Diffusion Coefficient Calculator
Compute diffusion coefficient using validated scientific equations. See step-by-step derivations, unit analysis, and reference values.
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The Stokes-Einstein equation calculates the diffusion coefficient D from Boltzmann constant kB, temperature T, viscosity eta, and particle radius r. For liquid-phase molecular diffusion, the Wilke-Chang correlation provides an empirical estimate.
Last reviewed: December 2025
Worked Examples
Example 1: Protein in Water
Example 2: Small Molecule in Water
Background & Theory
The Diffusion Coefficient Calculator applies the following established principles and formulas. Chemistry is the science of matter's composition, structure, properties, and transformations. At the heart of quantitative chemistry lies the mole concept. One mole of any substance contains exactly 6.022×10²³ entities (Avogadro's number, Nₐ), and the molar mass of an element or compound in grams per mole is numerically equal to its atomic or molecular mass in atomic mass units. This allows chemists to convert between measurable mass and the number of reacting particles. Stoichiometry uses balanced chemical equations to relate the amounts of reactants and products. A balanced equation conserves both mass and charge. Molarity, the most common concentration unit, is defined as M = n/V, where n is moles of solute and V is volume of solution in liters, giving units of mol/L. Acidity and basicity are quantified by the pH scale, defined as pH = −log₁₀[H⁺], where [H⁺] is the molar concentration of hydrogen ions. Pure water at 25°C has pH 7.00; acids have lower values and bases higher values. Each unit change represents a tenfold change in hydrogen ion concentration. Gas behavior is described by the ideal gas law PV = nRT, where P is pressure in pascals, V is volume in cubic meters, n is moles, R = 8.314 J/(mol·K), and T is temperature in kelvin. Special cases include Boyle's Law (P₁V₁ = P₂V₂ at constant temperature) and Charles's Law (V₁/T₁ = V₂/T₂ at constant pressure). Thermochemistry quantifies heat changes in reactions through enthalpy, H. Hess's Law states that the total enthalpy change for a reaction is the sum of enthalpy changes for any sequence of steps leading to the same overall reaction, making it possible to calculate enthalpies for reactions that cannot be measured directly. Electron configuration describes the distribution of electrons in atomic orbitals according to the Aufbau principle, Pauli exclusion principle, and Hund's rule. Periodic trends including atomic radius, ionization energy, and electronegativity arise systematically from electron configuration and nuclear charge, enabling chemists to predict and rationalize chemical behavior across the periodic table.
History
The history behind the Diffusion Coefficient Calculator traces back through the following developments. Chemistry's roots lie in alchemy, the medieval practice combining proto-scientific experimentation with mystical aims. Alchemists developed practical techniques including distillation, calcination, and the preparation of acids, building a body of empirical knowledge despite their theoretical misunderstandings. Modern chemistry is conventionally dated to Antoine Lavoisier (1743–1794), often called the father of modern chemistry. Lavoisier demonstrated the law of conservation of mass in 1789, showing that matter is neither created nor destroyed in chemical reactions. He identified oxygen's role in combustion, dismantling the phlogiston theory, and co-authored the first systematic chemical nomenclature, establishing the language still used today. John Dalton proposed the first modern atomic theory in 1803, asserting that all matter is composed of indivisible atoms, that atoms of the same element are identical in mass, and that compounds form from fixed ratios of different atoms. This provided a physical basis for Lavoisier's conservation law and Proust's law of definite proportions. Dmitri Mendeleev published his periodic table in 1869, arranging the 63 known elements by atomic mass and revealing repeating patterns of chemical behavior. He boldly left gaps for undiscovered elements and predicted their properties with remarkable accuracy, predictions confirmed by the subsequent discovery of gallium, scandium, and germanium. Ernest Rutherford's gold foil experiment in 1911 revealed the nuclear model of the atom: a tiny, dense, positively charged nucleus surrounded by electrons. Niels Bohr refined this in 1913 with a quantized model of electron orbits that explained the hydrogen emission spectrum. Quantum chemistry and molecular orbital theory, developed through the 1920s and 1930s, provided the full quantum mechanical description of chemical bonding. The latter 20th century saw the rise of computational chemistry, enabling molecular simulation at unprecedented scale. The green chemistry movement, articulated in the 12 Principles of Green Chemistry in 1998, reoriented the field toward sustainability, waste reduction, and benign chemical design, reflecting chemistry's growing awareness of its environmental responsibilities.
Frequently Asked Questions
Formula
D = kB * T / (6 * pi * eta * r)
The Stokes-Einstein equation calculates the diffusion coefficient D from Boltzmann constant kB, temperature T, viscosity eta, and particle radius r. For liquid-phase molecular diffusion, the Wilke-Chang correlation provides an empirical estimate.
Frequently Asked Questions
What is the diffusion coefficient?
The diffusion coefficient (D) quantifies how fast a substance spreads through another medium due to random molecular motion. It is measured in units of area per time, typically m²/s or cm²/s. A higher diffusion coefficient means molecules move through the medium more quickly. The value depends on temperature, the size of the diffusing particle, and the viscosity of the surrounding medium. For small molecules in water at room temperature, D is typically on the order of 1e-9 m²/s.
How does temperature affect diffusion?
Temperature has a direct proportional effect on the diffusion coefficient. As temperature increases, molecules gain more kinetic energy and move faster, leading to higher diffusion rates. The Stokes-Einstein equation shows D is linearly proportional to T. However, temperature also affects viscosity, which typically decreases as temperature rises, further increasing diffusion. A rule of thumb is that the diffusion coefficient roughly doubles for every 20-25 K increase in temperature for aqueous solutions.
What are typical diffusion coefficient values?
Diffusion coefficients vary enormously depending on the phase. In gases at atmospheric pressure, D is typically 1e-5 to 1e-4 m²/s. In liquids, D ranges from about 1e-10 to 1e-9 m²/s for small molecules and drops to 1e-11 m²/s or lower for large proteins. In solids, diffusion is extremely slow, with D values of 1e-14 m²/s or smaller. These values are critical for designing chemical reactors, drug delivery systems, and separation processes.
Is my data stored or sent to a server?
No. All calculations run entirely in your browser using JavaScript. No data you enter is ever transmitted to any server or stored anywhere. Your inputs remain completely private.
Does Diffusion Coefficient Calculator work offline?
Once the page is loaded, the calculation logic runs entirely in your browser. If you have already opened the page, most calculators will continue to work even if your internet connection is lost, since no server requests are needed for computation.
What inputs do I need to use Diffusion Coefficient Calculator accurately?
Each field is labelled with the required unit (metric or imperial). Gather your source values before starting — for example, a weight measurement in kilograms, a distance in metres, or a dollar amount — and enter them exactly as measured. The formula section on this page lists every variable and explains what each represents.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy