CHEM E 325 Energy and Entropy

2021-12-23

Contents

Thermodynamic Properties and Data

Description	Equations
Pressure (kinetic theory)	$P = \dfrac{F}{A}$
Ideal gas law	$PV = nRT \newline Pv = RT$
Quality	$q = \dfrac{n_v}{n_l + n_v}$
Fraction	$x_A = f^\alpha x_A^\alpha + (1-f^\alpha)x_A^\beta$
Lever rule for intensive properties	$v_{\text{total}} = v_v q + v_l (1-q)$
Gibbs phase rule	$\mathcal{F} = 2 + c - p - r$
Mole fraction	$x_i = \dfrac{n_i}{\sum_j n_j}$

Intermolecular Interactions

Intermolecular potentials

Description	Equations
Keesom potential (dipole-dipole)	$\Gamma_{ij} = -\dfrac{2}{3}\dfrac{\mu_i^2 \mu_j^2}{r_{ij}^6 k_B T}$
Debye potential (dipole-induced dipole)	$\Gamma_{ij} = -\dfrac{\alpha_i \mu_j^2}{r_{ij}^6}$
London dispersion potential (induced dipole-induced dipole)	$\Gamma_{ij} = -\dfrac{3}{2}\dfrac{\alpha_i \alpha_j}{r_{ij}^6}\dfrac{1}{\frac{1}{I_i} + \frac{1}{I_j}}$
Lennard-Jones potential	$\Gamma = 4\varepsilon \left[ \left(\dfrac{\sigma}{r}\right)^{12} - \left(\dfrac{\sigma}{r}\right)^{6} \right]$
Equilibrium intermolecular distance	$r(\Gamma_{\min}) = r(\varepsilon) = 2^{1/6}\sigma = 1.12 \sigma$

Molecular dynamics simulation

Description	Equations
Nondimensionalized distance	$“r” = \dfrac{r}{\sigma}$
Nondimensionalized temperature	$“T” = \dfrac{k_BT}{\varepsilon}$
Nondimensionalized pressure	$“P” = \dfrac{\sigma^3}{\varepsilon}P$
Nondimensionalized energy	$“E” = \dfrac{U}{\varepsilon}$
Nondimensionalized time	$“t” = \dfrac{t}{\sigma \sqrt{m/\varepsilon}}$
Kinetic energy and temperature	$\mathrm{KE} = nT$
Ideal gas pair potential energy	$\mathrm{PE} = 0$
Condensed phase interaction potential energy (with normalized energy unit of $\varepsilon$ )	$\mathrm{PE} = -N_{\text{inter}}$
Amount of interactions	$N_{\text{inter}} = \frac{1}{2}$ (# molecules)(# neighbors)

Equation of States

van der Waals EOS

Description	Equations
van der Waals EOS in terms of $P$	$P = \dfrac{RT}{v-b} - \dfrac{a}{v^2}$
van der Waals EOS in terms of $v$	$v^3 - \left(\dfrac{RT}{v-b}\right) v^2 + \dfrac{a}{P} v - \dfrac{a}{P}b = 0$
van der Waals parameter $a$	$a = \dfrac{27}{64}\dfrac{(RT_c)^2}{P_c}$
van der Waals parameter $b$	$b = \dfrac{RT_c}{8P_c}$
Molar potential energy	$e_p = -\dfrac{a}{v}$
Pressure at zero kinetic energy	$P = - \dfrac{a}{v^2}$
Reduced temperature	$T_r = \dfrac{T}{T_c}$
Reduced pressure	$P_r = \dfrac{P}{P_c}$
Compressibility factor	$z = \dfrac{v_{\text{real}}}{v_{\text{ideal}}} = \dfrac{Pv}{RT}$
Potential energy	$e_p = u_{\text{real}}(T, P) - u_{\text{ideal}}(T, P=0)$
Internal energy departure function	$\dfrac{e_p}{RT_c} = \dfrac{u_{\text{real}} - u_{\text{ideal}}}{RT_c}$
Internal energy departure function in van der Waals EOS	$\dfrac{e_p}{RT_c} = -\dfrac{27 P_r}{64T_r z}$

