FE Chemical Formula Sheet: Key Equations by Topic

Mathematics
6% of the exam

Analytic geometry, logarithms, and trigonometry, single-variable and integral calculus, ordinary and partial differential equations and Laplace transforms, matrix algebra and systems of equations, and numerical methods including error propagation, Taylor series, curve fitting, and Newton-Raphson.

Numerical Methods, Error and Significant Figures

Newton-Raphson roots, trapezoidal and Simpson integration, Taylor truncation, relative-error propagation, and significant-figure discipline for FE-Chem.

Newton-Raphson iteration: $x_{j+1} = x_j - \dfrac{f(x_j)}{f'(x_j)}$; Next root estimate; $f'$ is the derivative at the current guess. Quadratic convergence near a simple root; needs $f'\neq 0$ and a good seed.
Taylor series about a: $f(x) = \sum_{k=0}^{\infty}\dfrac{f^{(k)}(a)}{k!}(x-a)^k$; Basis of truncation-error analysis. Truncating after the linear term gives the Newton tangent. $a=0$ is the Maclaurin series.
Trapezoidal rule (n strips): $\int_a^b f\,dx \approx \dfrac{\Delta x}{2}\Big[f_0 + 2f_1 + 2f_2 + \cdots + 2f_{n-1} + f_n\Big]$; Straight-chord approximation; error $\propto \Delta x^2$. $\Delta x=(b-a)/n$.
Simpson's 1/3 rule: $\int_a^b f\,dx \approx \dfrac{\Delta x}{3}\Big[f_0 + 4f_1 + 2f_2 + \cdots + 4f_{n-1} + f_n\Big]$; Parabolic fit; exact for cubics, error $\propto \Delta x^4$. $n$ must be even (odd number of points).
Step width: $\Delta x = \dfrac{b-a}{n}$; Uniform spacing for $n$ subintervals between limits $a$ and $b$.
Relative error: $\varepsilon_{rel} = \dfrac{|x_{approx}-x_{true}|}{|x_{true}|}$; Dimensionless; multiply by 100 for percent. Use the previous iterate as the reference when the true value is unknown.
Error propagation (products): $\left|\dfrac{\Delta y}{y}\right| = \sum_i |a_i|\left|\dfrac{\Delta x_i}{x_i}\right|$; Worst-case relative error for $y=\prod x_i^{a_i}$; fractional errors add, weighted by exponents.
Error propagation (RSS): $\dfrac{\Delta y}{y} = \sqrt{\sum_i a_i^2\left(\dfrac{\Delta x_i}{x_i}\right)^2}$; Probable error when inputs are independent and random; always smaller than the linear sum.
Error propagation (sums): $y = \sum_i c_i x_i \;\Rightarrow\; \Delta y = \sqrt{\sum_i c_i^2\,(\Delta x_i)^2}$; For additive combinations the absolute errors (not relative) combine.

Where it lives: NCEES FE Reference Handbook — Mathematics (Numerical Methods, Numerical Integration, Taylor's Series) · NCEES FE Reference Handbook — Units and Conversion Factors (Significant Figures, p.2) · NCEES FE Reference Handbook — Engineering Probability and Statistics (Measurement Uncertainty / Propagation of Error, RSS form) · Chapra & Canale, Numerical Methods for Engineers

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Probability and Statistics
4% of the exam

Discrete, continuous, normal, and binomial distributions, expected value in decision making, hypothesis testing and design of experiments (t-test, ANOVA, outlier testing), measures of central tendency and dispersion with confidence intervals, regression and curve fitting, and statistical process-control limits.

Distributions, Confidence Intervals and Control Limits

Discrete and continuous distributions, the binomial and normal, mean/median/mode and spread, the confidence interval for a mean, and X-bar/R control limits.

Sample mean: $\bar{X}=\frac{1}{n}\sum_{i=1}^{n}X_i$; Arithmetic mean (balance point) of $n$ observations; the statistic every interval and chart centers on.
Sample variance and standard deviation: $s^2=\frac{1}{n-1}\sum_{i=1}^{n}\left(X_i-\bar{X}\right)^2,\qquad s=\sqrt{s^2}$; Spread of a sample; divide by $n-1$ (use $N$ for a full population). Units of $s$ match the data; $s^2$ is squared units.
Binomial mass function: $P(x)=C(n,x)\,p^{x}(1-p)^{\,n-x}$; Probability of exactly $x$ successes in $n$ independent trials, each with success probability $p$. $C(n,x)=n!/[x!(n-x)!]$.
Binomial mean and variance: $\mu=np,\qquad \sigma^{2}=npq,\quad q=1-p$; Expected count and its variance (a count, not a fraction). Standard deviation $\sigma=\sqrt{npq}$.
Normal probability density: $f(x)=\frac{1}{\sigma\sqrt{2\pi}}\,e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}}$; Gaussian curve, mean $\mu$, standard deviation $\sigma$. Never integrated by hand on the FE; standardize instead.
Standard-normal transform: $z=\frac{x-\mu}{\sigma}$; Converts any normal value to the unit normal so the handbook table gives $F(z)$ (left area), $R(z)$ (right area), $W(z)$ (within $\pm z$).
Within-band probability: $P(-z\le Z\le z)=W(z)=2F(z)-1$; Fraction of a normal population inside $\mu\pm z\sigma$; e.g. $W(1.96)\approx0.95$.
Standard error of the mean: $\sigma_{\bar X}=\frac{\sigma}{\sqrt{n}}\;\;(\text{or }s/\sqrt{n})$; Spread of the sample mean; shrinks only as $\sqrt{n}$. Drives both the confidence interval and the X-bar chart width.
Confidence interval, sigma known: $\bar{X}\pm z_{\alpha/2}\,\frac{\sigma}{\sqrt{n}}$; Interval for $\mu$ when the population $\sigma$ is known; $z_{0.025}=1.960$ for 95%, $1.645$ for 90%.
Confidence interval, sigma unknown: $\bar{X}\pm t_{\alpha/2,\,n-1}\,\frac{s}{\sqrt{n}}$; Interval for $\mu$ using sample $s$; $t$ has $n-1$ degrees of freedom and is wider than $z$ for small samples.
X-bar chart limits: $\text{UCL}=\overline{\overline{X}}+A_2\bar{R},\qquad \text{LCL}=\overline{\overline{X}}-A_2\bar{R}$; Three-sigma limits on subgroup averages; $\overline{\overline{X}}$ is the grand mean, $\bar{R}$ the mean range, $A_2$ from the handbook table for subgroup size $n$.
R chart limits and process sigma: $\text{UCL}_R=D_4\bar{R},\quad \text{LCL}_R=D_3\bar{R},\qquad \hat{\sigma}=\frac{\bar{R}}{d_2}$; Limits on subgroup range; $D_3,D_4,d_2$ from the table. $\hat{\sigma}=\bar{R}/d_2$ estimates the process standard deviation.

Where it lives: NCEES FE Reference Handbook — Engineering Probability and Statistics · NCEES FE Reference Handbook — Statistical Quality Control (Average and Range Charts) · Montgomery & Runger, Applied Statistics and Probability for Engineers

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Engineering Sciences
4% of the exam

Basic dynamics (friction, force, mass, acceleration, momentum), work, energy, and power for particles and rigid bodies, and electricity fundamentals (charge, current, voltage, power, Ohm's law, and Kirchhoff's laws).

Newtonian Dynamics, Work, Energy and Power

Newton's second law with friction, the work-energy theorem, and impulse-momentum for particles — the three bookkeeping methods that solve almost every FE dynamics question.

