kT}\right)$ |
| Pressure-based equilibrium constant | $K_p = \dfrac{p_C^c p_D^d}{p_A^a p_B^b} = \left(\dfrac{kT}{V}\right)^{(c+d) - (a+b)} \dfrac{q_C^c q_D^d}{q_A^a q_B^b} \exp\left(\dfrac{\Delta D}{kT}\right)$ |
| Chemical potential | $\mu = kT \ln\left(\dfrac{p}{p_{int}^\circ}\right) = \mu^\circ + kT \ln p$ |
| Internal pressure | $p_{int}^\circ = q_0' kT$ |
$T, P$ dependence of equilibrium
| Description | Equations |
|---|---|
| $\ln K_p = -\dfrac{\Delta\mu^\circ}{kT}$ | |
| vant’s Hoff equation ★ $\Delta h^\circ \not =\Delta h^\circ(T)$ |
$\dfrac{\partial (\ln K_p)}{\partial T} = \dfrac{\Delta h^\circ}{kT^2}$ |
| vant’s Hoff plot ★ $\Delta h^\circ \not =\Delta h^\circ(T)$ |
$\dfrac{\partial (\ln K_p)}{\partial (1/T)} = - \dfrac{\Delta h^\circ}{k}$ |
| vant’s Hoff equation extrapolation ★ $\Delta h^\circ \not =\Delta h^\circ(T)$ |
$\ln \left(\dfrac{K_{p2}}{K_{p1}}\right)=-\dfrac{\Delta h^{\circ }}{k}\left(\dfrac{1}{T_2}-\dfrac{1}{T_1}\right)$ |
| Gibbs-Helmholtz equation | $\dfrac{\partial }{\partial T}\left(\dfrac{G}{T}\right)=-\dfrac{H}{T^2}$ |
| Gibbs-Helmholtz equation | $\dfrac{\partial }{\partial T}\left(\dfrac{F}{T}\right)=-\dfrac{U}{T^2}$ |
| Pressure dependence of equilibrium constant | $\dfrac{\partial (\ln K)}{\partial p} = -\dfrac{\Delta v^\circ}{kT}$ |
Ch 14 Physical Equilibrium
| Description | Equations |
|---|---|
| Equilibrium condition | $\mu_{\mathrm{vapor}} = \mu_{\mathrm{condensed}}$ |
| Chemical potential of vapor | $\mu_{\mathrm{vapor}} = kT\ln \left(\dfrac{p}{p^{\circ}_{int}}\right)$ |
| Entropy of condensed phase | $\Delta S_{\mathrm{condensed}} = 0$ |
| Internal energy of condensed phase | $\Delta U_{\mathrm{condensed}} = \frac{1}{2} Nzw_{AA}$ |
| Free energy of condensed phase | $\Delta F_{\mathrm{condensed}} = \frac{1}{2} Nzw_{AA}$ |
| Chemical potential of condensed phase | $\mu_{\mathrm{condensed}} = \left(\dfrac{\partial F}{\partial N}\right)_{T,V} = \dfrac{1}{2}zw_{AA}$ |
| Description | Equations |
|---|---|
| Equilibrium vapor pressure | $p = p^{\circ}_{int} \exp \left(\dfrac{zw_{AA}}{2kT}\right)$ |
| Clausius-Clapyeron equation | $\ln \left(\dfrac{p_2^{\text{sat}}}{p_1^{\text{sat}}}\right)=-\dfrac{\Delta h}{R}\left(\dfrac{1}{T_2}-\dfrac{1}{T_1}\right)$ |
| Enthalpy of vaporization | $\Delta h_{\mathrm{vap}} = - \frac{1}{2}zw_{AA}$ |
| Internal energy to close a cavity | $\Delta U = \frac{1}{2}zw_{AA}$ |
| Internal energy to open a cavity | $\Delta U = - \frac{1}{2}zw_{AA}$ |
| Surface tension | $\sigma = -\dfrac{w_{AA}}{2a}$ |
| Free energy of adsorption | $F_{\text{ads}} = \sigma a = -\dfrac{w_{AA}}{2a}$ |
Ch 15 Mixtures
Ideal solutions
| Description | Equations |
|---|---|
| Entropy of solution for binary systems | $\Delta S_{\text{soln}} = -Nk(x_A \ln x_A + x_B \ln x_B)$ |
| Entropy of solution for multicomponent systems | $\Delta S_{\text{soln}} = -Nk \sum x_i \ln x_i$ |
| Internal energy of solution | $\Delta U_{\text{soln}} = 0$ |
| Free energy of solution | $\Delta F_{\text{soln}} = -T \Delta S_{\text{soln}}$ |
Regular solutions