Lee-Kesler EOS

Description	Equations
Lee-Kesler compressibility factor	$z = z^{(0)} + \omega z^{(1)}$
Acentric factor	$\omega = -1-\log_{10}(P^{\text{sat}}_r(T_r=0.7))$
General departure function in Lee-Kesler EOS	$\text{dep} = \text{dep}^{(0)} + \omega \ \text{dep}^{(1)}$
Internal energy and enthalpy departure function	$\dfrac{u_{\text{real}} - u_{\text{ideal}}}{RT_c} = \dfrac{h_{\text{real}} - h_{\text{ideal}}}{RT_c} + T_r (1-z)$

First Law of Thermodynamics

System Type	First Law of Thermodynamics
Isolated system	$\Delta U = 0$
Closed system	$\Delta U = Q + W$
Open system	$\dfrac{dU}{dt} = \sum\limits_{\text{in}} \dot{n}_i h_i - \sum\limits_{\text{out}} \dot{n}_i h_i + \sum \dot{Q_i} + \dot{W_s}$
Open system in steady state	$0 = \sum\limits_{\text{in}} \dot{n}_i h_i - \sum\limits_{\text{out}} \dot{n}_i h_i + \sum \dot{Q_i} + \dot{W_s}$

Description	Equations
Work	$W = \displaystyle\int P \ dV$
Enthalpy	$H = U + PV$
Constant volume molar heat capacity	$c_v = \left(\dfrac{\partial u}{\partial T}\right)_v$
Constant pressure molar heat capacity	$c_P = \left(\dfrac{\partial h}{\partial T}\right)_P$
Relationship between molar heat capacities	$c_P = c_v + R$

Second Law of Thermodynamics

System Type	Second Law of Thermodynamics
Isolated system $S_{\text{gen}} \ge 0$	$\Delta S = S_{\text{gen}}$
Closed system $S_{\text{gen}} \ge 0$	$\Delta S = \displaystyle\int \dfrac{\delta Q}{T} + S_{\text{gen}}$
Open system $\dot{S}_{\text{gen}} \ge 0$	$\dfrac{dS}{dt} = \sum\limits_{\text{in}} \dot{n}_i s_i - \sum\limits_{\text{out}} \dot{n}_i s_i + \sum \dfrac{\dot{Q_i}}{T_i} + \dot{S}_{\text{gen}}$
Open system in steady state $\dot{S}_{\text{gen}} \ge 0$	$0 = \sum\limits_{\text{in}} \dot{n}_i s_i - \sum\limits_{\text{out}} \dot{n}_i s_i + \sum \dfrac{\dot{Q_i}}{T_i} + \dot{S}_{\text{gen}}$

Counting configurations

Description	Equations
Ergotic hypothesis	$\lvert f \rvert = \langle f \rangle$
Equal probability postulate	$P_j = \dfrac{1}{\Omega}$
Entropy	$S = k_B \ln\Omega$
Permutability $N_A$ distinguishable particles in $N$ sites	$\Pi = \dfrac{N!}{(N-N_A)!}$
Multiplicity $N_A$ indistinguishable particles in $N$ sites	$\Omega = \dfrac{N!}{N_A!(N-N_A)!}$
Multiplicity $N_A, N_B, …$ indistinguishable particles in $N$ sites	$\Omega = \dfrac{N!}{N_A!N_B!N_C! \cdots}$
Stirling approximation	$\lim\limits_{a \to\infty} \ln(a!) = a \ln(a) - a$
Entropy of $N$ sites with fraction $x$ activated	$S = k_BN[-x\ln(x) - (1-x)\ln(1-x)]$

Fundamental properties

Description	Equations
Molar entropy $\mathcal{C}$ = constant	$s = R\ln(vu^{3/2}) + \mathcal{C}$
Molar internal energy $\mathcal{C}$ = constant	$u = \mathcal{C}v^{-2/3}\exp\left(\dfrac{2}{3}\dfrac{s}{R}\right)$
Temperature	$T = \left(\dfrac{\partial u}{\partial s}\right)_v$
Pressure	$P = -\left(\dfrac{\partial u}{\partial v}\right)_s$
Fundamental property relation	$du = T ds - P dv$
Reversible heat at constant $v$	$\Delta u = q = \int T \ ds$
Reversible work at constant $s$	$w = - \int P \ dv$

Entropy generation

Description	Equations
Reversible process	$s_{\text{gen}} = 0$
Irreversible process (caused by temperature gradient)	$s_{\text{gen}} > 0$

First and Second Law Analysis

Isothermal/Isoenergetic process

Isoenergetic process ( $\Delta u = 0 \implies \Delta T = 0$ ) of ideal gas has similar analysis.