Newton's second law (particle): $\sum \mathbf{F} = m\mathbf{a} = m\dfrac{d\mathbf{v}}{dt}$; Net external force equals mass times acceleration. SI: $F$ in N, $m$ in kg, $a$ in m/s$^2$. Apply per axis from a free-body diagram.
Newton's law in USCS (gc form): $F = \dfrac{m\,a}{g_c}, \quad g_c = 32.174\ \tfrac{\text{lbm}\cdot\text{ft}}{\text{lbf}\cdot\text{s}^2}$; Required whenever mass is in lbm and force in lbf; $g_c$ reconciles the two. Weight $W = mg/g_c$.
Coulomb friction: $F_f \le \mu_s N, \qquad F_f = \mu_k N \ (\text{sliding})$; Friction opposes relative motion. $N$ = actual normal force ($mg\cos\theta$ on an incline). $\mu_s,\mu_k$ dimensionless.
Constant-acceleration kinematics: $v = v_0 + at, \quad v^2 = v_0^2 + 2a(x-x_0)$; Links velocity, acceleration, time, and displacement for constant $a$. Use with $\sum F = ma$ to bridge force and motion.
Kinetic energy: $T = \tfrac{1}{2}mv^{2}$; Energy of motion of a particle. $v$ = speed (m/s), result in J. Note $v$ is squared — never linear.
Work of a force: $U = \int \mathbf{F}\cdot d\mathbf{r} = Fs\cos\theta\ (\text{constant } F)$; $\theta$ = angle between force and displacement; forces perpendicular to motion do zero work.
Work-energy theorem: $U_{1\to 2} = \tfrac{1}{2}mv_2^{2} - \tfrac{1}{2}mv_1^{2}$; Net work equals change in kinetic energy. Time-free — ideal for force-and-distance to speed problems.
Conservation of energy: $T_1 + V_1 + U_{1\\to 2}^{nc} = T_2 + V_2$; $V = mgh + \\tfrac{1}{2}ks^2$. $U_{1\\to 2}^{nc}$ = work of nonconservative forces only (negative for friction; zero for conservative systems). Distinct from the work-energy theorem $U_{1\\to 2}$, which is the net work of all forces — do not equate the two.
Linear impulse-momentum: $m\mathbf{v}_2 = m\mathbf{v}_1 + \int_{t_1}^{t_2}\mathbf{F}\,dt$; Impulse of net force equals change in momentum. For constant $F$: $F\,\Delta t = m\,\Delta v$.
Conservation of momentum (impact): $m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2'$; Holds when external impulse is negligible during impact. KE conserved only if $e=1$.
Coefficient of restitution: $e = \dfrac{(v_2' - v_1')_n}{(v_1 - v_2)_n}, \ 0\le e\le 1$; $e=1$ perfectly elastic, $e=0$ perfectly plastic (bodies stick). Applied along the line of impact.
Power and efficiency: $P = \mathbf{F}\cdot\mathbf{v}, \qquad \eta = \dfrac{P_{out}}{P_{in}}$; Power in W (or ft·lbf/s); $1\,\text{hp}=746\,\text{W}$. Force and velocity must be collinear.

Where it lives: NCEES FE Reference Handbook — Dynamics · NCEES FE Reference Handbook — Dynamics (Laws of Friction) · Hibbeler, Engineering Mechanics: Dynamics

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Materials Science
4% of the exam

Chemical, electrical, mechanical, and physical properties and the effects of temperature, pressure, stress, and strain, material types and compatibilities for ferrous, nonferrous, and engineered materials, corrosion mechanisms and control, and polymers, ceramics, and composites.

Material Properties and Corrosion Control

Read the tensile curve, apply Hooke's law and the elastic constants, account for temperature and time, and set up the electrochemical cell that drives corrosion.

Engineering stress and strain: $\sigma = \dfrac{F}{A_0}, \qquad \varepsilon = \dfrac{\Delta L}{L_0}$; Based on ORIGINAL area $A_0$ and length $L_0$. $\sigma$ in Pa (or MPa), $\varepsilon$ dimensionless.
Hooke's law: $\sigma = E\,\varepsilon$; Linear elastic region only. $E$ = elastic (Young's) modulus, Pa; slope of the tensile curve's straight portion.
Axial deformation: $\delta = \dfrac{PL}{AE}$; Elastic elongation of a bar of length $L$, area $A$, modulus $E$ under axial load $P$. Units m.
Shear and bulk modulus: $G = \dfrac{E}{2(1+\nu)}, \qquad K = \dfrac{E}{3(1-2\nu)}$; Isotropic elastic links. $\nu$ = Poisson's ratio ($\approx 0.3$ for steel). Only two of $E,G,K,\nu$ are independent.
Poisson's ratio: $\nu = -\dfrac{\varepsilon_{\text{lateral}}}{\varepsilon_{\text{longitudinal}}}$; Negative ratio of transverse to axial strain. Dimensionless; $0 \le \nu \le 0.5$ for stable isotropic solids.
True stress and true strain: $\sigma_T = \dfrac{F}{A}, \qquad \varepsilon_T = \ln(1+\varepsilon)$; Use instantaneous area $A$; needed in plastic/necking analysis. Differs from engineering values beyond yield.
Thermal deformation and stress: $\delta_t = \alpha L\,(T - T_0), \qquad \sigma_{\text{thermal}} = E\,\alpha\,\Delta T$; Free expansion (first) vs fully restrained stress (second). $\alpha$ = expansion coefficient, 1/K or 1/°C.
Thermal expansion coefficient: $\alpha = \dfrac{\varepsilon}{\Delta T}$; Engineering strain per degree of temperature change. Units 1/K (equivalently 1/°C for a difference).
Fracture toughness: $K_{IC} = Y\,\sigma\sqrt{\pi a}$; Critical stress intensity; $a$ = crack length, $Y$ = geometry factor ($1$ interior, $1.1$ surface). Units MPa·m$^{1/2}$.
Hardness–strength estimate: $\text{TS (MPa)} \approx 3.5\,\text{BHN}$; Plain-carbon steel approximation (or TS(psi) ≈ 500·BHN). BHN = Brinell hardness number.
Resistivity and capacitance: $\rho = \dfrac{RA}{L}, \qquad C = \dfrac{\varepsilon A}{d}$; $\rho$ = resistivity (Ω·m); $C$ = parallel-plate capacitance, $\varepsilon=\kappa\varepsilon_0$, $\varepsilon_0=8.85\times10^{-12}\,$F/m.
Diffusion coefficient (Arrhenius): $D = D_0\,e^{-Q/(RT)}$; Temperature dependence of diffusion/creep. $Q$ = activation energy, $R=8.314\,$J/(mol·K), $T$ in K.
Galvanic cell EMF: $E_{cell} = E^\circ_{ox,\,\text{anode}} + E^\circ_{red,\,\text{cathode}}$; Positive $E_{cell}$ means the couple corrodes spontaneously. Use handbook oxidation potentials (reverse sign for the cathode).
Nernst equation (25 °C): $E = E^\circ - \dfrac{0.0592}{n}\log_{10} Q$; Concentration correction to the standard potential; $n$ = electrons transferred, $Q$ = reaction quotient.

Where it lives: NCEES FE Reference Handbook — Materials Science/Structure of Matter · NCEES FE Reference Handbook — Mechanics of Materials · NCEES FE Reference Handbook — Chemistry and Biology · Callister & Rethwisch, Materials Science and Engineering: An Introduction

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Chemistry and Biology
7% of the exam

Inorganic chemistry (molarity, normality, acids and bases, redox, solubility product, pH and pK, electrochemistry), organic chemistry (nomenclature, structure, reactions, synthesis), analytical chemistry, biochemistry and microbiology (cell function, glycolysis and the Krebs cycle, enzymes, genetics), and bioprocessing (fermentation and aerobic/anaerobic treatment).

Solutions, pH, Buffers and Electrochemistry

Concentration units, weak-acid and buffer pH, the solubility product, and the Nernst equation — the aqueous-chemistry workhorses the FE rewards for speed.

