Chemistry formulas — 136 solvers

Absorbance and Transmittance

A = -\log_{10}\!\left(\frac{\%T}{100}\right)

ChemistryConverts a spectrophotometer's percent transmittance into absorbance and back, the logarithmic bridge between what a detector sees and what Beer's law needs.

Acid Feed to Reduce Alkalinity

\dot m = \frac{\Delta \mathrm{Alk} \, Q \times \frac{\mathrm{EW}}{50.04}}{p/100\%}

Water TreatmentChemistrySizes the acid feed needed to knock a target amount of alkalinity out of a stream, correcting for the acid's equivalent weight and its commercial strength.

Activation Energy from an Arrhenius Plot

E_a = -R \times \text{slope}

ChemistryConverts the slope of a ln k versus 1/T Arrhenius plot into an activation energy, the standard graphical method for kinetics data.

Alkalinity Remaining After Alum

A_f = A_0 - 0.45 \, D

Water & WastewaterWater TreatmentChemistryAlkalinity left after coagulation, since each mg/L of alum destroys 0.45 mg/L of alkalinity as CaCO₃ in forming aluminium hydroxide floc.

Antoine Equation (Vapour Pressure)

\log_{10} P = A - \frac{B}{C + T}

ChemistryThermodynamicsThe three-constant fit that every handbook uses for pure-component vapour pressure. The published constants A, B and C are for pressure in mmHg and temperature in degrees Celsius; enter pressure and temperature in whatever units you like and the conversion is handled inside.

Arrhenius Equation

k = A\,e^{-E_a/RT}

ChemistryGives a reaction's rate constant from its activation energy, pre-exponential factor, and absolute temperature — the core law of chemical kinetics.

Arrhenius Two-Temperature Form

\ln\frac{k_2}{k_1} = \frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)

ChemistryExtracts an activation energy from two rate constants measured at two temperatures, eliminating the pre-exponential factor entirely.

Beer–Lambert Law (A = εbc)

A = \varepsilon\, b\, c

ChemistryRelates a sample's absorbance to its molar absorptivity, path length, and concentration — the working equation of UV-Vis spectrophotometry.

Blood Alcohol Estimate (Widmark Equation)

\mathrm{BAC} = \frac{100\,A}{r\,m} - \beta t

Everyday & HealthChemistryWidmark's 1932 estimate of blood alcohol concentration from alcohol consumed, body mass, a distribution ratio and elapsed time. An estimate only, never a fitness-to-drive test.

BOD Mass Loading

W = Q \, C

Water & WastewaterWater TreatmentChemistryMass of BOD, COD or solids arriving per unit time from a flow and its concentration, on the sanitary basis of 1 mg/L = 1 g/m³.

Bohr Model Energy Levels

E_n = -\frac{13.606\ \mathrm{eV}}{n^{2}}

Modern PhysicsChemistryPhysicsEnergy of the hydrogen atom's nth level: −13.606 eV divided by n².

Boiler Blowdown Percent

\%B = \frac{\text{TDS}_{fw}}{\text{TDS}_b} \times 100

Water TreatmentChemistryBoiler blowdown as a percentage of feedwater, set by the ratio of feedwater TDS to the maximum TDS allowed in the drum.

Boiler Cycles of Concentration

\text{COC} = \frac{\text{TDS}_b}{\text{TDS}_{fw}}

Water TreatmentChemistryCycles of concentration in a steam boiler: boiler water TDS divided by feedwater TDS, since steam leaves the dissolved solids behind.

Boiling-Point Elevation

\Delta T_b = K_b \, b

ChemistryGives how far a dissolved solute raises a solvent's boiling point from the molality and the solvent's ebullioscopic constant.

Boyle's Law

P_1 V_1 = P_2 V_2

ThermodynamicsChemistryPhysicsAt constant temperature, pressure times volume stays constant for a fixed amount of gas.

Breakpoint Chlorine-to-Ammonia Ratio

R = \frac{\mathrm{Cl_2}}{\mathrm{NH_3\text{-}N}}

Water TreatmentChemistryThe weight ratio of chlorine dose to ammonia nitrogen, which must reach about 7.6 to 1 to pass breakpoint and produce a free chlorine residual.

Calorimeter Heat (q = C_cal ΔT)

q = C_{\text{cal}} \, \Delta T

ChemistryThermodynamicsGives the heat absorbed by a calorimeter from its calibrated heat capacity and the temperature rise it records.

