Science Formula Sheet: Physics, Chemistry and Biology

Mechanics All Mechanics tools

Projectile Motion Simulator
Animate a trajectory and read off range, apex and flight time as you change launch speed and angle.

Galileo, Two New Sciences (1638)

$R = \frac{v_0^{2}\sin 2\theta}{g}$
Elastic and Inelastic Collision Simulator
Collide two bodies and watch momentum survive every time while kinetic energy only survives a perfect bounce.

Conservation of momentum with Newton’s law of restitution

$m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$
Simple Pendulum Simulator
Swing a pendulum, add damping, and watch the small-angle approximation fail at large amplitude.

Huygens, Horologium Oscillatorium (1673)

$T \approx 2\pi\sqrt{\frac{L}{g}}$
Orbit Simulator
One launch speed slider walks an orbit from circle through ellipse to escape.

Kepler’s laws (1609, 1619) with Newton’s law of gravitation (1687)

$T^{2} = \frac{a^{3}}{M}\quad(\text{AU, yr, } M_{\odot})$
Mass Spring Simulator
Simple harmonic motion with damping and a driving force, plus the resonance curve.

Hooke’s law (1678) with simple harmonic motion

$T = 2\pi\sqrt{\frac{m}{k}}$
Density Calculator
Density, mass or volume, with g/cm³ and kg/m³ side by side.

Definition of density

$\rho = \frac{m}{V}$
Kinetic Energy Calculator
Kinetic energy, mass or speed from KE = ½mv², plus momentum.

Classical kinetic energy, from the work-energy theorem

$KE = \tfrac{1}{2}mv^{2}$
Force Calculator (F = ma)
Force, mass or acceleration from F = ma, with g-force and weight.

Newton, Principia (1687), second law of motion

$F = ma$
Pressure Calculator
Solve P = F/A in mixed units and read the answer in atm, bar and psi at once.

Definition of pressure

$P = \frac{F}{A}$
Work and Power Calculator
Work from force, distance and angle, plus the average power over a time.

Definitions of mechanical work and power

$W = Fd\cos\theta$
Gravitational Potential Energy Calculator
PE = mgh solved any way round, with the impact speed for that drop.

Potential energy in a uniform gravitational field

$PE = mgh$
Hooke’s Law Calculator
Force, stiffness or extension from F = kx, plus the energy stored.

Hooke, De Potentia Restitutiva (1678)

$F = kx$
Torque Calculator
Torque from arm, force and angle, in N·m, lbf·ft and lbf·in.

Definition of the moment of a force

$\tau = rF\sin\theta$
Momentum Calculator
Momentum from mass and speed, plus the impulse and force to stop it.

Newton, Principia (1687), definition of momentum

$p = mv$
Centripetal Force Calculator
Force and acceleration for a turn, with the result also in multiples of g.

Acceleration in uniform circular motion

$F = \frac{mv^{2}}{r}$
Kinematics Calculator
Any three of u, v, a, s, t gives the other two, and it names the equation.

Constant-acceleration kinematics

$v = u + at,\quad s = ut + \tfrac{1}{2}at^{2}$
Gravitational Force Calculator
Mutual gravity between two masses, and why you never notice it.

Newton, Principia (1687), law of universal gravitation

$F = G\frac{m_1 m_2}{r^{2}}$
Inclined Plane Simulator
Tilt a slope, set the friction and a push, and see whether the block slides, how fast, and where its energy goes.

Amontons (1699) and Coulomb (1785), laws of friction

$a = g(\sin\theta - \mu_k\cos\theta), \quad \mu_s = \tan\theta_c$
Conservation of Energy Simulator
A cart on a track you reshape, with kinetic, potential and thermal energy bars whose total never changes.

Conservation of energy, counting friction’s work as heat (Joule, 1850)

$\tfrac{1}{2}mv^{2} + mgh + E_{\text{th}} = mgh_0$
Double Pendulum Simulator
Two linked arms, a ghost released a millionth of a degree away, and the moment the two part company.

Lyapunov (1892), measured on a double pendulum by Shinbrot, Grebogi, Wisdom and Yorke (1992)

$\delta(t) \approx \delta_0\,e^{\lambda t}$
Buoyancy Simulator
Lower a block into a liquid on a spring balance and watch the buoyant force share its weight.

Archimedes, On Floating Bodies, Book I

$F_b = \rho_f V_{\text{sub}}\, g$
Free Fall Calculator
Fall time and impact speed from any height, with and without air resistance.

NASA Glenn Research Center, the drag equation and terminal velocity

$t = \sqrt{\frac{2h}{g}}, \quad v = \sqrt{2gh}, \quad v_t = \sqrt{\frac{2mg}{\rho C_d A}}$
Time Dilation Calculator
The Lorentz factor, time dilation and length contraction from a speed, γ, two times or two lengths.

Einstein (1905), On the Electrodynamics of Moving Bodies

$\Delta t = \gamma\,\Delta t_0, \quad L = \frac{L_0}{\gamma}, \quad \gamma = \frac{1}{\sqrt{1 - v^{2}/c^{2}}}$
Motion Graphs Simulator
Run a cart through up to five stages and watch its x-t, v-t and a-t graphs draw, with live gradients and areas.

OpenStax University Physics Volume 1, chapter 3, Motion Along a Straight Line

$v = \frac{dx}{dt},\quad a = \frac{dv}{dt},\quad s = \tfrac{1}{2}(u + v)t$
PID Controller Simulator
Tune Kp, Ki and Kd on a cart or an oven, with actuator limits, anti-windup and a sampled controller.

Åström and Murray, Feedback Systems (2008)

$u(t) = K_p e(t) + K_i \int_0^t e(\tau)\,d\tau + K_d \frac{de}{dt}$
Free-Body Diagram Maker
Draw the forces on a block, a hanging weight, a mass in a lift or your own set to scale, and read off the net force.

Newton, Principia (1687), Corollaries I and II: composition and resolution of forces

$F_{\text{net}} = \sqrt{(\sum F_x)^2 + (\sum F_y)^2}$

Chemistry All Chemistry tools

Collision Theory and Activation Energy Simulator
See how few molecules clear the activation barrier, and why heating changes that so violently.