Molarity and normality: $M = \dfrac{n_{solute}}{V_{soln}(\text{L})}, \qquad N = M\,z$; $M$ in mol/L; $z$ = equivalents per mole (protons for acids, electrons for redox). Normality is solution-context dependent.
Titration equivalence: $N_1 V_1 = N_2 V_2$; Equal equivalents at the endpoint. Volumes in any consistent unit; valid only in normality, not molarity (unless 1:1).
pH and water ion product: $\mathrm{pH} = -\log_{10}[\mathrm{H^+}], \quad \mathrm{pH}+\mathrm{pOH}=14, \quad K_w = [\mathrm{H^+}][\mathrm{OH^-}] = 10^{-14}$; At $25\,^\circ\mathrm{C}$. $[\mathrm{H^+}]$ in mol/L. Acids pH < 7, bases pH > 7.
Weak-acid dissociation: $K_a = \dfrac{[\mathrm{H^+}][\mathrm{A^-}]}{[\mathrm{HA}]}, \qquad \mathrm{p}K_a = -\log_{10}K_a$; Equilibrium constant for $\mathrm{HA}\rightleftharpoons \mathrm{H^+}+\mathrm{A^-}$. Smaller $K_a$ (larger $\mathrm{p}K_a$) = weaker acid.
Weak-acid pH approximation: $[\mathrm{H^+}] \approx \sqrt{K_a\,C}$; Valid when dissociation $<5\%$ of formal concentration $C$ (mol/L). Verify the fraction $[\mathrm{H^+}]/C$ afterward.
Henderson-Hasselbalch: $\mathrm{pH} = \mathrm{p}K_a + \log_{10}\dfrac{[\mathrm{A^-}]}{[\mathrm{HA}]}$; Buffer pH from the conjugate-base to acid ratio. Independent of dilution; $\mathrm{pH}=\mathrm{p}K_a$ when the ratio is 1.
Solubility product: $\mathrm{A}_m\mathrm{B}_n \rightleftharpoons m\mathrm{A}^{n+}+n\mathrm{B}^{m-}, \quad K_{sp} = [\mathrm{A}^{n+}]^m[\mathrm{B}^{m-}]^n$; Solids and pure liquids omitted (activity = 1). Compare $Q$ to $K_{sp}$ to test precipitation.
Equilibrium constant (general): $K_{eq} = \dfrac{[\mathrm{C}]^c[\mathrm{D}]^d}{[\mathrm{A}]^a[\mathrm{B}]^b}$; For $a\mathrm{A}+b\mathrm{B}\rightleftharpoons c\mathrm{C}+d\mathrm{D}$. Brackets are activities; = molar concentration in dilute solution, partial pressure for gases, 1 for pure solids/liquids.
Standard cell potential: $E^\circ_{cell} = E^\circ_{cathode} - E^\circ_{anode}$; Both values are reduction potentials (V). $E^\circ_{cell}>0$ means spontaneous as written.
Nernst equation (25 °C): $E = E^\circ - \dfrac{0.0592}{n}\log_{10}Q$; $n$ = electrons transferred; $Q$ = reaction quotient (products over reactants). Full form: $E=E^\circ-(RT/nF)\ln Q$, $F=96{,}485\,\mathrm{C/mol}$.
Faraday's law of electrolysis: $m = \dfrac{Q\,M}{zF}$; $m$ = mass deposited (g); $Q$ = charge (C = A·s); $M$ = molar mass (g/mol); $z$ = electrons per ion; $F=96{,}485\,\mathrm{C/mol}$.

Where it lives: NCEES FE Reference Handbook — Chemistry and Biology · NCEES FE Reference Handbook — Environmental Engineering · Felder & Rousseau, Elementary Principles of Chemical Processes

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Fluid Mechanics and Dynamics
8% of the exam

Fluid properties and dimensionless numbers (Reynolds), the mechanical energy balance with pipe, valve, fitting, and packed-bed losses, the Bernoulli equation and hydrostatics, laminar and turbulent flow, flow measurement (orifices and Venturi meters), pumps, compressors, and vacuum systems, and compressible and non-Newtonian flow.

The Mechanical Energy Balance and Bernoulli

Write extended Bernoulli as heads, add pipe, valve and fitting friction losses, account for elevation, and size pump head or pressure drop with confidence.

Bernoulli equation (no friction, no machine): $\frac{P_1}{\gamma}+\frac{v_1^{2}}{2g}+z_1=\frac{P_2}{\gamma}+\frac{v_2^{2}}{2g}+z_2$; Conservation of mechanical energy per unit weight along a streamline; each term in metres (or feet). $\gamma=\rho g$.
Extended (mechanical energy) balance: $\frac{P_1}{\gamma}+\frac{v_1^{2}}{2g}+z_1+h_p=\frac{P_2}{\gamma}+\frac{v_2^{2}}{2g}+z_2+h_f$; Adds pump head $h_p$ (energy in) and total head loss $h_f$ (energy out, always positive). The form used for pump and pressure-drop sizing.
Continuity: $\dot{m}=\rho Q=\rho A v,\qquad Q=A_1 v_1=A_2 v_2$; Mass and (for constant density) volumetric flow conserved; $A=\tfrac{\pi}{4}D^2$ for round pipe. Units $\mathrm{m^3/s}$, $\mathrm{m/s}$.
Darcy-Weisbach major loss: $h_f=f\,\frac{L}{D}\,\frac{v^{2}}{2g}$; Straight-pipe friction head; $f$ from Moody chart vs $\mathrm{Re}$ and $\varepsilon/D$. $L$, $D$ in m, $v$ in m/s.
Fanning–Darcy relation: $f_{\text{Darcy}}=4\,f_{\text{Fanning}}$; Chemical-engineering charts often give Fanning $f_F$; the Darcy form needs the factor of four (or use $h_f=4f_F\tfrac{L}{D}\tfrac{v^2}{2g}$).
Minor (fitting) losses: $h_{f,\text{minor}}=\sum C\,\frac{v^{2}}{2g}$; $C$ (or $K$) is the dimensionless loss coefficient per fitting/valve/entrance/exit; sum over the run.
Hydrostatic pressure: $P_2-P_1=-\gamma(z_2-z_1)=\rho g h$; Static-fluid pressure variation with depth $h$; basis of manometers. Use absolute or gauge consistently.
Pressure drop in a straight horizontal pipe: $P_1-P_2=\gamma\,h_f=f\,\frac{L}{D}\,\frac{\rho v^{2}}{2}$; Special case with $z_1=z_2$, $v_1=v_2$, no pump; converts head loss directly to a pressure drop in Pa.
Hydraulic (fluid) power: $\dot{W}_{\text{fluid}}=\rho g Q h_p=\gamma Q h_p$; Useful power delivered to fluid by a pump of head $h_p$; in watts when $Q$ in $\mathrm{m^3/s}$, $h_p$ in m.
Brake and electrical power: $\dot{W}_{\text{brake}}=\frac{\dot{W}_{\text{fluid}}}{\eta_{\text{pump}}},\qquad \dot{W}_{\text{elec}}=\frac{\dot{W}_{\text{brake}}}{\eta_{\text{motor}}}$; Shaft power and electrical draw; efficiencies are fractions $\le 1$.
Torricelli (frictionless efflux): $v=\sqrt{2gh}$; Free jet velocity from a head $h$ when friction is negligible; the friction case puts $1+f\tfrac{L}{D}+\sum C$ in the denominator inside the root (the leading $1$ is the exit velocity head).

Where it lives: NCEES FE Reference Handbook — Fluid Mechanics · NCEES FE Reference Handbook — Fluid Mechanics (Energy Equation, Bernoulli, Darcy-Weisbach, Minor Losses) · Felder & Rousseau, Elementary Principles of Chemical Processes

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Thermodynamics
8% of the exam

Thermodynamic properties of pure components and mixtures, property data and phase diagrams (steam tables, P-h, T-s, x-y), the first and second laws, isothermal, adiabatic, and isentropic processes, power and refrigeration cycles, phase equilibrium (Raoult's law, fugacity, activity coefficients), chemical equilibrium, and heats of reaction and mixing.

First and Second Law: Processes, Cycles and Efficiency

Closed- and open-system energy balances, entropy and the second law, the isothermal/adiabatic/isentropic process family, and Carnot ceilings on efficiency and COP.