Charles's Law

\frac{V_1}{T_1} = \frac{V_2}{T_2}

ThermodynamicsChemistryPhysicsAt constant pressure, gas volume is directly proportional to absolute temperature.

Chemical Feed Rate (lb/day = mg/L × MGD × 8.34)

\dot m = C \, Q

Water TreatmentChemistryConverts a target chemical dose and a plant flow into a mass feed rate, the classic pounds-per-day formula built on water weighing 8.34 lb per gallon.

Chemical Feed Rate from Dose

W = C \, Q \, \rho_w

Water TreatmentChemistryChemical feed rate in pounds per day from a ppm dose and a water flow, using the trade density of 8.34 lb per US gallon.

Chemical Mass Consumed Over a Period

m = W \, t

Water TreatmentChemistryTreatment chemical a feed rate consumes over a period — the step that turns pounds per day into the annual tonnage a contract is priced on.

Chemical Treatment Cost

C_p = m \, p_c

Water TreatmentChemistryInvoice for a treatment chemical: the mass of product consumed over a period times its unit price, the second line of a water budget.

Chemical-Consuming Loss Rate

L = B + D

Water TreatmentChemistryFluid MechanicsFlow that actually carries treatment out of a cooling tower — blowdown plus drift, because evaporation leaves every molecule of inhibitor behind.

Chlorine Dose from a Weight of Product

C = \frac{m \, p}{V}

Water TreatmentChemistryGives the free chlorine concentration produced by dissolving a known weight of hypochlorite product of known strength in a known volume of water.

Chlorine Dose, Demand and Residual

D = C_{\text{demand}} + C_{\text{res}}

Water TreatmentChemistryThe fundamental chlorination balance: the dose applied equals the chlorine consumed by the water's demand plus the residual left for disinfection.

Clausius–Clapeyron Equation (Two-Point Form)

\ln\!\left(\frac{P_2}{P_1}\right) = -\frac{\Delta H_{vap}}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)

ChemistryThermodynamicsRelates two points on a liquid's vapour-pressure curve to its molar enthalpy of vaporisation, assuming ΔH is constant over the interval and the vapour behaves ideally.

Closed Loop Slug Dose Volume

V_p = \frac{C \, V_s \, \rho_w}{\rho_p}

Water TreatmentChemistryVolume of liquid product needed for a one-shot slug dose to hit a target ppm in a closed loop of known volume and product density.

Combined Gas Law

\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}

ThermodynamicsChemistryPhysicsFor a fixed amount of gas, pressure times volume over absolute temperature stays constant between any two states.

Compressibility Factor (Z = PV/nRT)

Z = \frac{P V}{n R T}

ChemistryThermodynamicsHow far a real gas departs from ideal behaviour, as a single multiplier on the ideal gas law. Z = 1 is ideal; below 1 attraction dominates, above 1 the molecules' own volume does.

Cooling Tower LSI at Cycles of Concentration

\mathrm{LSI} = \mathrm{pH} - \mathrm{pH_s}\left(T_s,\ N\,\mathrm{Ca_m},\ N\,\mathrm{Alk_m},\ N\,\mathrm{TDS_m}\right)

Water TreatmentChemistryThe rigorous Langelier index of recirculating water: makeup calcium, alkalinity and TDS multiplied by the cycles, evaluated at the measured tower pH and the hottest surface temperature.

Corrosion Rate from Coupon Weight Loss

P = \frac{m \, Y}{\rho \, A \, t}

Water TreatmentChemistryUniform corrosion rate as metal thickness lost per year, from a coupon's weight loss, density, exposed area and exposure time.

CSTR Design Equation (First Order)

V = \frac{v_0\,X}{k\left(1 - X\right)}

ChemistryVolume of a perfectly mixed continuous tank reactor for a first-order liquid-phase reaction at a target conversion. The whole vessel sits at the outlet concentration, which is why the volume runs away as conversion approaches 100%.

Cycles of Concentration from Chloride

\text{COC} = \frac{\mathrm{Cl}_t}{\mathrm{Cl}_m}

Water TreatmentChemistryCycles of concentration from a chloride titration — the conservative tracer that neither precipitates nor gets dosed into the system.

Cycles of Concentration from Conductivity

\text{COC} = \frac{\sigma_t}{\sigma_m}

Water TreatmentChemistryCycles of concentration read straight off a conductivity meter: tower water conductivity divided by makeup water conductivity.