Description	Equations
Condition ★ Ideal gas	$\Delta T = 0$
Internal energy change	$\Delta u = 0$
Enthalpy change	$\Delta h = 0$
First law	$\Delta u = q + w = 0$
Work (changing volume)	$w = -\displaystyle\int \dfrac{RT}{v} dv = -RT\ln\left(\dfrac{v_2}{v_1}\right)$
Work (changing pressure)	$w = \displaystyle\int \dfrac{RT}{P} dP = RT\ln\left(\dfrac{P_2}{P_1}\right)$
Heat	$q = -w$
Entropy change	$\Delta s = \displaystyle\int \dfrac{\delta q}{T} = \dfrac{q}{T} = -\dfrac{w}{T}$
Entropy change (changing volume)	$\Delta s = R\ln\left(\dfrac{v_2}{v_1}\right)$
Entropy change (changing concentration)	$\Delta s = -R\ln\left(\dfrac{c_2}{c_1}\right)$
Entropy change (changing pressure)	$\Delta s = -R\ln\left(\dfrac{P_2}{P_1}\right)$

Adiabatic process

Description	Equations
Condition ★ Ideal gas	$q = 0$
First law	$\Delta u = w$
Enthalpy change	$\Delta h = \Delta u + R \Delta T$
Work (changing volume)	$w = -\displaystyle\int \dfrac{RT}{v} dv = -RT\ln\left(\dfrac{v_2}{v_1}\right)$
Work (changing pressure)	$w = \displaystyle\int \dfrac{RT}{P} dP = RT\ln\left(\dfrac{P_2}{P_1}\right)$
Entropy change	$\Delta s = 0$

Isochoric process

Description	Equations
Condition ★ Ideal gas	$\Delta v = 0$
Work	$w = 0$
Internal energy change	$\Delta u = \displaystyle\int c_v \ dT$
First law	$q = \Delta u$
Entropy change	$\Delta s = \displaystyle\int \dfrac{\delta q}{T} = \int \dfrac{du}{T} = \int \dfrac{c_v}{T} \ dT$

Isobaric process

Description	Equations
Condition ★ Ideal gas	$\Delta P = 0$
Internal energy change	$\Delta u = \displaystyle\int c_v \ dT$
Enthalpy change	$\Delta h = \displaystyle\int c_p \ dT$
Work	$w = -P\Delta v$
Heat	$q = \Delta h$
Entropy change	$\Delta s = \displaystyle\int \dfrac{\delta q}{T} = \int \dfrac{dh}{T} = \int \dfrac{c_p}{T} \ dT$

Other processes

Description	Equations
Incompressible condensed phases	$v_0$ is constant, small
Incompressible condensed phases at low pressure	$\Delta u = \Delta h = \displaystyle\int c_p \ dT$
Incompressible condensed phases at high pressure	$\Delta h = \displaystyle\int c_p \ dT + v_0(P_1 - P_0)$
Incompressible condensed phases	$\Delta s = \displaystyle\int \dfrac{c_p}{T} dT = c_p \ln\left(\dfrac{T_1}{T_2}\right)$
Phase change	$\Delta h = q \newline \Delta s = \dfrac{q}{T} = \dfrac{\Delta h}{T}$