Closed-system first law: $Q - W = \Delta U + \Delta KE + \Delta PE$; Energy balance with no mass flow. $Q$ positive when added to the system, $W$ positive when done by the system; energies in kJ or Btu.
Steady-flow energy equation: $\dot{Q}_{in} - \dot{W}_{net} = \sum_e \dot{m}_e\!\left(h_e + \tfrac{V_e^2}{2} + gZ_e\right) - \sum_i \dot{m}_i\!\left(h_i + \tfrac{V_i^2}{2} + gZ_i\right)$; Open-system rate balance at steady state. $h$ = specific enthalpy (kJ/kg), $\dot m$ = mass flow (kg/s); flow work is already inside $h = u + Pv$.
Ideal-gas relations: $Pv = RT, \quad \Delta u = c_v\,\Delta T, \quad \Delta h = c_p\,\Delta T, \quad c_p - c_v = R$; $R$ = specific gas constant (kJ/kg·K). Property changes depend only on $\Delta T$ for an ideal gas, independent of path.
Isentropic (reversible adiabatic) relations: $Pv^{k} = \text{const}, \qquad \frac{T_2}{T_1} = \left(\frac{P_2}{P_1}\right)^{\frac{k-1}{k}} = \left(\frac{v_1}{v_2}\right)^{k-1}$; $k = c_p/c_v$. Use absolute $T$ and $P$. Valid for an ideal gas with constant specific heats.
Isentropic ideal-gas work: $w_{closed} = \frac{R(T_2 - T_1)}{1 - k}, \qquad w_{open} = \frac{kR(T_2 - T_1)}{1 - k}$; Closed-system (boundary) work versus open-system (shaft) work for an isentropic process; the two differ by the factor $k$.
Isothermal ideal-gas work: $w = RT\ln\frac{v_2}{v_1} = RT\ln\frac{P_1}{P_2}$; Constant-temperature ideal-gas process; for a closed system $q = w$ since $\Delta u = 0$.
Ideal-gas entropy change: $\Delta s = c_p\ln\frac{T_2}{T_1} - R\ln\frac{P_2}{P_1} = c_v\ln\frac{T_2}{T_1} + R\ln\frac{v_2}{v_1}$; Entropy change between two states of an ideal gas (kJ/kg·K). Absolute $T$ throughout.
Increase-of-entropy principle: $\Delta S_{total} = \Delta S_{sys} + \Delta S_{surr} \ge 0$; Total entropy never decreases; equality only for a reversible process. Sets the direction of spontaneous change.
Thermal efficiency: $\eta = \frac{W}{Q_H} = 1 - \frac{Q_L}{Q_H}$; Fraction of supplied heat converted to net work. Denominator is the heat supplied $Q_H$, never $W$.
Carnot efficiency: $\eta_{Carnot} = 1 - \frac{T_L}{T_H}$; Maximum efficiency between reservoirs at absolute temperatures $T_H$, $T_L$ (K or °R). Upper bound no real engine exceeds.
Coefficients of performance: $\mathrm{COP}_{ref} = \frac{Q_L}{W}, \qquad \mathrm{COP}_{HP} = \frac{Q_H}{W} = \mathrm{COP}_{ref} + 1$; Cooling values $Q_L$; heating values $Q_H$. The two differ by exactly one because $Q_H = Q_L + W$.
Carnot COP ceilings: $\mathrm{COP}_{HP,\max} = \frac{T_H}{T_H - T_L}, \qquad \mathrm{COP}_{ref,\max} = \frac{T_L}{T_H - T_L}$; Maximum COP of a reversed-Carnot cycle between the two reservoirs; absolute temperatures only.
Reservoir entropy change: $\Delta S_{res} = \frac{Q}{T_{res}}$; Entropy change of a thermal reservoir exchanging heat $Q$ at constant absolute temperature $T_{res}$ (kJ/K).

Where it lives: NCEES FE Reference Handbook — Thermodynamics · Smith, Van Ness, Abbott & Swihart, Introduction to Chemical Engineering Thermodynamics · Moran, Shapiro, Boettner & Bailey, Fundamentals of Engineering Thermodynamics

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Material and Energy Balances
10% of the exam

Steady-state and unsteady-state mass balances, steady-state and unsteady-state energy balances, recycle and bypass processes, and reactive systems including combustion, with conversion, yield, and extent of reaction.

Material Balances and Degrees of Freedom

The steady-state total and component mass balance, degrees-of-freedom analysis, tie components, and a disciplined recipe for solving multi-unit flowsheets.

General material balance: $\dfrac{dm}{dt} = \dot{m}_{in} - \dot{m}_{out} + \dot{m}_{gen}$; Accumulation = in − out + generation. Per unit time for a defined control volume; $\dot m$ in kg/s (or mol/s for a species).
Steady-state total balance: $\sum \dot{m}_{in} = \sum \dot{m}_{out}$; Total mass is conserved even with reaction; accumulation and net generation of total mass are zero.
Component balance (no reaction): $\sum_{\text{in}} F_k\,x_{j,k} = \sum_{\text{out}} P_l\,x_{j,l}$; For each species $j$: mass (or mole) of $j$ in = out. $x_{j,k}$ is the fraction of $j$ in stream $k$.
Composition closure: $\sum_j x_j = 1$; Mass or mole fractions in any single stream sum to unity — one usable equation per stream.
Tie-component relation: $F\,x_{\text{tie},F} = P\,x_{\text{tie},P}$; An inert that enters and leaves in one stream each directly links the two stream flows $F$ and $P$.
Degrees of freedom: $DOF = N_{\text{unknowns}} - N_{\text{indep. equations}}$; $DOF=0$ solvable; $>0$ underspecified (set a basis); $<0$ over-specified. Independent equations = balances + specs + relations.
Independent balances: $N_{\text{indep. balances}} = S$; For $S$ species, only $S$ of the (total + $S$ component) balances are independent.
Mass–mole conversion: $n_j = \dfrac{m_j}{M_j}, \qquad y_j = \dfrac{n_j}{\sum_k n_k}$; Convert a stream between mass basis ($m_j$, mass fraction) and mole basis ($n_j$, mole fraction $y_j$) using molar masses $M_j$.

Where it lives: NCEES FE Reference Handbook — Chemical Engineering · NCEES FE Reference Handbook — Thermodynamics · Felder & Rousseau, Elementary Principles of Chemical Processes

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Heat Transfer
8% of the exam

Conductive, convective (natural and forced), and radiation heat transfer, overall, local, and fouling heat-transfer coefficients, and heat-transfer equipment design (double-pipe and shell-and-tube exchangers, log-mean temperature difference, the effectiveness-NTU method, and flow configuration).

Conduction, Thermal Resistance and the Overall U

Fourier conduction through walls and cylinders, the series-parallel resistance network, fouling, and the overall heat-transfer coefficient U.

Fourier's law (1-D): $\dot{Q} = -kA\,\dfrac{dT}{dx}$; Conduction heat rate (W). $k$ = thermal conductivity [W/(m·K)], $A$ = area normal to flow (m²); minus sign = flow down the gradient.
Plane-wall conduction: $\dot{Q} = \dfrac{kA}{L}\,(T_1 - T_2)$; Steady 1-D slab of thickness $L$ (m); $T_1>T_2$ are the two face temperatures (K or °C — a difference is the same in both).
Thermal-resistance (Ohm) form: $\dot{Q} = \dfrac{\Delta T_{\text{total}}}{\sum R}$; Heat rate equals total temperature drop over the sum of series resistances (K/W). The same $\dot{Q}$ flows through each.
Plane-wall resistance: $R = \dfrac{L}{kA}$; Conduction resistance of a slab (K/W). $L$ = thickness, $A$ = face area.
Convection resistance: $R = \dfrac{1}{hA}$; Film resistance (K/W). $h$ = convection coefficient [W/(m²·K)], $A$ = wetted area on that side.
Cylindrical-wall resistance: $R = \dfrac{\ln(r_2/r_1)}{2\pi k L}$; Radial conduction through a tube of length $L$ between radii $r_1$ (inner) and $r_2$ (outer).
Critical insulation radius: $r_{cr} = \dfrac{k_{\text{ins}}}{h_o}$; Outer radius that maximizes heat loss from an insulated cylinder; below it, adding insulation increases loss.
Fouling resistance: $R_{\text{foul}} = \dfrac{R_f}{A}$; $R_f$ is the tabulated per-area fouling factor [(m²·K)/W]; divide by the fouled-side area to get K/W.
Overall coefficient (UA): $\dfrac{1}{UA} = \dfrac{1}{h_i A_i} + \dfrac{R_{f,i}}{A_i} + \dfrac{\ln(D_o/D_i)}{2\pi k L} + \dfrac{R_{f,o}}{A_o} + \dfrac{1}{h_o A_o}$; Series sum of all tube-side resistances. $UA$ (W/K) carries the area, so $\dot{Q}=UA\,\Delta T$ needs no reference-area choice.
Overall U vs reference area: $U_i A_i = U_o A_o = \dfrac{1}{\sum R}$; $U$ must be quoted against a stated area (inside or outside); only the product $UA$ is unambiguous.
Interface temperature: $T_{n+1} = T_n - \dot{Q}\,R_n$; March the same $\dot{Q}$ through each resistance to recover any intermediate temperature.
Biot number (lumped check): $Bi = \dfrac{h V}{k A_s} = \dfrac{h L_c}{k}$; If $Bi < 0.1$ a body is nearly isothermal and the lumped-capacitance transient model applies; $L_c=V/A_s$.