Arrhenius (1889), with collision theory of reaction rates

$k = Ae^{-E_a/RT}$
Titration Curve Calculator
An exact titration curve, with the equivalence pH and the indicator to match it.

Acid-base equilibrium, after Henderson (1908) and Hasselbalch (1917)

$[\mathrm{H^+}] + [\mathrm{M^+}] = [\mathrm{A^-}] + [\mathrm{OH^-}]$
Radioactive Decay Simulator
Individual nuclei decaying at random, against the curve the formula predicts.

Rutherford and Soddy (1902), exponential decay law

$N = N_0 e^{-\lambda t}$
Molarity Calculator
Solve for any one of concentration, mass, molar mass or volume, with units from nM to M.

IUPAC definition of amount concentration

$c = \frac{n}{V} = \frac{m}{M \cdot V}$
Dilution Calculator
Stock and diluent volumes for any target concentration, plus full serial dilution plans.

Conservation of amount of substance on dilution

$c_1 V_1 = c_2 V_2$
Molar Mass Calculator
Type any formula, including hydrates, and get molar mass, percent composition and grams to moles or particles.

IUPAC standard atomic weights

$M = \sum_i n_i A_{r,i}$
pH Calculator
pH, pOH, [H⁺] and [OH⁻] for strong or weak acids and bases, solved exactly.

Sorensen (1909), definition of pH

$\mathrm{pH} = -\log_{10}[\mathrm{H^+}]$
Henderson-Hasselbalch Calculator
Buffer pH from the acid-base ratio, or the ratio needed for a target pH.

Henderson (1908) and Hasselbalch (1917)

$\mathrm{pH} = \mathrm{p}K_a + \log_{10}\frac{[A^-]}{[HA]}$
Percent Solution Calculator
Percent w/v to g/L, mg/mL or ppm, and the mass to weigh out.

Mass and volume concentration definitions

$\%\,\mathrm{w/v} = \frac{m_{\mathrm{g}}}{V_{\mathrm{mL}}} \times 100$
Beer-Lambert Law Calculator
Concentration from absorbance, or any other term in A = εlc.

Bouguer (1729), Lambert (1760) and Beer (1852)

$A = \varepsilon l c$
Half-Life Calculator
Remaining amount, elapsed time or half-life for any decay process.

Exponential decay, Rutherford and Soddy (1902)

$N = N_0 \left(\tfrac{1}{2}\right)^{t/t_{1/2}}$
Titration Calculator
Find an unknown concentration from endpoint volume, with the stoichiometric ratio applied.

Stoichiometric equivalence at the end point

$r\,c_1 V_1 = c_2 V_2$
Percent Yield Calculator
Percent yield from actual and theoretical mass, with the shortfall.

Definition of percentage yield

$\%\,\text{yield} = \frac{m_{\text{actual}}}{m_{\text{theoretical}}} \times 100$
Empirical Formula Calculator
Empirical formula from composition, plus the molecular formula from molar mass.

Law of definite proportions, Proust (1797)

$\text{ratio}_i = \frac{m_i / A_{r,i}}{\min_j (m_j / A_{r,j})}$
Limiting Reagent Calculator
Limiting reagent, theoretical yield and percent yield in one pass.

Reaction stoichiometry

$\text{extent} = \min_i \frac{n_i}{\nu_i}$
VSEPR Molecular Geometry
Lone pairs drawn, so the shape has a visible reason.

Gillespie and Nyholm (1957)

$\text{steric number} = n_{\text{bond}} + n_{\text{lone}}, \quad \vec{\mu}_{\text{net}} = \sum_i \hat{b}_i$
Crystal Lattice Explorer
Unit cells you can turn, and the contact line that sets the maths.

Close packing of spheres, after Kepler (1611) and Barlow (1883)

$\text{packing} = \frac{Z\,\pi f^{3}}{6}, \quad \rho = \frac{Z M}{N_A a^{3}}$
Titration Lab Simulator
A burette, an indicator and an exact pH meter, with your endpoint measured against equivalence.

Indicator equilibrium, after Henderson (1908) and Hasselbalch (1917)

$\mathrm{pH} = \mathrm{p}K_{\text{In}} + \log_{10}\frac{[\mathrm{In^-}]}{[\mathrm{HIn}]}$
Chemical Equation Balancer
Balance any equation, ions and half-equations included, with the working and an atom count.

Conservation matrix Aν = 0, IUPAC Green Book (2007) after Alberty (1991)

$\sum_j A_{ij}\,\nu_j = 0, \quad \sum_j z_j\,\nu_j = 0$
Rutherford Scattering Simulator
Alphas fired at a nucleus: their exact paths, angles and closest approach, and the counts against Rutherford’s formula.

Rutherford (1911), tested by Geiger and Marsden (1913)

$d = \frac{2Ze^{2}}{4\pi\varepsilon_0K}, \quad \tan\frac{\theta}{2} = \frac{d}{2b}, \quad N(\theta) \propto \frac{1}{\left(\sin\frac{\theta}{2}\right)^{4}}$
Galvanic Cell Simulator
Two metals, two solutions and a salt bridge, run down while the voltage falls and the electrodes change mass.

Nernst (1889), and Faraday’s laws of electrolysis (1834)

$E = E^\circ - \frac{RT}{nF}\ln Q, \quad \Delta m = \frac{ItM}{zF}$
Arrhenius Equation Calculator
Find a reaction’s activation energy from two rate constants, or its rate constant at a new temperature.

Arrhenius (1889), rate constants and temperature

$k = A\,e^{-E_a/(RT)}, \quad \ln\frac{k_2}{k_1} = \frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)$
Bohr Model and Hydrogen Spectrum Simulator
Jump an electron between Bohr orbits and read the photon’s energy, wavelength and series.