Process Efficiency

Description	Equations
Maximum work of heat engine	$W_{\max} = Q_{\text{in}} \left(1 - \dfrac{T_c}{T_h}\right)$
Minimum work of heat heat pump	$W_{\max} = Q_{\text{out}} \left(1 - \dfrac{T_c}{T_h}\right)$
Reversible work	$W_{\text{rev}} = Q_{1} \left(1 - \dfrac{T_c}{T_h}\right)$
Carnot efficiency	$\eta = \dfrac{W_{\text{rev}}}{Q_1} = 1 - \dfrac{T_c}{T_h}$
Ideal gas entropy	$\Delta s(T, v) = \displaystyle\int \dfrac{c_v}{T} dT + R \ln\left(\dfrac{v}{v_0}\right)$
Ideal gas entropy	$\Delta s(T, P) = \displaystyle\int \dfrac{c_p}{T} dT - R \ln\left(\dfrac{P}{P_0}\right)$
Lost work	$W_{\text{lost}} = T_c s_{\text{gen}}$
Exthalpy for multi-stream	$\dot{E} = \dot{W}_{\text{rev}} \newline = \sum\limits_{i}^{\text{source}} \dot{n}_i (h_i - T_0 s_i) - \sum\limits_{i}^{\text{ground}} \dot{n}_i (h_i^\circ - T_0 s_i^\circ)$
Exthalpy for single stream	$\dot{W}_{\text{rev}} = \Delta h - T_0 \Delta s$

Phase equilibrium

Single component equilibrium

Description	Equations
Gibbs free energy (constant $T, P$ )	$G = H - TS$
Helmholtz free energy (constant $T, V$ )	$F = A = U - TS$
Entropy change of universe	$\Delta S_{\text{univ}} \ge 0$
Gibbs free energy change of spontaneous process	$\Delta G \le 0$
Helmholtz free energy change of spontaneous process	$\Delta F \le 0$
Thermal equilibrium	$T^\alpha = T^\beta$
Mechanical equilibrium	$P^\alpha = P^\beta$
Chemical equilibrium	$g^\alpha = g^\beta$
Clausius-Clapeyron equation	$\dfrac{d \ln P_{\text{sat}}}{d (1/T)} = -\dfrac{1}{R}\Delta h_{\text{vap}}(T)$
Clausius-Clapeyron equation ★ modest pressure, incompressible liquid, ideal gas, constant $\Delta h_{\text{vap}}$	$\ln\left(\dfrac{P_{\text{sat}}}{P_0}\right) = -\dfrac{\Delta h_{\text{vap}}^\circ}{R} \left(\dfrac{1}{T} - \dfrac{1}{T_0}\right)$
Antoine equation	$\ln(P_{\text{sat}}) = A - \dfrac{B}{C + T}$

Multicomponent equilibrium

Description	Equations
Partial molar properties	$\bar{x}_i = \left(\dfrac{\partial x}{\partial n_i}\right)_{\mathrm{others}, n_{j \not = i}}$
Partial molar gibbs free energy	$\bar{g}_i = \left(\dfrac{\partial G}{\partial n_i}\right)_{T, P, n_{j \not = i}}$
Partial molar gibbs free energy	$\bar{g}_i = \bar{h}_i - T\bar{s}_i$
Entropy	$S = -\left(\dfrac{\partial G}{\partial T}\right)_{P, n_{j}}$
Volume	$V = \left(\dfrac{\partial G}{\partial P}\right)_{T, n_{j}}$
Total derivative of gibbs free energy	$dG = -S \ dT + V \ dP + \sum\limits_i \bar{g}_i \ dn_i$
Chemical potential	$\mu_i = \bar{g}_i$
Chemical equilibrium	$\mu_i^\alpha = \mu_i^\beta$
Raoult’s law	$P_A = x_A^g P = x_A^l P_A^{\text{sat}}$
Condensation curve	$P = (P_A^{\text{sat}} - P_B^{\text{sat}})x_A^l + P_B^{\text{sat}}$
Boiling curve	$P = \dfrac{P_A^{\text{sat}}P_B^{\text{sat}}}{P_A^{\text{sat}} - (P_A^{\text{sat}} - P_B^{\text{sat}})x_A^g}$
Henry’s law ★ Low pressure, dilute solution	$C_A = K_H(T) P_A$
Gibbs-Duhem Equations	$\sum \mu_i \ dn_i = 0 \newline \sum n_i \ d\mu_i = 0$
Colligative property	$\mu_{\text{solvent}} = RT \ln(1 - x_A) + \mu_{\text{solvent}}^\circ$