Dalton's Law of Partial Pressures

P_{\text{total}} = P_1 + P_2 + P_3

ChemistryThermodynamicsPhysicsStates that each gas in a mixture exerts its own pressure independently, so the total pressure is the sum of the partial pressures.

Days Between Softener Regenerations

t = \frac{m_{\text{cap}}}{C \, Q}

Water TreatmentChemistryGives the run time a softener achieves between regenerations from its rated hardness capacity, the raw water hardness and the average water usage rate.

Density

\rho = \tfrac{m}{V}

PhysicsChemistryWater TreatmentMass per unit volume.

Dilution Equation (C1V1 = C2V2)

C_1 V_1 = C_2 V_2

ChemistryWater TreatmentStates that concentration times volume is conserved when a solution is diluted, since the moles of solute do not change.

Dose Achieved from Chemical Added

C = \frac{m}{V \, \rho_w}

Water TreatmentChemistryConcentration reached in a system of known volume after adding a measured mass of chemical, at 8.34 lb of water per US gallon.

Empirical Formula Mole Ratio from Percent Composition

\frac{n_X}{n_Y} = \frac{\%X / M_X}{\%Y / M_Y}

ChemistryConverts two elements' mass percentages into the ratio of their atom counts, the step that turns a percent-composition analysis into an empirical formula.

Equilibrium Constant Kc (A + B ⇌ C + D)

K_c = \frac{[\mathrm{C}][\mathrm{D}]}{[\mathrm{A}][\mathrm{B}]}

ChemistryComputes the equilibrium constant or reaction quotient for a one-to-one reaction from the four species concentrations, the law of mass action in its simplest form.

Equivalent Weight from Molar Mass and Valence

\mathrm{EW} = \frac{M}{z}

Water TreatmentChemistryGives the gram-equivalent weight of an ion or compound as its molar mass divided by the number of charges or replaceable hydrogens it carries.

Excess Reagent Remaining

n_{\text{excess}} = n_B - \frac{b}{a} \, n_A

ChemistryGives how much of the reactant in excess is left unreacted once the limiting reagent A has been entirely consumed.

Faraday's Law of Electrolysis (m = QM/nF)

m = \frac{Q M}{n F}

ChemistryFaraday's law of electrolysis: the mass plated out at an electrode from the charge passed, the molar mass, and the electrons transferred per ion.

Fenske Equation (Minimum Stages)

N_{min} = \frac{\ln\!\left[\frac{x_D}{1 - x_D}\cdot\frac{1 - x_B}{x_B}\right]}{\ln \alpha}

ChemistryThe number of theoretical stages a distillation column would need at total reflux — the absolute floor, before any reflux ratio is chosen. A real column needs roughly twice this many trays.

Fick's First Law of Diffusion

J = D\,\frac{C_1 - C_2}{L}

ChemistryPhysicsSteady-state diffusive flux down a concentration gradient. Written here for a slab of thickness L with a fixed concentration on each face, which is the form used for membranes, coatings and stagnant films. The flux J is in mol per square metre per second.

First-Order Integrated Rate Law

[\mathrm{A}] = [\mathrm{A}]_0\,e^{-kt}

ChemistryGives the concentration remaining in a first-order reaction, the exponential decay that governs radioactive decay and most drug clearance.

Flow-Weighted Blending Concentration

C = \frac{Q_1 C_1 + Q_2 C_2}{Q_1 + Q_2}

Water & WastewaterWater TreatmentChemistryThe concentration you land on when two streams of known flow and known strength join — each stream's mass rate added, then divided by the combined flow. It is the arithmetic behind blending a hard well against a soft surface source, and behind every question about what a side stream does to a plant's influent.

Freezing-Point Depression

\Delta T_f = K_f \, b

ChemistryGives how far a dissolved solute lowers a solvent's freezing point from the molality and the solvent's cryoscopic constant.

Freezing-Point Depression with the van 't Hoff Factor

\Delta T_f = i \, K_f \, b

ChemistryGives the freezing-point depression of an electrolyte solution, where each dissolved formula unit releases i particles into the solvent.

Gas Density from Molar Mass

\rho = \frac{PM}{RT}

ChemistryThermodynamicsPhysicsGives an ideal gas's density from its molar mass, pressure, and absolute temperature using R = 8.314462618 J/(mol·K).

Gas Volume at STP

V = n\,V_m

ChemistryConverts between moles of an ideal gas and its volume at STP using the molar volume Vm = 22.414 L/mol.