Bohr (1913), with the Rydberg formula (1888)

$E_n = -13.6\,\frac{Z^{2}}{n^{2}}\ \text{eV}, \quad \frac{1}{\lambda} = R Z^{2}\left(\frac{1}{n_1^{2}} - \frac{1}{n_2^{2}}\right)$
Atomic Orbitals Visualiser
Orbital shapes from the exact wavefunction, turnable, with every node counted.

Schrödinger (1926), the hydrogen atom solved exactly

$\psi = R_{n\ell}(r)\,Y_{\ell m}(\theta, \phi), \quad P(r) = r^{2}R_{n\ell}(r)^{2}, \quad n - \ell - 1 \text{ radial nodes}, \quad \ell \text{ angular nodes}$
Integrated Rate Law Calculator
Concentration, time, rate constant and half-life at zero, first and second order, and the order from data.

Integrated rate laws, OpenStax Chemistry 2e (2019), section 12.4

$[A] = [A]_0 - kt, \quad \ln[A] = \ln[A]_0 - kt, \quad \frac{1}{[A]} = \frac{1}{[A]_0} + kt$
Electron Configuration Calculator
Configurations of any element or ion in full and noble gas notation, with orbital boxes and unpaired electrons.

Madelung (1936) and Hund (1925), with ion ground states from the NIST Atomic Spectra Database

$N = 2(2\ell + 1), \quad \mu = \sqrt{n(n + 2)}\,\mu_{\text{B}}$
Freezing Point Depression Calculator
ΔT = i K m for freezing and boiling points, solved for the change, molality, i or molar mass.

Raoult (1882) and van ’t Hoff (1887)

$\Delta T_f = i K_f m, \quad \Delta T_b = i K_b m$
Protons, Neutrons and Electrons Calculator
Protons, neutrons and electrons for any isotope or ion, with the nuclide symbol.

IUPAC definitions of atomic number and mass number

$p = Z, \quad n = A - Z, \quad e = Z - q$
Nernst Equation Calculator
Cell potential from two half-reactions at any concentration and temperature, with the free energy, K and the working.

Nernst (1889), with the IUPAC Green Book’s cell conventions

$E = E^\circ - \frac{RT}{nF}\ln Q, \quad \Delta G^\circ = -nFE^\circ = -RT\ln K$
Stoichiometry Calculator
Grams to grams, moles and gas volumes through a balanced equation.

IUPAC Green Book, stoichiometric numbers and amount of substance

$n_{\text{B}} = n_{\text{A}}\frac{\nu_{\text{B}}}{\nu_{\text{A}}}, \quad n = \frac{m}{M} = \frac{PV}{RT} = cV$
ICE Table Calculator
Fill in an ICE table, solve for x exactly and test the small-x shortcut, or find K from equilibrium amounts.

Guldberg and Waage (1864), law of mass action

$K_c = \frac{[\mathrm{C}]^{c}[\mathrm{D}]^{d}}{[\mathrm{A}]^{a}[\mathrm{B}]^{b}}$
Born-Haber Cycle Calculator
Lattice enthalpy or enthalpy of formation, with the Born-Haber cycle drawn to scale and every step labelled.

Born and Haber (1919); data from NIST, with oxygen’s second electron affinity from Huheey’s Inorganic Chemistry

$\Delta_{\text{f}} H = \sum \Delta_{\text{at}} H + \sum \text{IE} + \sum \text{EA} + \Delta_{\text{latt}} H$
NMR Splitting Pattern Visualiser
Choose neighbours and coupling constants to draw the splitting tree and see the multiplet it makes, line by line.

Gutowsky, McCall and Slichter, J. Chem. Phys. 21, 279 (1953)

$\text{lines} = (n_A + 1)(n_B + 1)(n_C + 1)$
Molecular Orbital Diagram Visualiser
Molecular orbital diagrams from H2 to Ne2, filled by Aufbau and Hund’s rule, with bond order and magnetism read off.

Mulliken (1928) and Lennard-Jones (1929), with the orbital order of OpenStax Chemistry 2e

$\text{bond order} = \tfrac{1}{2}\left(n_{\text{bonding}} - n_{\text{antibonding}}\right)$
HPLC and GC Chromatography Simulator
Watch two to four analytes separate on an HPLC or GC column, with retention times, plate count and resolution.

IUPAC chromatography nomenclature (Ettre, 1993)

$R_s = \frac{2(t_{R2} - t_{R1})}{w_1 + w_2}$
Cyclohexane Chair Conformation Explorer
Flip a substituted cyclohexane chair and see which chair wins, by how many kJ/mol, and the ratio at any temperature.

A-values from McMurry, Organic Chemistry (OpenStax, 2023), Table 4.1

$\Delta G^\circ = G_{\text{B}} - G_{\text{A}} = -RT\ln K$
R and S Configuration Explorer
Rank four groups by the CIP rules, turn the stereocentre until group 4 points away, and read R or S from the arrow.

Cahn, Ingold and Prelog (1966); IUPAC 2013, rule P-92

$(\vec{r}_1 \times \vec{r}_2) \cdot \vec{r}_3 < 0 \;\Rightarrow\; \text{R}, \qquad > 0 \;\Rightarrow\; \text{S}$
Crystal Field Splitting Visualiser
Split the d orbitals in three geometries, fill them with electrons and see when Δ beats the pairing energy.

Bethe (1929) and Van Vleck (1932), crystal field theory

$\mathrm{CFSE} = (-0.4x + 0.6y)\,\Delta_o + mP$
Photoelectron Spectroscopy Visualiser
One peak per subshell for any atom from H to Kr, read off as a configuration, with a second element to compare.

Einstein (1905); binding energies from Carlson (1975), Table A1.A, after Lotz (1970)

$E_{\text{binding}} = h\nu - E_{\text{kinetic}}$

Biology All Biology tools

Cell Doubling Time Calculator
Doubling time and growth rate from two counts, or project forward.

Exponential growth kinetics

$T_d = \frac{t \ln 2}{\ln (N/N_0)}$
Centrifuge RCF Calculator
Convert RPM to × g and back for your rotor’s radius.

