FE Chemical cheat sheet

Material balance on a steady-state nonreacting system: Σ(mass in) = Σ(mass out); for a reacting system add generation minus consumption, and for each element the total atoms in equal total atoms out.

Ideal gas law PV = nRT with R = 8.314 J/(mol·K) = 0.08206 L·atm/(mol·K) = 10.73 psia·ft³/(lb-mol·R); use absolute temperature and absolute pressure in every gas-law calculation.

Conversion X = (moles of limiting reactant reacted)/(moles fed); for a single reaction A → products, n_A = n_A0(1 − X) and extent of reaction ξ = n_A0·X.

Equilibrium constant from Gibbs free energy: ΔG° = −RT ln K, and van't Hoff relation d(ln K)/dT = ΔH°/(RT²); K depends only on temperature for an ideal-gas reaction.

Le Chatelier rules: raising T favors the endothermic direction, raising P favors the side with fewer gas moles, and adding an inert at constant T and P shifts equilibrium toward more gas moles.

For an ideal binary vapor-liquid system, Raoult's law gives y_i·P = x_i·P_i^sat; relative volatility α_ij = P_i^sat/P_j^sat and bubble point satisfies Σ x_i·P_i^sat = P.

Liquid-phase reaction kinetics: first-order −r_A = k·C_A with C_A = C_A0·e^(−kt); second-order −r_A = k·C_A² gives 1/C_A − 1/C_A0 = kt; Arrhenius k = A·e^(−E/RT).

Batch reactor design time t = N_A0·∫dX/(−r_A·V); CSTR volume V = F_A0·X/(−r_A); PFR volume V = F_A0·∫dX/(−r_A); the CSTR requires the largest volume for positive-order kinetics.

Heat of reaction at temperature T from a reference value: ΔH_r(T) = ΔH_r(T_ref) + ∫ΔC_p dT, where ΔC_p = Σν_i·C_p,i with ν positive for products and negative for reactants.

Conduction through a plane wall: q = kA·ΔT/L; for a composite wall the resistances R = L/(kA) add in series, and cylindrical resistance is R = ln(r2/r1)/(2πkL).

Convection and overall coefficient: q = hA·ΔT and 1/(UA) = 1/(h_i·A_i) + R_wall + 1/(h_o·A_o); the controlling (smallest) coefficient dominates the overall resistance.

Log-mean temperature difference for a heat exchanger: ΔT_lm = (ΔT1 − ΔT2)/ln(ΔT1/ΔT2), with q = U·A·ΔT_lm and F correction for multipass or crossflow arrangements.

Effectiveness-NTU method: ε = q/q_max, q_max = C_min·(T_h,in − T_c,in), NTU = UA/C_min; for a counterflow exchanger ε = [1 − exp(−NTU(1 − C_r))]/[1 − C_r·exp(−NTU(1 − C_r))].

Radiation: q = ε·σ·A·(T_s⁴ − T_surr⁴) with σ = 5.67×10⁻⁸ W/(m²·K⁴); view factors obey reciprocity A_i·F_ij = A_j·F_ji and summation Σ F_ij = 1 over an enclosure.

Fourier number Fo = α·t/L² with thermal diffusivity α = k/(ρ·C_p); Biot number Bi = hL/k compares internal conduction resistance with external convection resistance.

LMTD requires ΔT1 ≠ ΔT2; if ΔT1 = ΔT2 the log-mean equals the arithmetic mean, and if ΔT1/ΔT2 < 1 the exchanger may approach temperature cross and must be checked for feasibility.

Bernoulli mechanical energy balance for incompressible flow: P1/ρ + v1²/2 + g·z1 = P2/ρ + v2²/2 + g·z2 + h_f, with h_f the friction loss in J/kg.

Reynolds number Re = ρ·v·D/μ = D·v/ν; laminar flow below Re ≈ 2100, transitional 2100–4000, turbulent above 4000 in a circular pipe.

Darcy-Weisbach friction loss h_f = f·(L/D)·v²/2; for laminar flow f = 64/Re, and for turbulent flow use the Colebrook equation or Moody chart with roughness ε/D.

Pump power: W_pump = ρ·g·H·Q/η, where H is total head in meters, Q volumetric flow in m³/s, and η the pump efficiency; brake horsepower = Q·ΔP/η in consistent units.

NPSH available = (P_suction/ρg) + z − (P_vapor/ρg) − h_f,suction; cavitation occurs when NPSH available falls below the pump's NPSH required.