Gay-Lussac's Law

\frac{P_1}{T_1} = \frac{P_2}{T_2}

ThermodynamicsChemistryPhysicsAt constant volume, gas pressure is directly proportional to absolute temperature.

Gibbs Free Energy and the Equilibrium Constant

\Delta G^{\circ} = -RT\ln K

ChemistryThermodynamicsConverts between a reaction's standard free energy change and its equilibrium constant, the bridge joining thermodynamics to equilibrium tables.

Gibbs Free Energy Change (ΔG = ΔH − TΔS)

\Delta G = \Delta H - T\,\Delta S

ChemistryThermodynamicsCombines a reaction's enthalpy and entropy changes at a given temperature to decide whether it can happen spontaneously.

Gibbs Free Energy from Cell Potential (ΔG° = −nFE°)

\Delta G^{\circ} = -n F E^{\circ}

ChemistryThermodynamicsConverts a cell's standard potential into the standard free energy change of its reaction, using the Faraday constant F = 96485.33212 C/mol.

Glycol Dilution

C_1 V_1 = C_2 V_2

Water TreatmentChemistryDilution equation for mixing glycol (or any concentrate) to a target strength.

Graham's Law of Effusion

\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}

ChemistryPhysicsRelates the effusion rates of two gases to the inverse square root of their molar masses.

Grains per Gallon ↔ ppm Hardness

H = 17.118\,G

Water TreatmentChemistryConverts water hardness between grains per US gallon and mg/L as CaCO₃ using the exact 17.118 mg/L per grain trade factor.

Half-Life and Decay Constant

t_{1/2} = \frac{\ln 2}{\lambda}

ChemistryPhysicsConverts between a half-life and the exponential decay constant via the factor ln 2 = 0.6931471806.

Half-Life Decay

N = N_0 \left(\frac{1}{2}\right)^{t/t_{1/2}}

ChemistryPhysicsGives the quantity remaining after repeated halvings over an elapsed time measured in half-lives.

Half-Life of a Second-Order Reaction

t_{1/2} = \frac{1}{k\,[\mathrm{A}]_0}

ChemistryGives the half-life of a second-order reaction, which unlike a first-order half-life depends on the starting concentration.

Hardness Load Removed per Regeneration

m = C \, V

Water TreatmentChemistryGives the mass of hardness as CaCO₃ a softener must remove between regenerations from the raw hardness and the volume of water treated.

Hardness Removal Efficiency and Leakage

R = \frac{C_{\text{in}} - C_{\text{out}}}{C_{\text{in}}}

Water TreatmentChemistryGives the fraction of hardness a softener or membrane removes from the inlet and outlet concentrations, with the remainder being leakage to service.

Heat of Reaction

q = n \Delta H

ChemistryThermodynamicsScales a reaction's molar enthalpy change by the amount reacted to give the total heat released or absorbed.

Henderson–Hasselbalch Equation (Weak Acid Buffer)

\mathrm{pH} = \mathrm{p}K_a + \log_{10}\!\frac{[\mathrm{A^-}]}{[\mathrm{HA}]}

ChemistryGives the pH of a buffer from the acid's pKa and the ratio of conjugate base to undissociated acid, the master equation of buffer preparation.

Henderson–Hasselbalch Equation (Weak Base Buffer)

\mathrm{pOH} = \mathrm{p}K_b + \log_{10}\!\frac{[\mathrm{BH^+}]}{[\mathrm{B}]}

ChemistryGives the pOH of a weak base buffer from the base's pKb and the ratio of conjugate acid to free base, the mirror image of the acid form.

Henry's Law (Gas Solubility)

C = H\,P

ChemistryWater & WastewaterAt a fixed temperature the concentration a gas reaches in a liquid is proportional to its partial pressure above the liquid. H is the solubility form of the Henry constant, in moles per cubic metre per pascal.

Hess's Law (Three-Step Sum)

\Delta H_{\text{rxn}} = \Delta H_1 + \Delta H_2 + \Delta H_3

ChemistryThermodynamicsHess's law: the enthalpy change of a target reaction is the sum of the enthalpy changes of the steps you route it through.

Holding Time Index

\text{HTI} = \frac{\ln 2 \; V}{B}

Water TreatmentChemistryHolding time index: the half-life of a chemical in a bled system, the time for half the treatment to wash out at a given blowdown rate.