Definition of relative centrifugal force

$\mathrm{RCF} = 1.118 \times 10^{-5} \, r \, \mathrm{RPM}^{2}\quad(r\text{ in cm})$
Nucleic Acid Quantification Calculator
Turn an A260 reading into ng/µL of DNA or RNA, and get the volume needed for a target mass.

Beer’s law (1852) with standard A260 conversion factors

$c = A_{260} \times \varepsilon \times d$
Action Potential Simulator
Hodgkin-Huxley, with threshold and refractoriness emerging.

Hodgkin and Huxley (1952)

$C_m\frac{dV}{dt} = I - I_{Na} - I_K - I_L$
Oxygen Dissociation Curve Simulator
The sigmoid curve, and what moves it left or right.

Hill (1910), with Severinghaus (1979) standard curve

$S = \frac{(P/P_{50})^{n}}{1 + (P/P_{50})^{n}}$
Enzyme Kinetics Simulator
Km and Vmax, and how each inhibitor changes them.

Michaelis and Menten (1913), Briggs and Haldane (1925)

$v = \frac{V_{max}[S]}{K_m + [S]}$
Acid Base Interpreter
A blood gas worked through, one reasoned step at a time.

Henderson-Hasselbalch, with Albert, Dell and Winters (1967)

$\mathrm{pH} = 6.1 + \log_{10}\!\left(\frac{[\mathrm{HCO_3^-}]}{0.03\,P_{\mathrm{CO_2}}}\right)$
eGFR and Creatinine Clearance Calculator
Three estimates of kidney function, and where they diverge.

Cockcroft and Gault (1976), MDRD Levey (2006), CKD-EPI Inker (2021)

$C_{Cr} = \frac{(140 - \text{age})\times \text{mass}}{72 \times S_{Cr}}$
Mean Arterial Pressure Calculator
MAP and pulse pressure from a blood pressure.

Diastolic pressure plus one third of pulse pressure

$\text{MAP} = P_{dia} + \frac{P_{sys} - P_{dia}}{3}$
Albumin Corrected Calcium Calculator
Total calcium corrected for albumin, both units.

Payne (1973), albumin-adjusted calcium

$\text{Ca}_{corr} = \text{Ca} + 0.8\,(4.0 - \text{albumin})$
Serum Osmolality and Osmolar Gap Calculator
Calculated osmolality, the gap, and tonicity.

Smithline and Gardner (1976), calculated osmolality

$\text{Osm} = 2\,[\text{Na}^+] + \frac{[\text{glucose}]}{18} + \frac{[\text{BUN}]}{2.8}$
A-a Gradient and Alveolar Gas Equation Calculator
Alveolar oxygen, the A-a gradient and P/F ratio.

Alveolar gas equation, Riley and Cournand (1949)

$P_AO_2 = F_iO_2\,(P_{atm} - P_{H_2O}) - \frac{P_aCO_2}{R}$
QTc Calculator
Four QT corrections at once, with thresholds.

Bazett (1920), Fridericia (1920), Hodges (1983), Sagie (1992)

$QTc = \frac{QT}{\sqrt{RR}}, \quad RR = \frac{60}{HR}$
Body Surface Area Calculator
Four BSA formulas, and how far apart they land.

Du Bois (1916), Gehan and George (1970), Haycock (1978), Mosteller (1987)

$\text{BSA} = \sqrt{\frac{h_{cm} \times w_{kg}}{3600}}$
DNA Double Helix Explorer
The grooves, and the strand offset that creates them.

Watson and Crick (1953), with Franklin and Gosling (1953)

$\text{bp per turn} = \frac{360^\circ}{\text{twist}}, \quad \text{pitch} = \text{rise} \times \text{bp per turn}$
Nephron Explorer
The loop, the gradient it builds, and what ADH can do with it.

Countercurrent multiplication, Kuhn and Ryffel (1942)

$\dot{V}_{\text{urine}} = \frac{\dot{n}_{\text{osm}}}{U_{\text{osm}}}, \quad C_{\mathrm{H_2O}} = \dot{V} - \frac{\dot{V}\,U_{\text{osm}}}{P_{\text{osm}}}$
Resting Membrane Potential Simulator
Nernst per ion, Goldman for the membrane.

Nernst (1889), with Goldman (1943) and Hodgkin and Katz (1949)

$V_m = \frac{RT}{F}\ln\frac{P_K[\text{K}^+]_o + P_{Na}[\text{Na}^+]_o + P_{Cl}[\text{Cl}^-]_i}{P_K[\text{K}^+]_i + P_{Na}[\text{Na}^+]_i + P_{Cl}[\text{Cl}^-]_o}$
Primer Tm Calculator
Primer Tm from the sequence and buffer by nearest neighbours, with the Wallace and GC rules beside it.

SantaLucia (1998), with the Mg²⁺ term of von Ahsen et al. (2001)

$T_m = \frac{\Delta H^\circ}{\Delta S^\circ + 0.368\,(N - 1)\ln[\mathrm{Na^+}] + R\ln(C_T/4)}\quad(\Delta H^\circ\text{ in cal/mol, }[\mathrm{Na^+}]\text{ and }C_T\text{ in mol/L, }T_m\text{ in K})$
Ligation Calculator
Nanograms of insert for any insert:vector molar ratio, from vector mass and both lengths.

Insert to vector molar ratio, as given in Promega’s pGEM-T manual (TM042)

$m_{\text{insert}} = R \times m_{\text{vector}} \times \frac{L_{\text{insert}}}{L_{\text{vector}}}$
Microscope Magnification Calculator
Magnification, actual size, field of view and cell size for a light microscope.

Definition of magnification, AQA GCSE Biology specification 8461, section 4.1.1.5

$M = \frac{\text{image size}}{\text{actual size}}$
Punnett Square Calculator
Genotype and phenotype ratios for crosses of up to three genes, with the square drawn.

Mendel (1866), segregation and independent assortment

$P(AaBb) = P(Aa) \times P(Bb)$
Reverse Complement Calculator
Reverse complement, complement or reverse of DNA or RNA, with IUPAC codes, length and GC content.

