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 |
$\mu_{\text{solvent}} = RT \ln(1 - x_A) + \mu_{\text{solvent}}^\circ$ |