Hypochlorite Product Mass from Available Chlorine

m_{\text{prod}} = \frac{m_{\mathrm{Cl}} \times 100\%}{p}

Water TreatmentChemistryConverts a required mass of available chlorine into the mass of hypochlorite product to weigh out, given the product's percent available chlorine.

Ideal Gas Law

P V = n R T

ChemistryPhysicsRelates pressure, volume, amount, and temperature of a gas. R = 8.314 J/(mol·K).

Ion Concentration as CaCO₃ Equivalent

C_{\mathrm{CaCO_3}} = C_{\mathrm{ion}} \times \frac{50.04}{\mathrm{EW}}

Water TreatmentChemistryRestates any ion's concentration in the trade's common currency of CaCO₃ by scaling with the ratio of equivalent weights, 50.04 over the ion's own.

Jar Test Dose Scale-Up

D = \frac{V_{st} \, C_{st}}{V_{s}}

Water & WastewaterWater TreatmentChemistryConverts millilitres of stock solution added to a jar test beaker into the equivalent plant dose in mg/L of raw water.

Ka and Kb Relation through Kw

K_a \, K_b = K_w

ChemistryLinks a weak acid's dissociation constant to that of its conjugate base through the ion product of water, Kw = 1.0 × 10⁻¹⁴ at 25 °C.

Kp from Kc (Kp = Kc(RT)^Δn)

K_p = K_c (RT)^{\Delta n}

ChemistryThermodynamicsConverts a gas-phase equilibrium constant between pressure and concentration bases using the change in moles of gas, with R = 0.08206 L·atm/(mol·K).

Langelier Saturation Index (LSI)

\mathrm{LSI} = \mathrm{pH} - \mathrm{pH_s}

Water TreatmentChemistryThe classic calcium-carbonate scaling index: measured pH minus the saturation pH, positive for scaling and negative for corrosive water.

Langelier Saturation Index from a Full Water Analysis

\mathrm{LSI} = \mathrm{pH} - \left(\mathrm{p}K_2 - \mathrm{p}K_{sp} + \mathrm{p[Ca^{2+}]} + \mathrm{p[HCO_3^-]} + 5\,\mathrm{p}f_m\right)

Water TreatmentChemistryLSI in one step from pH, temperature, calcium, total alkalinity and TDS, with temperature-dependent equilibria, activity correction and the alkalinity corrected for carbonate and hydroxide.

Larson–Skold Index

\mathrm{LS} = \dfrac{\frac{\mathrm{Cl}}{35.45} + \frac{\mathrm{SO_4}}{48.03}}{\frac{\mathrm{Alk}}{50.04}}

Water TreatmentChemistryRatio of aggressive chloride and sulphate equivalents to protective bicarbonate alkalinity, predicting pitting of mild steel in distribution water.

Lime Dose for Softening (as CaCO₃)

L = \mathrm{CO_2} + \mathrm{Alk} + \mathrm{Mg} + \mathrm{Ex}

Water TreatmentChemistryWater & WastewaterThe stoichiometric lime requirement for excess-lime softening, in mg/L as CaCO₃: every acid the lime has to neutralise before it can precipitate anything, plus the magnesium it must convert, plus the excess that makes the reaction finish in a real basin.

Mass Percent of a Solution

c = \frac{m_{\text{solute}}}{m_{\text{solution}}} \times 100\%

ChemistryExpresses a solution's concentration as the solute's share of the total solution mass in percent.

Mass-to-Mass Stoichiometry

m_B = \frac{m_A}{M_A} \cdot \frac{b}{a} \cdot M_B

ChemistryGives the mass of one substance produced or consumed from the mass of another, routing through moles and the balanced equation's coefficient ratio.

Mixing Two Solutions (C₁V₁ + C₂V₂ = C_f V_f)

C_f = \frac{C_1 V_1 + C_2 V_2}{V_1 + V_2}

ChemistryWater TreatmentGives the concentration you land on when two solutions of known strength and volume are poured together, by conserving the moles of solute and letting the volumes add.

Molality (b = n/m)

b = \frac{n}{m_{\text{solvent}}}

ChemistryDefines molality as moles of solute per kilogram of solvent.

Molarity (C = n/V)

C = \frac{n}{V}

ChemistryDefines molar concentration as moles of solute per volume of solution.

Mole Fraction

x_1 = \frac{n_1}{n_1 + n_2}

ChemistryGives the fraction of all moles in a two-component mixture contributed by the solute.