Watson and Crick (1953), with the NC-IUB codes of Cornish-Bowden (1985)

$r_i = c(s_{n+1-i}), \quad \mathrm{GC} = \frac{G + C + S}{A + C + G + T + U + S + W} \times 100\%$
SIR Epidemic Model Simulator
SIR and SEIR epidemics with R₀, herd immunity and vaccination.

Kermack and McKendrick (1927)

$\frac{dS}{dt} = -\beta\frac{SI}{N}, \quad \frac{dI}{dt} = \beta\frac{SI}{N} - \gamma I$
Predator-Prey Simulator
Lotka-Volterra cycles over time and in the phase plane, with the equilibrium and exact period.

Lotka (1925) and Volterra (1926)

$\frac{dx}{dt} = \alpha x - \beta xy,\quad \frac{dy}{dt} = \delta xy - \gamma y$
Photosynthesis Rate Simulator
Light, CO₂ and temperature against the rate, with the limiting factor named and pondweed bubbles counted.

Farquhar, von Caemmerer and Berry (1980)

$A = \min(A_c, A_j) - R_d$
Cardiac Cycle Simulator
Pressures, volume, valves and ECG through one heartbeat.

Time-varying elastance, Suga and Sagawa (1974)

$P_{\text{LV}} = e(t)\,E_{\text{es}}(V - V_0) + \left(1 - e(t)\right)P_0\,e^{\beta V}$
DNA to Protein Translator
Paste DNA or mRNA and read off the protein, codon by codon, in every reading frame.

Crick, Barnett, Brenner and Watts-Tobin (1961), with NCBI translation table 1

$n_{\text{aa}} = \frac{L_{\text{ORF}}}{3} - 1, \quad 4^{3} = 64 \text{ codons}$
Diffusion and Osmosis Simulator
Diffusion or osmosis across one membrane, counted against Fick’s law.

Fick (1855) and van ’t Hoff (1887)

$J = -D\frac{dc}{dx}, \quad \Pi = cRT$
Hardy-Weinberg Calculator
Allele and genotype frequencies from one known value or from counts, with a chi-square test of equilibrium.

Hardy (1908) and Weinberg (1908)

$p^{2} + 2pq + q^{2} = 1, \quad p + q = 1$
Cell Structure Explorer
Labelled animal and plant cells, with what every organelle is, does and measures.

Surface area and volume of a sphere, as in OpenStax Biology 2e (2018), section 4.2

$\frac{A}{V} = \frac{4\pi r^{2}}{\tfrac{4}{3}\pi r^{3}} = \frac{3}{r}$
Mitosis and Meiosis Explorer
Mitosis and meiosis stage by stage, with chromosome and chromatid counts for your 2n.

Sutton (1903), The Chromosomes in Heredity

$N_{\text{gametes}} = 2^{n}, \quad N_{\text{zygotes}} = 2^{n} \times 2^{n} = 4^{n}$
Protein Synthesis Simulator
Type a gene and watch RNA polymerase, a ribosome and tRNAs build its protein, step by step.

Elongation rates from Milo and Phillips (2015), Cell Biology by the Numbers

$t = \frac{L}{v}$
Water Potential Calculator
Solute and water potential for a cell and the solution around it, the way water moves, and the state the cell ends in.

van ’t Hoff (1887) and the AP Biology equation sheet

$\psi = \psi_{s} + \psi_{p}, \quad \psi_{s} = -iCRT$
Mark-Recapture Population Calculator
Population size from marked, caught and recaptured counts by Lincoln-Petersen and Chapman, with a 95% interval.

Petersen (1896), Lincoln (1930) and Chapman (1951)

$N = \frac{MC}{R}, \quad N_{C} = \frac{(M+1)(C+1)}{R+1} - 1$
Simpson’s Diversity Index Calculator
Simpson’s D, 1 − D and 1/D from species counts, with Shannon’s index, evenness and a second habitat to compare.

Simpson (1949), Measurement of diversity, Nature 163: 688; Shannon (1948)

$D = \frac{\sum n(n-1)}{N(N-1)}, \quad H' = -\sum p \ln p$
qPCR and ΔΔCt Simulator
Four Ct values to a fold change by 2^−ΔΔCt or Pfaffl, with amplification curves, a threshold and a standard curve.

Livak and Schmittgen (2001); Pfaffl (2001) for unequal efficiencies

$\text{fold change} = 2^{-\Delta\Delta C_t},\quad \Delta\Delta C_t = (C_{t,\text{target}} - C_{t,\text{ref}})_{\text{treated}} - (C_{t,\text{target}} - C_{t,\text{ref}})_{\text{control}}$
Muscle Twitch and Tetanus Simulator
Fire a muscle at any frequency and watch single twitches sum into unfused and then fused tetanus.

Milner-Brown, Stein and Yemm (1973) twitch, summed as in Fuglevand, Winter and Patla (1993)

$A(t) = \sum_k \frac{t - t_k}{T}\, e^{1 - (t - t_k)/T}, \quad F = F_{\text{max}}\left[1 - (1 - \rho)^{A}\right]$
Pharmacokinetics Dosing Simulator
Build repeated doses up to steady state and read the peak, trough, average, accumulation factor and loading dose.

Gibaldi and Perrier, Pharmacokinetics (1982); Rowland and Tozer (2011)

$C_{\mathrm{ss}} = \frac{F \cdot D}{\mathrm{CL} \cdot \tau}$
ECG Rhythm and Heart Block Simulator
A sweeping lead II strip and ladder diagram for sinus rhythm, heart blocks, AF, flutter and ectopic beats.

Kligfield et al. (2007) ECG recording standards; Kusumoto et al. (2018) ACC/AHA/HRS guideline

$\text{HR} = \frac{60}{RR} = \frac{1500}{RR_{\text{mm}}}$
Bacterial Unknown Identification Lab
Run standard tests on 15 teaching bacteria, or enter your own results, and watch a dichotomous key narrow the field.