Where it lives: NCEES FE Reference Handbook — Heat Transfer · Incropera & DeWitt, Fundamentals of Heat and Mass Transfer · Perry's Chemical Engineers' Handbook — Heat Transfer

Read the Heat Transfer chapter

Mass Transfer and Separation
8% of the exam

Molecular and convective mass transfer with diffusion and mass-transfer coefficients, separation systems (distillation, absorption, extraction, membranes, adsorption), equilibrium-stage methods (McCabe-Thiele, stage efficiency), continuous-contact methods (NTU, HTU, HETP), and humidification, drying, and evaporation.

Molecular Diffusion and Convective Mass Transfer

Fick's law, equimolar counter-diffusion versus diffusion through stagnant B, the two-film model and its coefficients, and the momentum-heat-mass analogy.

Fick's first law: $J_A = -D\,\dfrac{dC_A}{dx}$; Steady 1-D molar flux ($\text{mol·m}^{-2}\text{s}^{-1}$) of A; $D$ = diffusivity ($\text{m}^2/\text{s}$), $C_A$ = concentration ($\text{mol/m}^3$). Minus sign: flux goes down the gradient.
Fick's second law: $\dfrac{\partial C}{\partial t} = D\,\dfrac{\partial^2 C}{\partial x^2}$; Unsteady 1-D diffusion (constant $D$). The mass-transfer twin of the transient heat-conduction equation.
Semi-infinite solution (erf): $\dfrac{C - C_s}{C_0 - C_s} = \operatorname{erf}\!\left(\dfrac{x}{2\sqrt{Dt}}\right)$; Surface held at $C_s$, bulk initially $C_0$. Form $z = x/(2\sqrt{Dt})$, then read $\operatorname{erf}(z)$ from the handbook table.
Equimolar counter-diffusion (gases): $N_A = \dfrac{D_m}{RT\,\Delta z}\,(p_{A1}-p_{A2})$; Use when $N_B=-N_A$ (no net molar flow), e.g., binary distillation. $\Delta z$ = path length.
Diffusion through stagnant B: $N_A = \dfrac{D_m\,P}{RT\,\Delta z\,(p_B)_{lm}}\,(p_{A1}-p_{A2})$; Use when B is inert and motionless (gas absorption). The $(p_B)_{lm}$ term adds the Stefan drift; always gives a larger flux than equimolar.
Log-mean partial pressure of B: $(p_B)_{lm} = \dfrac{p_{B2}-p_{B1}}{\ln(p_{B2}/p_{B1})}, \quad p_B = P - p_A$; Log mean of the inert's partial pressures at the two film faces.
Two-film flux: $N_A = k_G'(p_{AG}-p_{Ai}) = k_L'(C_{Ai}-C_{AL})$; Gas-film and liquid-film rate expressions; equal at steady state. $k_G'$, $k_L'$ are individual coefficients, interface in equilibrium.
Overall coefficients: $\dfrac{1}{K_G'} = \dfrac{1}{k_G'} + \dfrac{H}{k_L'}, \quad \dfrac{1}{K_L'} = \dfrac{1}{H k_G'} + \dfrac{1}{k_L'}$; Series resistances. Large $H$ (insoluble) -> liquid film controls; small $H$ (soluble) -> gas film controls.
Henry's law (interface): $p_A^{*} = H\,C_{AL}, \qquad C_A^{*} = p_{AG}/H$; Links a liquid concentration to the equilibrium gas partial pressure; $H$ has units of pressure per concentration.
Sherwood correlation (turbulent tube): $\mathrm{Sh} = \dfrac{k_m D}{D_m} = 0.023\,\mathrm{Re}^{0.8}\,\mathrm{Sc}^{1/3}$; Here $D$ = tube diameter (NOT the diffusivity $D_m$); $\mathrm{Sc}=\mu/(\rho D_m)$, $\mathrm{Re}=\rho V D/\mu$. Mass-transfer twin of Dittus-Boelter.
Chilton-Colburn analogy: $j_H = \dfrac{\mathrm{Nu}}{\mathrm{Re}\,\mathrm{Pr}^{1/3}} = j_M = \dfrac{\mathrm{Sh}}{\mathrm{Re}\,\mathrm{Sc}^{1/3}} = \dfrac{f}{8}$; Estimates $k_m$ from a friction factor or heat coefficient when a direct correlation is missing. The $f/8$ form uses the DARCY friction factor (with the Fanning factor it is $f/2$).

Where it lives: NCEES FE Reference Handbook — Chemical Engineering (Mass Transfer) · Welty, Wicks, Wilson & Rorrer, Fundamentals of Momentum, Heat, and Mass Transfer · Geankoplis, Transport Processes and Separation Process Principles

Read the Mass Transfer and Separation chapter

Solids Handling
3% of the exam

Particle properties and size distributions, surface and bulk forces, solids processing (crushing, grinding, and crystallization), and transportation and storage (belt and pneumatic conveying, slurries, tanks, and hoppers).

Particle Size, Distributions and Comminution

How to describe a particle-size distribution, compute the right mean diameter, relate surface area to size, and apply the Rittinger/Kick/Bond grinding-energy laws and crystallization basics.

Number-length (mean-length) diameter: $X_{ML}=\int_0^1 x\,dF_N=\sum_{i=1}^{m} x_i\,\Delta F_{Ni}$; Plain number-average particle size. $x_i$ = mean size of sub-range $i$ ($\mu$m or m); $\Delta F_{Ni}$ = number fraction in that sub-range (dimensionless, summing to 1).
Sauter (surface-mean) diameter: $X_{SM}=\dfrac{\sum_i x_i^{3}\,\Delta F_{Ni}}{\sum_i x_i^{2}\,\Delta F_{Ni}}=\dfrac{1}{\sum_i \frac{1}{x_i}\,\Delta F_{Vi}}$; Diameter of a sphere with the same surface-to-volume ratio as the sample; use for surface-rate processes (dissolution, drying, catalysis). $\Delta F_{Vi}$ = volume/mass fraction in sub-range $i$.
Mean-volume diameter: $X_{MV}=\left(\sum_i x_i^{3}\,\Delta F_{Ni}\right)^{1/3}$; Volume-weighted mean; governs mass distribution. Always $X_{ML}\le X_{MV}\le X_{SM}$ for the same powder.
Specific surface area of a sphere: $a_v=\dfrac{6}{d}, \qquad a_m=\dfrac{6}{\rho_p\,d}$; Surface area per unit volume ($a_v$, $\text{m}^2/\text{m}^3$) or per unit mass ($a_m$, $\text{m}^2/\text{kg}$). Scales as $1/d$: halving size doubles area. $\rho_p$ = particle density.
Specific surface area with sphericity: $a_v=\dfrac{6}{\Phi_s\,d}$; Non-spherical correction. $\Phi_s$ = sphericity (0–1): area of equal-volume sphere divided by actual surface area; the same $\Phi_s$ used in the Ergun equation.
General comminution law: $\dfrac{dE}{dx}=-C\,x^{-n}$; Energy per unit mass to reduce size; $n=2$ Rittinger, $n=1$ Kick, $n=1.5$ Bond. $x$ = particle size; $C$ = material constant.
Rittinger's law: $E=K_R\left(\dfrac{1}{x_2}-\dfrac{1}{x_1}\right)$; Energy $\propto$ new surface created; best for fine grinding. $x_1$ feed, $x_2$ product size; $K_R$ Rittinger constant.
Kick's law: $E=K_K\,\ln\dfrac{x_1}{x_2}$; Energy depends only on size reduction ratio; best for coarse crushing. $K_K$ = Kick constant.
Bond's law (size form): $E=K_B\left(\dfrac{1}{\sqrt{x_2}}-\dfrac{1}{\sqrt{x_1}}\right)$; Intermediate law for rod/ball mills. $K_B$ = Bond constant tied to the work index.
Bond work-index equation: $W = 10\,W_i\left(\dfrac{1}{\sqrt{P_{80}}}-\dfrac{1}{\sqrt{F_{80}}}\right)$; Specific energy in kWh/short ton; $W_i$ = Bond work index (kWh/short ton), $F_{80},P_{80}$ = 80%-passing feed and product sizes in $\mu$m. Mill power $=W\times$ mass throughput.
Supersaturation: $S=\dfrac{C}{C^{*}}, \qquad \Delta C=C-C^{*}$; Crystallization driving force. $C$ = actual concentration, $C^{*}$ = equilibrium solubility at the operating temperature. $S>1$ required for nucleation and growth.
Stokes terminal settling velocity: $v_t=\dfrac{g\,(\rho_p-\rho_f)\,d^{2}}{18\,\mu}$; Laminar ($Re\le1$) settling/classification of a particle of diameter $d$; $\rho_p,\rho_f$ = particle and fluid density, $\mu$ = fluid viscosity. Ties particle size to separation.