Mole Ratio from a Balanced Equation

n_B = n_A \, \frac{b}{a}

ChemistryConverts moles of one substance in a balanced equation into moles of another using the ratio of their stoichiometric coefficients.

Moles from Mass (n = m/M)

n = \frac{m}{M}

ChemistryConverts a measured mass into an amount of substance by dividing by the molar mass.

Nernst Equation

E = E^{\circ} - \frac{RT}{nF}\ln Q

ChemistryCorrects a cell's standard potential for non-standard concentrations, the equation behind every pH meter, ion-selective probe, and battery voltage curve.

Normality from Molarity

N = M \times n_{\text{eq}}

ChemistryConverts molarity into normality by multiplying by the number of reactive equivalents each mole of solute supplies.

Osmotic Pressure (Π = MRT)

\Pi = M R T

ChemistryGives the osmotic pressure of a dilute solution from its molar concentration and absolute temperature using the van 't Hoff equation.

Partial Pressure from Mole Fraction

P_i = x_i \, P_{\text{total}}

ChemistryThermodynamicsGives a gas component's partial pressure as its mole fraction times the total pressure, the practical form of Dalton's law of partial pressures.

Particles from Moles (Avogadro's Number)

N = n\,N_A

ChemistryConverts an amount in moles into an actual particle count using the Avogadro constant NA = 6.02214076 × 10²³ per mole.

Percent Composition of an Element

\%X = \frac{a\,M_X}{M_{\text{compound}}} \times 100\%

ChemistryGives the mass percent an element contributes to a compound from the formula subscript and molar masses.

Percent Ionization of a Weak Acid

\%\,\text{ion} = \frac{[\mathrm{H^+}]}{C} \times 100\%

ChemistryExpresses what fraction of a weak acid has actually donated its proton, comparing the equilibrium hydrogen ion concentration to the formal concentration.

Percent Yield

\%\,\text{yield} = \frac{m_{\text{actual}}}{m_{\text{theoretical}}} \times 100\%

ChemistryCompares the mass actually isolated from a reaction to the maximum mass stoichiometry predicts.

PFR Design Equation (First Order)

V = \frac{v_0}{k}\ln\!\left(\frac{1}{1 - X}\right)

ChemistryVolume of a plug-flow (tubular) reactor for a first-order liquid-phase reaction. Concentration falls smoothly along the tube instead of dropping to the outlet value at the inlet, so a PFR always needs less volume than a CSTR for the same conversion.

pH and pOH Relation

\mathrm{pH} + \mathrm{pOH} = 14

ChemistryLinks the acidity and basicity scales of any aqueous solution at 25 °C, where pKw = 14.

pH from Hydrogen Ion Concentration

\mathrm{pH} = -\log_{10}\,[\mathrm{H^+}]

ChemistryWater TreatmentExpresses acidity as the negative base-10 logarithm of the hydrogen ion concentration in mol/L.

pH of a Weak Acid from Ka

\mathrm{pH} = -\log_{10}\sqrt{K_a\,C}

ChemistryEstimates the pH of a weak monoprotic acid solution from its dissociation constant and formal concentration using the standard x-is-small approximation.

pH of a Weak Base from Kb

\mathrm{pH} = 14 + \tfrac{1}{2}\log_{10}\left(K_b\,C\right)

ChemistryEstimates the pH of a weak monoprotic base from its ionisation constant and formal concentration, via [OH⁻] = √(Kb·C) and pH = 14 − pOH.

pKa from Acid Dissociation Constant

\mathrm{p}K_a = -\log_{10} K_a

ChemistryConverts an acid dissociation constant into its logarithmic pKa form and back, compressing a huge range of acid strengths onto one readable scale.

pKb from Base Dissociation Constant

\mathrm{p}K_b = -\log_{10} K_b

ChemistryConverts a base dissociation constant into its logarithmic pKb form and back, the basic-side counterpart of the pKa scale.

pOH from Hydroxide Ion Concentration

\mathrm{pOH} = -\log_{10}\,[\mathrm{OH^-}]

ChemistryWater TreatmentExpresses basicity as the negative base-10 logarithm of the hydroxide ion concentration in mol/L.

Pounds of Active Chemical in a Tank

m = V \times \mathrm{SG} \times \rho_w \times p

Water TreatmentChemistryGives the mass of active chemical held in a storage tank from its volume, the solution's specific gravity and the percent active ingredient.