Reference reactions from Farmer et al., J. Clin. Microbiol. 21:46 (1985), and Bergey’s Manual of Systematic Bacteriology

$o \text{ fits} \iff x(t) \in R(o, t) \text{ for every test } t \text{ run}$
Gel Electrophoresis Simulator
Run a virtual agarose gel with a ladder and digests, and size an unknown band from the semi-log standard curve.

Southern (1979); Helling, Goodman and Boyer (1974); Stellwagen, Gelfi and Righetti (1997)

$d = \frac{\mu_0 E t}{1 + L / L_{1/2}}, \quad \log_{10} L \approx a - b\,d$
Spirometry and Flow-Volume Loop Simulator
Watch normal, obstructive, restrictive and upper airway flow-volume loops form, with FEV1/FVC read out.

ATS/ERS interpretive strategies: Pellegrino et al. (2005), Stanojevic et al. (2022)

$\frac{\mathrm{FEV_1}}{\mathrm{FVC}} < \mathrm{LLN}$

Electricity All Electricity tools

RC Charge and Discharge Simulator
Charge a capacitor through a resistor and watch where the time constant comes from.

Exponential RC response, from Kirchhoff’s laws

$V_C(t) = V_0\left(1 - e^{-t/RC}\right)$
RLC Circuit Simulator
A series RLC transient, from ringing through critical damping to sluggish.

Damped LC oscillation, after Thomson (1853)

$L\frac{d^{2}q}{dt^{2}} + R\frac{dq}{dt} + \frac{q}{C} = V$
LED Resistor Calculator
Series resistor value, next standard part up and power for any LED.

Ohm’s law applied across the diode forward voltage

$R = \frac{V_s - V_f}{I}$
RC Time Constant Calculator
Time constant, cutoff frequency and settling time for any RC pair.

RC time constant

$\tau = RC$
Ohm’s Law Calculator
Voltage, current or resistance from V = IR, plus power dissipation.

Ohm, Die galvanische Kette (1827)

$V = IR$
Capacitor Energy Calculator
Energy and charge stored in a capacitor from capacitance and voltage.

Energy stored in a charged capacitor

$E = \tfrac{1}{2}CV^{2}$
Wire Resistance Calculator
Conductor resistance from length and cross-section, plus the voltage drop.

Resistivity and Pouillet’s law

$R = \frac{\rho L}{A}$
Resistor Network Calculator
Series and parallel networks, with the current in each resistor shown.

Kirchhoff’s circuit laws (1845)

$\frac{1}{R_p} = \sum_i \frac{1}{R_i}, \quad R_s = \sum_i R_i$
Coulomb’s Law Calculator
Force between two point charges, plus the field one of them produces.

Coulomb (1785)

$F = k\frac{q_1 q_2}{r^{2}}$
Solenoid Magnetic Field Simulator
The field through and around a real coil, and the halving at its ends.

Biot and Savart (1820), with Ampere (1823)

$B(z) = \frac{\mu_0 n I}{2}\left[\frac{\ell/2 + z}{\sqrt{R^2 + (\ell/2+z)^2}} + \frac{\ell/2 - z}{\sqrt{R^2 + (\ell/2-z)^2}}\right]$
Resistor Colour Code Calculator
Resistor bands to ohms and back, with tolerance, temperature coefficient and the nearest E-series value.

IEC 60062:2016, Marking codes for resistors and capacitors

$R = (10a + b) \times 10^{n}, \quad R = (100a + 10b + c) \times 10^{n}$
Electric Field Simulator
Point charges you can drag, with field lines, equipotentials and a probe reading E and V.

Coulomb (1785), with the superposition principle

$E = \frac{k|q|}{r^{2}}, \quad V = \sum_i \frac{kq_i}{r_i}$
Electromagnetic Induction Simulator
A magnet moved through a coil: induced EMF, current and Lenz’s law direction.

Faraday (1831) and Lenz (1834), with the flux of a magnetic dipole

$\varepsilon = -N\frac{d\Phi}{dt}, \quad \Phi = \frac{\mu_0 m a^{2}}{2(a^{2} + z^{2})^{3/2}}$
Capacitors in Series and Parallel Calculator
Series, parallel and mixed capacitor networks, with the charge on each capacitor.

Kirchhoff’s circuit laws (1845) and conservation of charge

$\frac{1}{C_s} = \sum_i \frac{1}{C_i}, \quad C_p = \sum_i C_i$
Series and Parallel Circuit Simulator
Bulbs, switches and meters in series, parallel and bridge circuits, solved exactly.

Kirchhoff’s circuit laws (1845), by modified nodal analysis

$R_{\text{s}} = R_1 + R_2 + R_3, \quad \frac{1}{R_{\text{p}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}, \quad P = I^{2}R$
Voltage Divider Calculator
Two resistors in series: the output voltage, or the resistor that gives a target output, with loading shown.

Ohm (1827), with Kirchhoff’s voltage law (1845)

$V_{\text{out}} = V_{\text{in}}\frac{R_2}{R_1 + R_2}$
Charged Particle in a Magnetic Field Simulator
Circles, helices, a velocity selector and a mass spectrometer, from the exact Lorentz force.

Heaviside (1889) and Lorentz (1895), the force on a moving charge

$F = qvB\sin\theta, \quad r = \frac{mv}{qB}, \quad T = \frac{2\pi m}{qB}, \quad v = \frac{E}{B}$
Oscilloscope Simulator
Two signal generators on a scope screen: set the volts/div, time/div and trigger, then read the trace off the grid.

Tektronix, XYZs of Oscilloscopes (primer)

$V_{\text{pp}} = n_{y} \times \text{V/div}, \quad T = n_{x} \times \text{s/div}, \quad f = \frac{1}{T}$
Bode Plot and Filter Visualiser
Gain and phase of RC and RLC filters on log axes, with the straight-line asymptotes, the −3 dB point and Q.

Horowitz and Hill, The Art of Electronics, 3rd edition (2015)

$G = -10\log_{10}\left[1 + \left(\frac{f}{f_c}\right)^{2}\right],\quad f_c = \frac{1}{2\pi RC}$

Waves & Optics All Waves & Optics tools

Standing Wave Simulator
Harmonics on a string, with the two travelling waves that superpose to make them.