Where it lives: NCEES FE Reference Handbook — Chemical Engineering (Solids Processing: particle-size distributions and mean diameters) · NCEES FE Reference Handbook — Fluid Mechanics (sphericity, packed beds, drag/terminal velocity) · Perry's Chemical Engineers' Handbook · McCabe, Smith & Harriott, Unit Operations of Chemical Engineering

Read the Solids Handling chapter

Chemical Reaction Engineering
7% of the exam

Reaction rates and order, the Arrhenius rate constant, conversion, yield, and selectivity, series, parallel, homogeneous, heterogeneous, and biological reactions, reactor types (batch, semibatch, CSTR, plug flow, gas and liquid phase), and catalysis.

Rate Laws, Integrated Forms and Arrhenius

Reaction order and rate laws, the zero/first/second-order integrated forms and half-life, and the Arrhenius temperature dependence — the kinetic backbone of every reactor problem.

Rate of reaction (constant volume): $-r_A = -\dfrac{1}{V}\dfrac{dN_A}{dt} = -\dfrac{dC_A}{dt}$; Moles of $A$ consumed per volume per time; positive for a disappearing reactant. $C_A$ in $\text{mol/L}$, $t$ in $\text{s}$ or $\text{min}$.
Power-law rate and order: $-r_A = kC_A^{x}C_B^{y}, \quad n = x+y$; Order $x,y$ are empirical; overall order $n$ sets the units of $k$.
Fractional conversion: $C_A = C_{A0}(1-X_A)$; Links concentration and conversion at constant $V$; $X_A$ = mol $A$ reacted per mol $A$ fed.
Zero-order integrated form: $C_A = C_{A0}-kt, \quad t_{1/2}=\dfrac{C_{A0}}{2k}$; $k$ in $\text{mol·L}^{-1}\text{time}^{-1}$. $C_A$ linear in $t$; half-life grows with $C_{A0}$.
First-order integrated form: $\ln\dfrac{C_{A0}}{C_A}=kt,\quad C_A=C_{A0}e^{-kt}$; $k$ in $\text{time}^{-1}$. Equivalent: $-\ln(1-X_A)=kt$. $\ln C_A$ linear in $t$.
First-order half-life: $t_{1/2}=\dfrac{\ln 2}{k}$; Independent of $C_{A0}$ — the diagnostic signature of first-order kinetics.
Second-order integrated form: $\dfrac{1}{C_A}-\dfrac{1}{C_{A0}}=kt,\quad t_{1/2}=\dfrac{1}{kC_{A0}}$; $k$ in $\text{L·mol}^{-1}\text{time}^{-1}$. $1/C_A$ linear in $t$; half-life shrinks with $C_{A0}$.
Arrhenius equation: $k = A\,e^{-E_a/RT}$; $A$ = pre-exponential factor (same units as $k$), $E_a$ = activation energy ($\text{J/mol}$), $T$ in K, $R=8.314\ \text{J/(mol·K)}$.
Two-temperature Arrhenius: $\ln\dfrac{k_2}{k_1}=\dfrac{E_a}{R}\dfrac{T_2-T_1}{T_1 T_2}$; Solve for $E_a$ from two $(k,T)$ pairs; $A$ cancels. $T$ in K.
Arrhenius linear form: $\ln k = \ln A - \dfrac{E_a}{R}\dfrac{1}{T}$; Slope $-E_a/R$, intercept $\ln A$ on a $\ln k$ vs $1/T$ plot.
Differential method for order: $\ln(-r_A)=\ln k + n\ln C_A$; Slope of $\ln(-r_A)$ vs $\ln C_A$ gives the order $n$.

Where it lives: NCEES FE Reference Handbook — Chemical Engineering · NCEES FE Reference Handbook — Chemistry and Biology · Fogler, Elements of Chemical Reaction Engineering

Read the Chemical Reaction Engineering chapter

Engineering Economics
4% of the exam

Time value of money (present, annual, and future worth, rate of return), economic analyses (break-even, benefit-cost, optimal economic life), uncertainty with expected value and risk, and project selection with unequal lives, depreciation, and discounted cash flow.

Time Value of Money and Interest Factors

The six interest factors, gradients, nominal-vs-effective rate, and rate of return — the toolkit that converts cash flows across time on the FE.

Single-payment compound amount (F/P): $F = P\,(1+i)^n$; Future worth of a present lump sum. $i$ = rate per period, $n$ = number of periods (same period as $i$).
Single-payment present worth (P/F): $P = F\,(1+i)^{-n}$; Discounts a single future amount $F$ to the present. The base building block of all discounting.
Uniform-series present worth (P/A): $P = A\,\dfrac{(1+i)^n-1}{i(1+i)^n}$; Present worth of $n$ equal end-of-period payments $A$. Reciprocal is capital recovery $(A/P)$.
Capital recovery (A/P): $A = P\,\dfrac{i(1+i)^n}{(1+i)^n-1}$; Equal payment that repays a present amount $P$ (loan payment); also used to annualize a present cost.
Uniform-series compound amount (F/A): $F = A\,\dfrac{(1+i)^n-1}{i}$; Future worth of $n$ equal deposits $A$. Reciprocal is the sinking-fund factor $(A/F)$.
Sinking fund (A/F): $A = F\,\dfrac{i}{(1+i)^n-1}$; Equal deposit needed to accumulate a target future fund $F$ in $n$ periods.
Gradient present worth (P/G): $P = G\left[\dfrac{(1+i)^n-1}{i^2(1+i)^n}-\dfrac{n}{i(1+i)^n}\right]$; Present worth of an arithmetic gradient $G$ (first increment at end of period 2). Add to base-annuity $P$.
Gradient to uniform series (A/G): $(A/G,i,n)=\dfrac{1}{i}-\dfrac{n}{(1+i)^n-1}$; Converts a gradient $G$ to an equivalent level annuity: $A_\text{eq}=A_\text{base}+G\,(A/G,i,n)$.
Effective interest rate: $i_e=\left(1+\dfrac{r}{m}\right)^m-1$; Annual effective rate from nominal annual $r$ with $m$ compoundings/yr. Continuous limit: $i_e=e^r-1$.
Equivalence among PW, AW, FW: $AW = PW\,(A/P,i,n),\quad FW = PW\,(F/P,i,n)$; The three worth measures are interconvertible; sign and accept/reject conclusion are identical.
Capitalized cost (perpetuity): $P_\infty=\dfrac{A}{i}$; Present worth of an infinite uniform series — capitalized cost of perpetual service or maintenance.
Rate of return condition: $PW(i^*)=\sum_{t=0}^{n}\dfrac{CF_t}{(1+i^*)^t}=0$; $i^*$ = internal rate of return. Accept if $i^*\ge MARR$. Beware multiple roots with sign changes.