Predicted Cooling Tower LSI and Maximum Cycles, from the Makeup Analysis

\mathrm{LSI} = \mathrm{pH_{eq}} - \mathrm{pH_s}, \quad \mathrm{pH_{eq}} = 1.465\,\log_{10}(N\,\mathrm{Alk_m}) + 4.54

Water TreatmentChemistryProjects a tower's Langelier index from its makeup water and cycles, estimating tower pH from the cycled alkalinity, and solves for the cycles at which the index reaches a chosen limit.

Product Dose from Active Strength

D_p = \frac{100 \, D_a}{A}

Water TreatmentChemistryConverts a required active-ingredient dose into the as-supplied product dose when the drum is only a given percent active.

Puckorius (Practical) Scaling Index

\mathrm{PSI} = 2\,\mathrm{pH_s} - \mathrm{pH_{eq}}, \quad \mathrm{pH_{eq}} = 1.465\,\log_{10}\mathrm{Alk} + 4.54

Water TreatmentChemistryA Ryznar variant that replaces measured pH with an equilibrium pH derived from alkalinity, giving a truer scaling call in poorly buffered cooling water.

Radioactive Activity (A = λN)

A = \lambda N

Modern PhysicsChemistryPhysicsDecays per second: the decay constant times the number of undecayed nuclei.

Raoult's Law

P = x \, P^{0}

ChemistryGives the vapor pressure of a solvent above an ideal solution as its mole fraction times the pure solvent's vapor pressure.

Reaction Quotient Q (aA + bB ⇌ cC)

Q = \frac{[\mathrm{C}]^{c}}{[\mathrm{A}]^{a}\,[\mathrm{B}]^{b}}

ChemistryReaction quotient Q from any concentrations, raised to stoichiometric powers — the number you compare with K to predict which way a mixture shifts.

Relative Volatility (Binary)

\alpha = \frac{y\left(1 - x\right)}{x\left(1 - y\right)}

ChemistryThe single number that says how easy a binary mixture is to distil: the ratio of the light component's vapour-to-liquid enrichment against the heavy component's. α = 1 means the two boil identically and no simple column will part them.

Ryznar Stability Index (RSI)

\mathrm{RSI} = 2\,\mathrm{pH_s} - \mathrm{pH}

Water TreatmentChemistryAn empirical scaling index scaled so that values below about 6 predict scale and values above about 7 predict corrosion in the same water.

Salt Dose per Regeneration

m_{\text{salt}} = D \, V

Water TreatmentChemistryGives the mass of sodium chloride a softener draws each regeneration from the resin volume and the programmed salt dosage per unit volume.

Saturation pH (pHs) for Langelier's Index

\mathrm{pH_s} = (9.3 + A + B) - (C + D)

Water TreatmentChemistryComputes the pH at which water is saturated with calcium carbonate from its TDS, temperature, calcium hardness and total alkalinity as CaCO₃.

Saturation pH (pHs), Rigorous — from the Carbonate Equilibria

\mathrm{pH_s} = \mathrm{p}K_2 - \mathrm{p}K_{sp} + \mathrm{p[Ca^{2+}]} + \mathrm{p[HCO_3^-]} + 5\,\mathrm{p}f_m

Water TreatmentChemistryThe pH at which water is exactly saturated with calcite, from temperature-dependent K₂ and Ksp and a Davies activity correction, not the four-term handbook shortcut.

Second-Order Integrated Rate Law

\frac{1}{[\mathrm{A}]} = \frac{1}{[\mathrm{A}]_0} + kt

ChemistryGives the concentration remaining in a second-order reaction, where the reciprocal of concentration rises linearly with time.

Sensible Heat (Q = mcΔT)

Q = m c \Delta T

ThermodynamicsPhysicsChemistryHeat needed to change a mass's temperature: mass times specific heat times the temperature change.

Serum Anion Gap

\mathrm{AG} = \mathrm{Na^+} - (\mathrm{Cl^-} + \mathrm{HCO_3^-})

Biomedical & ClinicalChemistryThe difference between the measured cation and the measured anions in serum, the standard first split in the workup of a metabolic acidosis.

Soda Ash Dose for Softening (as CaCO₃)

S = \mathrm{TH} - \mathrm{Alk}

Water TreatmentChemistryWater & WastewaterSoda ash makes up the difference between the hardness a water carries and the alkalinity it carries — the noncarbonate hardness, which lime cannot remove because no bicarbonate is paired with it. Expressed, as everything in softening is, in mg/L as CaCO₃.