Normal modes of a vibrating string, Mersenne (1636)

$f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}}$
Ray Diagram Simulator
The three principal rays, with the image located where they cross.

Gaussian optics, thin-lens and mirror equations

$\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$
Wavelength and Frequency Calculator
Wavelength to frequency and back, for light, sound or any wave speed.

Wave relation between speed, frequency and wavelength

$v = \lambda f$
Photon Energy Calculator
Photon energy from wavelength, in eV, joules and kJ per mole.

Planck (1900) and Einstein (1905)

$E = \frac{hc}{\lambda}$
Snell’s Law Calculator
Solve n₁sinθ₁ = n₂sinθ₂ for any term, with the critical angle shown for the pair of media.

Snell (1621), published by Descartes (1637)

$n_1 \sin\theta_1 = n_2 \sin\theta_2$
Thin Lens Calculator
Solve the thin lens equation and get magnification with the image type spelled out.

Thin-lens equation, Gaussian optics

$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$
Decibel Calculator
Sound level from intensity, with distance falloff and safe exposure time.

Logarithmic level definition, the decibel

$L = 10\log_{10}\!\left(\frac{I}{I_0}\right)$
Double Slit Simulator
Interference fringes, and the single-slit envelope shaping them.

Young (1801), double-slit interference

$d\sin\theta = m\lambda$
Doppler Effect Simulator
Wavefronts from a moving source, and the shift they produce.

Doppler (1842)

$f' = f\,\frac{v \pm v_o}{v \mp v_s}$
Electromagnetic Spectrum Explorer
Radio waves to gamma rays on one scale, with wavelength, frequency and photon energy linked.

Wave relation c = λf, with Planck (1900) and Einstein (1905) for E = hf

$c = \lambda{}f, \quad E = hf = \frac{hc}{\lambda}$
Wave Interference Simulator
Two coherent sources, their nodal lines, and the path difference at any point.

Young (1807), interference of two sets of circular waves

$|r_1 - r_2| = m\lambda \text{ (constructive)}, \quad |r_1 - r_2| = \left(m + \tfrac{1}{2}\right)\lambda \text{ (destructive)}$
Refraction Simulator
Rays through a boundary, a glass block and a prism, with total internal reflection and white light split into colours.

Snell (1621) and Descartes (1637), with Fresnel (1823) for reflection

$n_1\sin\theta_1 = n_2\sin\theta_2, \quad \sin\theta_c = \frac{n_2}{n_1}$
Photoelectric Effect Simulator
Which light frees electrons from a metal, how fast they leave, and what stops them.

Einstein (1905), photoelectric equation

$K_{\text{max}} = eV_0 = hf - \phi, \quad f_0 = \frac{\phi}{h}$
De Broglie Wavelength Calculator
Matter wavelength, h/p, from speed, kinetic energy, voltage or momentum.

de Broglie (1924), matter waves

$\lambda = \frac{h}{p} = \frac{h}{mv}, \quad \lambda = \frac{h}{\sqrt{2meV}}$
Sampling and Aliasing Visualiser
A sine sampled at any rate, and the slower sine its samples describe once the rate falls below twice its frequency.

Oppenheim and Schafer, Discrete-Time Signal Processing, 3rd ed., ch. 4

$f_{\text{alias}} = \left| f - k f_s \right|$

Thermodynamics All Thermodynamics tools

Heat Engine and Carnot Cycle Simulator
Run a gas around a thermodynamic cycle and read the work off the enclosed area.

Carnot, Réflexions sur la puissance motrice du feu (1824)

$\eta_{\text{Carnot}} = 1 - \frac{T_C}{T_H}$
Ideal Gas Law Calculator
Solve PV = nRT for any variable, with kelvin conversion handled for you.

Clapeyron (1834), combining the Boyle, Charles and Avogadro laws

$PV = nRT$
Calorimetry Calculator
Heat, mass, specific heat or ΔT from q = mcΔT, in joules or calories.

Specific heat capacity, after Joseph Black (c. 1760)

$q = mc\Delta T$
Heating Curve Calculator
Energy from ice to steam, counting every stage a single m c dT misses.

Specific and latent heat, after Joseph Black (c. 1760)

$Q = \sum m c \Delta T + \sum m L$
Latent Heat Calculator
Heat to melt or boil a mass, and why the thermometer stops moving.

Latent heat, Joseph Black (c. 1761)

$Q = mL$
Thermal Expansion Calculator
Length change from a temperature change, plus the stress if it cannot move.

Linear thermal expansion coefficient

$\Delta L = \alpha L_0 \Delta T$
Combined Gas Law Calculator
Relate pressure, volume and temperature between two states of a sealed gas.

Boyle (1662), Charles (1787) and Gay-Lussac (1802) combined

$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$
Kinetic Theory Gas Simulator
Atoms in a box, with the gas law measured rather than assumed.

Kinetic theory of gases, Maxwell (1860) and Boltzmann (1872)

$P A = N k T,\quad f(v) \propto v\,e^{-mv^{2}/2kT}$
Boyle’s Law Calculator
Squeeze or expand a gas at one temperature and get its new pressure or volume, plus the work done.

Boyle (1662), pressure and volume at constant temperature

$P_1V_1 = P_2V_2$
States of Matter Simulator
Atoms that freeze, melt and evaporate as you heat and cool them.

Lennard-Jones (1931), the 12-6 potential between atoms

$U(r) = 4\varepsilon\left[\left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6}\right]$
Charles’s Law Calculator
Warm or cool a gas at constant pressure and get its new volume or temperature, with kelvin handled.

Gay-Lussac (1802), crediting unpublished work by Charles (1787)

$\frac{V_1}{T_1} = \frac{V_2}{T_2}$
Gibbs Free Energy Calculator
ΔG from ΔH, T and ΔS, or any one of them, with the spontaneity verdict, the crossover temperature and K.