Where it lives: NCEES FE Reference Handbook — Engineering Economics · Newnan, Eschenbach & Lavelle, Engineering Economic Analysis · Blank & Tarquin, Engineering Economy

Read the Engineering Economics chapter

Process Design
7% of the exam

Process flow diagrams and piping and instrumentation diagrams, equipment selection, sizing, and scale-up, equipment and facilities cost estimation with cost indices, process design and optimization (sustainability, efficiency, green engineering, inherently safer design), and design standards (regulatory, ASTM, ISO, OSHA).

PFDs, P&IDs and Cost Estimation

Read a PFD versus a P&ID, then turn an equipment list into a capital-cost number with the six-tenths rule, a cost index, and Lang factors.

Cost index update: $C_{2} = C_{1}\left(\dfrac{I_{2}}{I_{1}}\right)$; Escalate a past cost $C_1$ (year-1 index $I_1$) to a present cost $C_2$ (current index $I_2$). CEPCI base $1957\text{-}59=100$. Time only — never size.
Capacity scaling (power law): $\dfrac{C_{A}}{C_{B}} = \left(\dfrac{S_{A}}{S_{B}}\right)^{n}$; $S$ is a capacity measure (area, flow, volume, power); $n$ is the cost-capacity exponent. Interpolate within a size class only.
Six-tenths rule: $C_{A} = C_{B}\left(\dfrac{S_{A}}{S_{B}}\right)^{0.6}$; Default when the specific exponent is unknown: cost rises as capacity to the $0.6$ power.
Back-out the exponent: $n = \dfrac{\ln\!\left(C_{2}/C_{1}\right)}{\ln\!\left(S_{2}/S_{1}\right)}$; Fit $n$ from two cost-capacity data points; slope of the cost line on log-log axes.
Combined escalate-and-scale: $C = C_{B}\left(\dfrac{S_{A}}{S_{B}}\right)^{n}\left(\dfrac{I_{2}}{I_{1}}\right)$; Move a base cost across both size and time in one expression.
Lang factor (installed/fixed-capital cost): $C_{\text{FCI}} = f_{L}\,C_{p,\text{delivered}}$; $f_L$: fluid plant $5.0$ (fixed) / $6.0$ (total), solid-fluid $4.3/5.0$, solid $4.0/4.7$. $C_p$ = total delivered purchased-equipment cost.
Module (factored) estimate: $C_{\text{TM}} = \sum_{i} C_{p,i}\,F_{BM,i}$; Each item's purchased cost times its bare-module factor $F_{BM}$ (raises with alloy, pressure); summed for the total module cost.
Total capital investment: $C_{\text{TCI}} = C_{\text{FCI}} + C_{\text{WC}}$; Fixed capital (built, depreciable) plus working capital (inventory, receivables; typically $\sim 15\%$ of TCI), recovered at end of life.

Where it lives: NCEES FE Reference Handbook — Chemical Engineering (Cost Estimation: cost indexes, scaling of equipment costs, capital cost / Lang factors) · NCEES FE Reference Handbook — Engineering Economics (cost indices, capital recovery, working capital) · Peters, Timmerhaus & West, Plant Design and Economics for Chemical Engineers · ISA-5.1 Instrumentation Symbols and Identification

Read the Process Design chapter

Process Control
4% of the exam

Process dynamics (first- and second-order processes, gains and time constants, stability, damping, transfer functions), control strategies (feedback, feedforward, cascade, ratio, PID tuning), and control-loop design and hardware (sensors, control valves, interlocks, and conceptual DCS and PLC programming).

Process Dynamics: First- and Second-Order Response

Process gain and time constant, the 63.2% first-order step, second-order damping and overshoot, transfer functions, and what makes a loop stable.

First-order transfer function: $\dfrac{Y(s)}{R(s)} = \dfrac{K}{\tau s + 1}$; $K$ = steady-state gain (output units / input units); $\tau$ = time constant (s). Single pole at $s=-1/\tau$.
First-order step response: $y(t) = y_0 e^{-t/\tau} + KM\left(1 - e^{-t/\tau}\right)$; Response to a step of magnitude $M$. In deviation variables $y_0=0$. Reaches $63.2\%$ of $KM$ at $t=\tau$.
Percent-complete landmarks: $63.2,\ 86.5,\ 95.0,\ 98.2,\ 99.3\,\% \text{ at } t = 1,2,3,4,5\,\tau$; Fraction of the final change reached after each time constant; practically settled by $5\tau$.
First-order plus dead time (FOPDT): $\dfrac{Y(s)}{R(s)} = \dfrac{K e^{-\theta s}}{\tau s + 1}$; $\theta$ = dead time (transport lag, s). The step response is delayed by $\theta$ before the exponential begins.
Second-order standard form: $\dfrac{Y(s)}{R(s)} = \dfrac{K\,\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}$; $\omega_n$ = undamped natural frequency (rad/s); $\zeta$ = damping ratio (dimensionless); $K$ = steady-state gain.
Second-order (process form): $\dfrac{Y(s)}{R(s)} = \dfrac{K}{\tau^2 s^2 + 2\zeta\tau s + 1}$; Equivalent form with $\tau = 1/\omega_n$. Common in the chemical-process literature.
Damping classification: $\zeta > 1\ \text{(over)}, \quad \zeta = 1\ \text{(critical)}, \quad 0<\zeta<1\ \text{(under)}$; Only the underdamped band overshoots and oscillates; critically damped is fastest with no overshoot.
Damped natural frequency: $\omega_d = \omega_n\sqrt{1-\zeta^2}$; Actual ringing frequency of an underdamped response (rad/s); always less than $\omega_n$.
Percent overshoot: $\%\text{OS} = 100\,\exp\!\left(\dfrac{-\zeta\pi}{\sqrt{1-\zeta^2}}\right)$; Overshoot above the final value for a unit step; depends on $\zeta$ only, not $\omega_n$.
Peak time: $t_p = \dfrac{\pi}{\omega_d} = \dfrac{\pi}{\omega_n\sqrt{1-\zeta^2}}$; Time to the first (largest) peak of an underdamped step response (s).
Two-percent settling time: $T_s = \dfrac{4}{\zeta\omega_n}$; Time for the envelope $e^{-\zeta\omega_n t}$ to decay within $2\%$ of final value (s).
DC gain (Final Value Theorem): $\text{dc gain} = \lim_{s\to 0} G(s)$; Steady-state gain, valid only when all poles have negative real parts. Equals $K$ for the first-order form.

Where it lives: NCEES FE Reference Handbook — Instrumentation, Measurement, and Control (Control Systems; First- and Second-Order Control System Models) · Seborg, Edgar, Mellichamp & Doyle — Process Dynamics and Control · Coughanowr & LeBlanc — Process Systems Analysis and Control

Read the Process Control chapter

Safety, Health, and Environment
5% of the exam

Hazardous material properties and safety data sheets, industrial hygiene (toxicity, noise, PPE, ergonomics), process safety and hazard analysis (LOPA, HAZOP, fault and event trees, dispersion modeling), overpressure and underpressure protection (relief and inherently safer design), waste minimization, treatment, and regulation (RCRA, CWA, EPA, OSHA), and reactivity hazards (inerting, runaway reactions, compatibility).

Process Safety: HAZOP, LOPA and Risk Assessment

Generate deviations with HAZOP guide words, multiply independent-protection-layer PFDs in LOPA, and combine fault- and event-tree gates with the AND-multiply, OR-add rules.