| Description | Equations |
|---|---|
| Exchange parameter (dimensionless free energy) |
$\chi_{AB} = \dfrac{z}{kT} (w_{AB}-\frac{1}{2}\left(w_{AA}+w_{BB}\right))$ |
| Exchange parameter | $\chi_{AB} = -\ln K_{\text{exch}}$ |
| Exchange parameter interpretation | $\chi_{AB} > 0$, mixing unfavorable $\chi_{AB} < 0$, mixing favorable |
| Exchange energy | $RT\chi_{AB}$ |
| Constant | $\begin{aligned}c_1 &= \chi_{AB}T \\ &= \dfrac{z}{k} (w_{AB}-\frac{1}{2}\left(w_{AA}+w_{BB}\right))\end{aligned}$ |
| Entropy of solution for binary systems | $\Delta S_{\text{soln}} = -Nk(x_A \ln x_A + x_B \ln x_B)$ |
| Entropy of solution for multicomponent systems | $\Delta S_{\text{soln}} = -Nk \sum x_i \ln x_i$ |
| Internal energy of solution of binary systems | $\Delta U_{\text{soln}} = NkT\chi_{AB}x_A x_B$ |
| Internal energy of solution of multicomponent systems | $\Delta U_{\text{soln}} = NkT \sum \chi_{ij} x_i x_j$ |
| Free energy of solution | $\Delta F_{\text{soln}} = \Delta U_{\text{soln}} - T \Delta S_{\text{soln}}$ |
| Free energy of solution of binary systems | $\Delta F_{\text{soln}} = NkT (x_A \ln x_A + x_B \ln x_B + \chi_{AB}x_A x_B)$ |
| Free energy of solution of multicomponent systems | $\Delta F_{\text{soln}} = NkT (\sum x_i \ln x_i + \sum\chi_{ij}x_i x_j)$ |
| Chemical potentials of binary systems | $\mu_A = \ln x_A + \dfrac{zw_{AA}}{2kT} + \chi_{AB} x_B^2 \\ \ \\ \mu_B = \ln x_B + \dfrac{zw_{BB}}{2kT} + \chi_{AB} x_A^2$ |
| Chemical potentials of multicomponent systems | $\begin{aligned} \mu_i & = \ln x_i + \dfrac{zw_{ii}}{kT} + \chi_{ij} (1-x_i)^2 \\ &=\mu^\circ + kT \ln(\gamma x) \end{aligned}$ |
Surface tension
| Description | Equations |
|---|---|
| Interfacial tension | $\begin{aligned} \sigma_{AB} &= \dfrac{1}{a} (w_{AB}-\frac{1}{2}\left(w_{AA}+w_{BB}\right)) \\ &= \dfrac{kT}{za} \chi_{AB} \end{aligned}$ |
| Free energy of adsorption | $F_{\text{ads}} = \sigma a = \dfrac{kT}{z} \chi_{AB}$ |
Ch 16 Solvation and Phase Transfer
Lewis/Randall rule
| Description | Equations |
|---|---|
| Notation ★ Solvent, pure limit $x_B \to 1$ |
A - non-volatile solute (e.g. $\ce{NaCl}$) B - volatile solvent (e.g. $\ce{H2O}$) |
| Lewis/Randall rule reference state | $\begin{aligned}p_B &= p_{B, int}^\circ x_B \exp \left(\chi _{AB}x_A^2+\dfrac{zw_{BB}}{2kT}\right) \\ &= p_{B}^\circ x_B \exp \left(\chi _{AB}x_A^2\right)\end{aligned}$ |
| Raoult’s law ★ Ideal solution $\chi_{AB} = 0$ |
$p_B = p_B^\circ x_B$ |
| Vapor pressure of B | $p_B^\circ = p_{B, int}^\circ \exp \left(\dfrac{zw_{BB}}{2kT}\right)$ |
Henry’s law
| Description | Equations |
|---|---|
| Notation ★ Solute, dilute limit $x_B \to 0$ |
A - non-volatile solvent (e.g. $\ce{H2O}$) B - volatile solute (e.g. $\ce{CO_2}$) |
| Henry’s law reference state | $\begin{aligned} p_B &= p_{B, int}^\circ x_B \exp \left(\chi_{AB} + \dfrac{aw_{BB}}{2kT}\right) \\ &= p_{B, int}^\circ x_B \exp \left(w_{AB} - \dfrac{w_{AA}}{2}\right) \\ &=\mathcal{H}_B x_B \end{aligned}$ |