Gibbs (1876), On the Equilibrium of Heterogeneous Substances

$\Delta G = \Delta H - T\Delta S, \quad \Delta G^\circ = -RT\ln K$
Gay-Lussac’s Law Calculator
Heat or cool a gas in a sealed, rigid container and get its new pressure or temperature.

Amontons (1702) and Gay-Lussac (1802)

$\frac{P_1}{T_1} = \frac{P_2}{T_2}$
Psychrometric Chart Explorer
Drag a state on the psychrometric chart to read its dew point, wet bulb and enthalpy, then heat, cool or mix it.

ASHRAE Handbook Fundamentals (2017), chapter 1, with Hyland and Wexler (1983)

$W = 0.621945\,\frac{p_w}{p - p_w}, \quad h = 1.006\,t + W\,(2501 + 1.86\,t)$

Maths & Data All Maths & Data tools

Riemann Sum and Integral Explorer
Watch rectangles converge on an integral, and compare how fast each rule gets there.

Riemann, Habilitationsschrift (1854)

$\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i)\,\Delta x$
Derivative and Tangent Line Explorer
Sweep a tangent along a curve and watch the derivative draw itself underneath, one gradient at a time.

Differential calculus, Newton and Leibniz

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$
Derivative Calculator
Exact derivatives with every rule named, from the product rule to the chain rule.

Leibniz (1684), rules for differentiating products and quotients

$\frac{d}{dx}\left[u v\right] = u' v + u v'$
Standard Deviation Calculator
Mean, SD, SEM and a t-based confidence interval from pasted measurements.

Pearson (1894), standard deviation

$s = \sqrt{\frac{\sum_i (x_i - \bar{x})^2}{n-1}}$
Error Propagation Calculator
Uncertainty through sums, differences, products, quotients, powers and multipliers, properly rounded.

First-order propagation of uncertainty, ISO/IEC Guide 98-3 (GUM)

$\sigma_f = \sqrt{\sum_i \left(\frac{\partial f}{\partial x_i}\right)^{2}\sigma_{x_i}^{2}}$
Percent Error Calculator
Compare a measurement to the accepted value and see percent, absolute and signed error.

Definition of relative error

$\%\,\text{error} = \frac{|x_{\text{meas}} - x_{\text{true}}|}{|x_{\text{true}}|} \times 100$
Unit Converter
Convert between every unit this site uses, with the whole group shown at once.

SI Brochure, BIPM

$x_{\text{to}} = \frac{x_{\text{from}} f_{\text{from}} + o_{\text{from}} - o_{\text{to}}}{f_{\text{to}}}$
3D Vector and Cross Product Explorer
See where the cross product points, and why its length is an area.

Vector analysis, Gibbs and Wilson (1901)

$\vec{a} \times \vec{b} = \begin{pmatrix} a_y b_z - a_z b_y \\ a_z b_x - a_x b_z \\ a_x b_y - a_y b_x \end{pmatrix}, \quad |\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta$
Fourier Series Visualiser
Harmonics added one at a time and drawn by turning circles, overshoot and all.

Fourier, Théorie analytique de la chaleur (1822)

$f(x) = \frac{4A}{\pi}\sum_{n\,\text{odd}} \frac{1}{n}\sin nx$
Significant Figures Calculator
Count sig figs digit by digit, round to any number of them, and carry them through arithmetic.

NIST Guide to the SI, SP 811 (2008), sections 7.9 and B.7.1

$x_n = \mathrm{round}\left(\frac{x}{10^{k}}\right) \times 10^{k}, \quad k = \mathrm{floor}(\log_{10}|x|) - n + 1$
Unit Circle Explorer
Drag a point round the circle and read sin, cos and tan as lengths, with exact values at the special angles.

Unit circle definitions, NCERT Class XI Mathematics, chapter 3

$(x, y) = (\cos\theta, \sin\theta), \quad \tan\theta = \frac{y}{x}, \quad x^{2} + y^{2} = 1$
Linear Regression Calculator
Least-squares line of best fit from pasted data, with uncertainties on the slope and intercept.

Method of least squares, Legendre (1805) and Gauss (1809)

$m = \frac{n\sum xy - \sum x \sum y}{n\sum x^{2} - (\sum x)^{2}}, \quad c = \frac{\sum y - m\sum x}{n}$
Percent Difference Calculator
The gap between two values as a percentage of their average, with the percent change each way.

Relative percent difference, US EPA SW-846 Chapter One (2014)

$\%\,\text{difference} = \frac{|a - b|}{(|a| + |b|)/2} \times 100$
Chi-Square Calculator
Goodness of fit and contingency tables, with expected counts, p-value, critical values and working.

Pearson (1900), chi-squared test

$\chi^{2} = \sum \frac{(O - E)^{2}}{E}$
Scientific Notation Converter
Scientific, E and engineering notation both ways, with working, plus arithmetic in scientific notation.

NCERT Class XI Chemistry, section 1.4.1, Scientific Notation

$x = a \times 10^{n}, \quad n = \mathrm{floor}(\log_{10}|x|), \quad a = \frac{x}{10^{n}}$
Frequency Density and Histogram Calculator
Frequency density for each class, a histogram where area is frequency, and the grouped mean, median and modal class.

Freedman, Pisani and Purves, Statistics, 4th ed. (2007), ch. 3

$\text{frequency density} = \frac{\text{frequency}}{\text{class width}}$
Error Bars and Worst-Fit Line Calculator
Steepest and shallowest lines through every error bar, with the uncertainties in gradient and intercept they give.

IB Diploma Physics guide (2014), topic 1.2: uncertainty of gradient and intercepts

$\Delta m = \frac{m_{\text{max}} - m_{\text{min}}}{2}, \quad \text{percentage uncertainty} = \frac{\Delta m}{m} \times 100$
Sensitivity and Specificity Visualiser
Move a test’s cut-off and watch sensitivity, specificity and PPV change, on two bell curves and 1,000 people.

Yerushalmy, Public Health Reports (1947)

$\text{sensitivity} = \frac{\text{TP}}{\text{TP} + \text{FN}}, \quad \text{specificity} = \frac{\text{TN}}{\text{TN} + \text{FP}}$