Ergun equation for packed beds: ΔP/L = 150·μ·v·(1−ε)²/(ε³·d_p²) + 1.75·ρ·v²·(1−ε)/(ε³·d_p); the first term is viscous, the second inertial.

Orifice and venturi meters: Q = C_d·A_o·√(2·ΔP/ρ) with discharge coefficient C_d near 0.6 for orifice and 0.98 for venturi; flow is proportional to the square root of the pressure drop.

Fick's first law for a binary mixture: J_A = −D_AB·dC_A/dz; steady diffusion across a stagnant film gives N_A = D_AB·(C_A1 − C_A2)/(z·x_BM) with x_BM the log-mean of the inert mole fraction.

Mass transfer coefficient correlations use dimensionless groups: Sherwood Sh = k_c·L/D_AB, Reynolds Re, and Schmidt Sc = μ/(ρ·D_AB); analogous to heat transfer with Nu, Re, and Pr.

Two-film theory: overall coefficient 1/K_G = 1/k_G + H/k_L, where H is the Henry's law constant; the more soluble gas has the liquid-film resistance dominate.

Henry's law p_A = H·x_A for dilute dissolved gas; the partition ratio m = y/x = H/P makes the equilibrium line y = m·x for absorption and stripping calculations.

Packed absorption column height Z = H_OG·N_OG, where N_OG is the number of transfer units and H_OG = G/(K_y·a) is the height of an overall gas-phase transfer unit.

Fick's second law ∂C_A/∂t = D_AB·∂²C_A/∂z²; unsteady diffusion in a slab has a characteristic time τ ≈ L²/D_AB, and the penetration depth scales as √(D·t).

Drying and humidification use humidity ratio Y = 0.622·p_w/(P − p_w), percentage humidity, and the adiabatic saturation temperature; the wet-bulb temperature approximates the adiabatic saturation temperature for air-water.

Distillation minimum stages from the Fenske equation: N_min = ln[(x_D/(1−x_D))·((1−x_B)/x_B)]/ln α, applied at total reflux with constant relative volatility.

Minimum reflux from Underwood equations: Σ α_i·x_iF/(α_i − θ) = 1 − q and Σ α_i·x_iD/(α_i − θ) = R_min + 1; Gilliland correlation converts N_min and R_min to actual stages and reflux.

McCabe-Thiele construction uses the operating lines y = (R/(R+1))·x + x_D/(R+1) for the rectifying section and the q-line with slope q/(q−1) to locate the feed stage.

Kremser equation for absorption or extraction with constant factors: N = ln[(1 − 1/A)·((y_in − m·x_in)/(y_out − m·x_in)) + 1/A]/ln A, where A = L/(m·V) is the absorption factor.

Leaching and extraction stage calculations use the same Kremser or graphical triangular approach; the solvent-to-feed ratio sets the operating line slope and the number of equilibrium stages.

Relative volatility α = (y_A/x_A)/(y_B/x_B) = K_A/K_B; α close to 1 requires many stages and high reflux, while α large means easy separation.

Azeotropes occur when the equilibrium curve crosses the diagonal (y = x); simple distillation cannot cross the azeotrope, so pressure-swing or extractive distillation is required.

Control valve sizing uses the liquid sizing coefficient C_v: Q = C_v·√(ΔP/SG), with Q in gpm, ΔP in psi, and SG relative to water; gas flow uses the expansion factor and absolute pressures.

PID controller output: u(t) = K_c·e(t) + (K_c/τ_I)·∫e dt + K_c·τ_D·de/dt; proportional band PB = 100/K_c percent, and integral action removes offset.

First-order process transfer function G = K/(τs + 1); a step input gives 63.2% of final response at t = τ and 95% at 3τ; dead time appears as e^(−θs).

Second-order underdamped response: natural frequency ω_n and damping ratio ζ; overshoot depends only on ζ, with ζ < 1 producing oscillation and ζ ≥ 1 no overshoot.

Closed-loop stability by Routh-Hurwitz on the characteristic equation 1 + G_c·G_p·G_v·G_m = 0; roots in the right half plane indicate instability, and Bode gain margin and phase margin quantify robustness.

Ziegler-Nichols tuning from the ultimate gain K_u and ultimate period P_u: PID settings K_c = 0.6·K_u, τ_I = P_u/2, τ_D = P_u/8 (open-loop reaction-curve method uses L and T).