| Henry’s constant | $\begin{aligned}\mathcal{H}_B &= p_{B, int}^\circ \exp\left(\dfrac{z}{kT}w_{AB} - \dfrac{w_{AA}}{2}\right) \\ &= p_{B, int}^\circ \exp\left(\dfrac{\Delta h^\circ_{\mathrm{soln}}}{kT}\right)\end{aligned}$ |
| Enthalpy of solution | $\Delta h^\circ_{\mathrm{soln}} = z\left(w_{AB} - \dfrac{w_{AA}}{2}\right)$ |
Activity coefficient
| Description | Equations |
|---|---|
| Standard state chemical potential | $\Delta \mu_B^\circ = \mu_B^\circ(\text{liquid}) - \mu_B^\circ(\text{gas})$ |
| Activity coefficient | $\gamma_B = \dfrac{p_B}{x_B}\exp\left(-\dfrac{\Delta \mu_B^\circ}{kT}\right)$ |
| Activity coefficient in Lewis/Randall solvent convection | $\gamma_B = \exp[\chi_{AB} (1 - x_B)^2]$ |
| Activity coefficient in Henry’s solute convection | $\gamma_B = \exp[\chi_{AB} x_B (x_B - 2)]$ |
Colligative properties
| Description | Equations |
|---|---|
| Boiling point elevation | $\Delta T_b = \dfrac{RT_b^2 x_A}{\Delta h_{\text{vap}}^\circ}$ |
| Freezing point depression | $\Delta T_f = \dfrac{RT_f^2 x_A}{\Delta h_{\text{fus}}^\circ}$ |
| Osmotic pressure | $\pi = \dfrac{RTx_A}{v_B} = RTc_A$ |
Solute partition
| Description | Equations |
|---|---|
| Notation | A - immiscible solvent B - immiscible solvent s - solute |
| Partition coefficient from solvent A to B | $K_A^B = \dfrac{x_{sB}}{x_{sA}}$ |
| Free energy of transfer | $\Delta \mu^\circ = \mu_s^\circ(B) - \mu_s^\circ(A)$ |
| Statistical mechanical interpretation | $\ln K_A^B = \chi_{sA} (1 - x_{sA})^2 - \chi_{sB} (1 - x_{sB})^2$ |
| Thermodynamical interpretation | $\ln K_A^B = - \dfrac{\mu_s^\circ(B) - \mu_s^\circ(A)}{kT} - \ln \left(\dfrac{\gamma_{sB}}{\gamma_{sA}}\right)$ |
| Partition coefficient at infinite dilution ★ Infinite dilution of solute in both phases $x_{sa} \ll 1$, $x_{sB} \ll 1$, $\gamma_{sa} \to 1$, $\gamma_{sB} \to 1$ |
$\begin{aligned}\ln K_A^B &= - \dfrac{\mu_s^\circ(B) - \mu_s^\circ(A)}{kT} \\ &= \chi_{sA} - \chi_{sB} \end{aligned}$ |
Ch 25 Phase Transitions
| Description | Equations |
|---|---|
| Fractions | $f^\alpha = \dfrac{n^\alpha}{n}$ |
| Lever rule | $f' = \dfrac{x'' - x_0}{x'' - x'} \\ f'' = \dfrac{x_0 - x'}{x'' - x'}$ |
| Lever rule | $v_A = f^\alpha v_A^\alpha + f^\beta v_A^\beta$ |
| Binodal curve | $\dfrac{\partial (\Delta F_{\text{mix}})}{\partial x} = 0 = NkT\left[\ln \left(\dfrac{x}{1-x}\right)+\chi _{AB}\left(1-2x\right)\right]$ |
| Binodal curve ★ Dilute solute $x' \ll 1$, large $\chi_{AB}$ |
$\chi_{AB} = -\ln x'$ |
| Spinodal curve | $\dfrac{\partial^2 F}{\partial x^2} = 0 = NkT\left[\frac{1}{x}+\frac{1}{1-x}-2\chi _{AB}\right]$ |
| Spinodal curve ★ Dilute solute $x' \ll 1$, large $\chi_{AB}$ |
$x' = \dfrac{1}{2\chi_{AB} - 1}$ |
| Critical point | $\dfrac{\partial^3 F}{\partial x^2} = 0$ |
| Critical composition | $x_c = \dfrac{1}{2}$ |
| Critical exchange parameter | $\chi_c = 2$ |
| Critical exchange temperature | $T_c = \dfrac{c_1}{2}$ |
| van der Waals EOS | $\left(p + \dfrac{a}{v^2}\right)(v-b) = RT$ |
| Reduced form of van der Waals EOS | $\left(p_r + \dfrac{3}{v_r^2}\right)\left(v_r - \dfrac{1}{3}\right) = \dfrac{8}{3}T_r$ |