Risk definition: $\text{Risk} = \text{Hazard}\times\text{Probability}$; Risk combines the inherent capacity to harm with the likelihood (and severity) of the harmful event.
HAZOP deviation: $\text{Deviation} = \text{Guide word}\times\text{Parameter}$; Guide words (NO, MORE, LESS, REVERSE, etc.) crossed with flow, pressure, temperature, level, composition generate credible deviations.
LOPA mitigated frequency: $f_{\text{mit}} = f_{\text{init}}\prod_{j}\mathrm{PFD}_j$; $f_{\text{init}}$ in events/yr; each independent IPL contributes a dimensionless PFD. Result in events/yr is compared to a tolerable target.
Risk reduction factor: $\mathrm{RRF} = \dfrac{1}{\mathrm{PFD}}$; Demand-failure reciprocal; an $\mathrm{RRF}=100$ layer has $\mathrm{PFD}=0.01$ (SIL 2).
Required risk reduction: $\mathrm{RRF}_{\text{req}} = \dfrac{f_{\text{unmitigated}}}{f_{\text{tolerable}}}$; Total reduction the protection layers must collectively supply; product of all IPL RRFs must meet or exceed it.
Fault-tree AND gate: $P_{\text{AND}} = \prod_i p_i$; Top event needs all inputs to fail (redundant safeguards). Independent probabilities multiply.
Fault-tree OR gate: $P_{\text{OR}} = 1-\prod_i(1-p_i)\approx \sum_i p_i$; Top event needs any one input to fail (series). Approximate sum valid for small, independent $p_i$.
Event-tree outcome frequency: $f_{\text{outcome}} = f_{\text{init}}\prod_k p_k$; Multiply the initiating frequency by the branch probability (success or failure) along each path.
Gaussian plume centerline: $C = \dfrac{\dot{m}}{\pi\,u\,\sigma_y \sigma_z}$; Ground-level downwind concentration for a continuous neutral release; inverse in wind speed $u$ — calm air is worst case.

Where it lives: NCEES FE Reference Handbook — Safety · CCPS (AIChE), Guidelines for Hazard Evaluation Procedures and Layer of Protection Analysis · IEC 61511 Functional Safety — Safety Instrumented Systems for the Process Industry

Read the Safety, Health, and Environment chapter

Ethics and Professional Practice
3% of the exam

Codes of ethics of professional and technical societies, agreements, contracts, and contract law (noncompete, nondisclosure, memoranda of understanding), public health, safety, and welfare with licensing and professional liability, and intellectual property (copyrights, trade secrets, patents, and trademarks).

Codes of Ethics and Public Welfare

The NCEES Model Rules, the paramount duty to public health and safety, competence and conflict-of-interest obligations, licensure, and a method for resolving ethical dilemmas.

Priority of obligations: $\text{Public} \;\gg\; \text{Employer/Client} \;\gtrsim\; \text{Other licensees}$; Only the public-paramount rule is codified: the duty to public health, safety, and welfare outranks every other duty (Model Rules, Obligations to the Public). The employer/client-over-other-licensee ordering is a practical exam heuristic, not an explicit Model Rules ranking.
Paramount duty: $\text{First responsibility} = \text{safeguard public health, safety, and welfare}$; Model Rules §240.15(A)(1): the licensee's first and foremost responsibility when serving clients and employers.
Sealing rule: $\text{Seal} \iff (\text{prepared by you or under your responsible charge}) \;\wedge\; (\text{within your competence})$; A licensee may sign and seal a document only if both conditions hold; sealing work outside competence or not in responsible charge is a violation.
Conflict-of-interest test: $\text{Disclose if it could influence } \textbf{or appear to influence} \text{ professional judgment}$; Both actual and apparent conflicts must be disclosed to the employer or client; appearance alone triggers the duty.
Dual compensation rule: $\text{Accept pay from } \ge 2 \text{ parties on one project} \iff \text{full disclosure} \wedge \text{written agreement of all}$; Compensation from more than one party for the same project requires disclosure to and written consent of every interested party.
Escalation duty: $\text{Judgment overruled} \;\wedge\; \text{public endangered} \Rightarrow \text{notify employer/client AND appropriate authority}$; When a safety-critical judgment is overruled, the obligation extends beyond the employer to an outside authority.
Licensure path: $\text{ABET degree} \to \text{FE exam (EI)} \to \sim 4\text{ yr experience} \to \text{PE exam} \to \text{PE license}$; Typical Model Law route to a Professional Engineer license; experience is four years after a qualifying bachelor's degree.
Competence boundary: $\text{Undertake assignment} \iff \text{qualified by education or experience in that specific field}$; Model Rules: accept only work for which you are technically qualified.

Where it lives: NCEES FE Reference Handbook — Ethics and Professional Practice · NCEES Model Rules, Section 240.15 — Rules of Professional Conduct · NCEES Model Law, Sections 110.20, 130.10, 150.10 — Definitions, Licensure, Disciplinary Action · NSPE Code of Ethics for Engineers

Read the Ethics and Professional Practice chapter

FE Chemical formula sheet

Mathematics
6% of the exam

Numerical Methods, Error and Significant Figures

Probability and Statistics
4% of the exam

Distributions, Confidence Intervals and Control Limits

Engineering Sciences
4% of the exam

Newtonian Dynamics, Work, Energy and Power

Materials Science
4% of the exam

Material Properties and Corrosion Control

Chemistry and Biology
7% of the exam

Solutions, pH, Buffers and Electrochemistry

Fluid Mechanics and Dynamics
8% of the exam

The Mechanical Energy Balance and Bernoulli

Thermodynamics
8% of the exam

First and Second Law: Processes, Cycles and Efficiency

Material and Energy Balances
10% of the exam

Material Balances and Degrees of Freedom

Heat Transfer
8% of the exam

Conduction, Thermal Resistance and the Overall U

Mass Transfer and Separation
8% of the exam

Molecular Diffusion and Convective Mass Transfer

Solids Handling
3% of the exam

Particle Size, Distributions and Comminution

Chemical Reaction Engineering
7% of the exam

Rate Laws, Integrated Forms and Arrhenius

Engineering Economics
4% of the exam

Time Value of Money and Interest Factors

Process Design
7% of the exam

PFDs, P&IDs and Cost Estimation

Process Control
4% of the exam

Process Dynamics: First- and Second-Order Response

Safety, Health, and Environment
5% of the exam

Process Safety: HAZOP, LOPA and Risk Assessment

Ethics and Professional Practice
3% of the exam

Codes of Ethics and Public Welfare

Now use them on real questions

Mathematics6% of the exam

Numerical Methods, Error and Significant Figures

Probability and Statistics4% of the exam

Distributions, Confidence Intervals and Control Limits

Engineering Sciences4% of the exam

Newtonian Dynamics, Work, Energy and Power

Materials Science4% of the exam

Material Properties and Corrosion Control

Chemistry and Biology7% of the exam

Solutions, pH, Buffers and Electrochemistry

Fluid Mechanics and Dynamics8% of the exam

The Mechanical Energy Balance and Bernoulli

Thermodynamics8% of the exam

First and Second Law: Processes, Cycles and Efficiency

Material and Energy Balances10% of the exam

Material Balances and Degrees of Freedom

Heat Transfer8% of the exam

Conduction, Thermal Resistance and the Overall U

Mass Transfer and Separation8% of the exam

Molecular Diffusion and Convective Mass Transfer

Solids Handling3% of the exam

Particle Size, Distributions and Comminution

Chemical Reaction Engineering7% of the exam

Rate Laws, Integrated Forms and Arrhenius

Engineering Economics4% of the exam

Time Value of Money and Interest Factors

Process Design7% of the exam

PFDs, P&IDs and Cost Estimation

Process Control4% of the exam

Process Dynamics: First- and Second-Order Response

Safety, Health, and Environment5% of the exam

Process Safety: HAZOP, LOPA and Risk Assessment

Ethics and Professional Practice3% of the exam

Codes of Ethics and Public Welfare

Now use them on real questions

Mathematics
6% of the exam

Probability and Statistics
4% of the exam

Engineering Sciences
4% of the exam

Materials Science
4% of the exam

Chemistry and Biology
7% of the exam

Fluid Mechanics and Dynamics
8% of the exam

Thermodynamics
8% of the exam

Material and Energy Balances
10% of the exam

Heat Transfer
8% of the exam

Mass Transfer and Separation
8% of the exam

Solids Handling
3% of the exam

Chemical Reaction Engineering
7% of the exam

Engineering Economics
4% of the exam

Process Design
7% of the exam

Process Control
4% of the exam

Safety, Health, and Environment
5% of the exam

Ethics and Professional Practice
3% of the exam