Process safety: pressure relief device area from API 520 sizing Q = C·A·√(P·ΔP) style relations; relief valve capacity must exceed the worst-case relieving rate, and a rupture disc is used for fast pressure rise.

Economics: cash flow = (revenue − operating cost − depreciation)·(1 − t) + depreciation, where t is the tax rate; depreciation is a non-cash expense that reduces taxable income.

Straight-line depreciation = (installed cost − salvage)/n; MACRS uses declining-balance rates with a half-year convention and no salvage value.

Present worth of a future amount: P = F/(1 + i)^n; for an annuity P = A·[(1 + i)^n − 1]/[i·(1 + i)^n]; capital recovery factor is the reciprocal.

Payback period = fixed capital investment / annual after-tax cash flow (simple, ignoring time value); discounted payback uses present worth of cash flows.

Net present value = Σ cash flow_n/(1 + i)^n − initial investment; a positive NPV at the minimum acceptable rate of return means the project is acceptable.

Break-even point occurs where total revenue equals total cost; contribution margin per unit = selling price − variable cost per unit, and break-even units = fixed cost / contribution margin.

Cost indices scale equipment cost: C_2 = C_1·(I_2/I_1), and the six-tenths rule scales capacity C_2 = C_1·(Q_2/Q_1)^0.6 for equipment of the same type.

Reaction equilibrium and phase equilibrium are independent requirements; a reacting system reaches equilibrium when the reaction Gibbs energy ΔG = ΔG° + RT·ln Q equals zero, giving Q = K.

Phase rule F = C − P + 2 for a nonreacting system; for a reacting system F = C − P + 2 − r, where r is the number of independent reactions.

For a binary VLE at low pressure, dew point satisfies Σ y_i/ K_i = 1 and bubble point satisfies Σ x_i·K_i = 1 with K_i = P_i^sat/P; the dew-point temperature is higher than the bubble-point temperature at fixed pressure.

Fugacity and activity: a_i = γ_i·x_i, and the equilibrium constant in terms of activity is K = Π a_i^ν_i; for ideal solutions γ = 1 and Raoult's law applies.

Chemical potential μ_i = μ_i° + RT·ln a_i; at equilibrium the chemical potential of each species is equal in every phase present.

Solubility of a sparingly soluble salt: K_sp = [cation]^m·[anion]^n; precipitation begins when the ion product exceeds K_sp.

Partial molar property: V̄_i = (∂V/∂n_i) at constant T, P, and other moles; the Gibbs-Duhem equation Σ x_i·dμ_i = 0 at constant T and P constrains activity coefficient changes.

Material balance on a steady-state nonreacting system: Σ(mass in) = Σ(mass out); for a reacting system add generation minus consumption, and for each element the total atoms in equal total atoms out.

Ideal gas law PV = nRT with R = 8.314 J/(mol·K) = 0.08206 L·atm/(mol·K) = 10.73 psia·ft³/(lb-mol·R); use absolute temperature and absolute pressure in every gas-law calculation.

Conversion X = (moles of limiting reactant reacted)/(moles fed); for a single reaction A → products, n_A = n_A0(1 − X) and extent of reaction ξ = n_A0·X.

Equilibrium constant from Gibbs free energy: ΔG° = −RT ln K, and van't Hoff relation d(ln K)/dT = ΔH°/(RT²); K depends only on temperature for an ideal-gas reaction.

Le Chatelier rules: raising T favors the endothermic direction, raising P favors the side with fewer gas moles, and adding an inert at constant T and P shifts equilibrium toward more gas moles.

For an ideal binary vapor-liquid system, Raoult's law gives y_i·P = x_i·P_i^sat; relative volatility α_ij = P_i^sat/P_j^sat and bubble point satisfies Σ x_i·P_i^sat = P.

Liquid-phase reaction kinetics: first-order −r_A = k·C_A with C_A = C_A0·e^(−kt); second-order −r_A = k·C_A² gives 1/C_A − 1/C_A0 = kt; Arrhenius k = A·e^(−E/RT).

Batch reactor design time t = N_A0·∫dX/(−r_A·V); CSTR volume V = F_A0·X/(−r_A); PFR volume V = F_A0·∫dX/(−r_A); the CSTR requires the largest volume for positive-order kinetics.

Heat of reaction at temperature T from a reference value: ΔH_r(T) = ΔH_r(T_ref) + ∫ΔC_p dT, where ΔC_p = Σν_i·C_p,i with ν positive for products and negative for reactants.