Softener Resin Volume Required

V = \frac{m_{\text{cap}}}{q}

Water TreatmentChemistrySizes the ion-exchange resin bed from the hardness capacity required per regeneration and the resin's rated capacity per unit volume.

Solubility Product of a 1:1 Salt

K_{sp} = s^{2}

ChemistryLinks the solubility product of an AB salt such as AgCl or BaSO4 to its molar solubility, since each formula unit releases one cation and one anion.

Solubility Product of an AB₂ Salt

K_{sp} = 4s^{3}

ChemistryLinks the solubility product of an AB2 or A2B salt such as CaF2 or Mg(OH)2 to its molar solubility, with the factor 4 from the doubled ion.

Space Time and Space Velocity

\tau = \frac{V}{v_0}

ChemistryWater & WastewaterHow long the feed nominally spends in the reactor: vessel volume divided by volumetric feed rate. Space velocity is simply its reciprocal, 1/τ, and is the number catalyst vendors quote as LHSV or GHSV in reciprocal hours.

Specific Gravity

SG = \frac{\rho}{\rho_{water}}

Fluid MechanicsWater TreatmentChemistryDensity expressed as a multiple of water's 1000 kg/m³.

Standard Cell Potential from Half-Cells

E^{\circ}_{\text{cell}} = E^{\circ}_{\text{cathode}} - E^{\circ}_{\text{anode}}

ChemistryBuilds a galvanic cell's standard voltage by subtracting the anode's standard reduction potential from the cathode's, using tabulated half-cell values.

Standard Enthalpy of Reaction from Formation Enthalpies

\Delta H^{\circ}_{\text{rxn}} = \sum \Delta H^{\circ}_{f,\text{prod}} - \sum \Delta H^{\circ}_{f,\text{react}}

ChemistryThermodynamicsThe tabulated form of Hess's law: standard enthalpy of reaction equals the summed formation enthalpies of the products minus those of the reactants.

TDS Estimated from Conductivity (TDS = k × EC)

\mathrm{TDS} = k \times \mathrm{EC}

Water TreatmentChemistryEstimates total dissolved solids in mg/L from a conductivity reading in µS/cm using an empirical factor k, typically 0.55 to 0.70 for natural waters.

Titration: Concentration of an Unknown

C_a = \frac{n\,C_b V_b}{V_a}

ChemistryFinds an analyte's concentration from the titre volume, titrant concentration, and the reaction's mole ratio at the equivalence point.

Total Alkalinity as CaCO₃ from Species

\mathrm{TA} = 0.8202\,\mathrm{HCO_3} + 1.6679\,\mathrm{CO_3} + 2.9425\,\mathrm{OH}

Water TreatmentChemistrySums bicarbonate, carbonate and hydroxide reported as their own ions into one total alkalinity expressed as milligrams per litre of CaCO₃.

Total Hardness as CaCO₃

\mathrm{TH} = 2.497\,\mathrm{Ca} + 4.118\,\mathrm{Mg}

Water TreatmentChemistryConverts a lab report's calcium and magnesium ion results into a single total hardness expressed as milligrams per litre of CaCO₃.

Total Water Treatment Program Cost

C = C_w + C_p + C_e

Water TreatmentChemistryFluid MechanicsTotal operating cost of a treated cooling system: the water and sewer bill, the chemical invoice and the energy bill added together.

van 't Hoff Equation (K at Two Temperatures)

\ln\frac{K_2}{K_1} = -\frac{\Delta H^{\circ}}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)

ChemistryThermodynamicsMoves an equilibrium constant from one temperature to another using the standard enthalpy of reaction, with R = 8.314462618 J/(mol·K).

Van der Waals Equation of State

\left(P + \frac{a n^{2}}{V^{2}}\right)\left(V - n b\right) = n R T

ChemistryThermodynamicsThe first equation of state to describe a real gas, adding a term for molecular attraction (a) and one for the volume the molecules themselves occupy (b). Constant a is entered in Pa·m⁶/mol² and b in volume per mole.

Water Resistivity and Conductivity

\rho = \frac{1}{\sigma}

Water TreatmentChemistryRelates the resistivity and conductivity of ultrapure and deionized water as exact reciprocals, the pair of scales used on DI and RO polishing loops.

Zero-Order Integrated Rate Law

[\mathrm{A}] = [\mathrm{A}]_0 - kt

ChemistryGives the concentration remaining in a zero-order reaction, where the rate is constant and concentration falls in a straight line with time.

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