Electrical Mobility Calculator - Einstein Relation Solver

An electrical mobility calculator turns a measured diffusion constant, a known carrier charge, and an absolute temperature into the drift velocity per unit electric field that the carrier would show in the host material. The result comes back in m^2/(V s) for physics work and in cm^2/(V s) and mm^2/(V s) for electrolyte and semiconductor tables, so the same answer lines up with every reference unit in the lab notebook.

• • Electron Mobility in Metals: Working out the drift response of conduction electrons in copper or aluminum from a measured D.
• • Ionic Mobility in Electrolytes: Converting an ionic diffusion coefficient into the ionic mobility used for H+, Na+, K+, and Cl-.
• • Semiconductor Carrier Response: Checking the electron and hole mu that goes into drift-diffusion models.
• • Drift Velocity in a Known Field: Reading the drift velocity when the electric field is known, so the answer lines up with drift-velocity tables.

Mobility is the bridge between random thermal motion and the directed drift an applied field produces. The calculator covers the full bridge: D on one end, the electric field on the other, and mu in the middle.

When the mobility result needs a sanity check against the basic V = I * R relationship in the same conductor, Ohm's Law calculator shows how the same current, voltage, and resistance values line up against the mu value the Einstein relation recommends.

The mobility formula divides the diffusion constant times the carrier charge by the Boltzmann constant times the absolute temperature. The same form in reverse gives D from a known mu.

• Diffusion constant D: Carrier diffusion constant in m^2/s. Copper electrons at 293.15 K sit near 7.577e-5 m^2/s; Na+ in water near 1.31e-9 m^2/s.
• Carrier charge q: Signed carrier charge in coulombs. 1 e = 1.602176634e-19 C.
• Temperature T: Absolute temperature in kelvin. 293.15 K is 20 C; 298.15 K is 25 C.
• Boltzmann constant k_B: Fixed at 1.380649 x 10^-23 J/K.
• Electric field E: Optional electric field in V/m. Multiplied by mu to give the drift velocity u = mu * E.

The mm^2/(V s) result lines up with electron-mobility tables in semiconductor physics, and the cm^2/(V s) result is the same number for electrolyte chemistry.

According to Omni Calculator, Electrical Mobility, the Einstein-Smoluchowski relation is D = mu k_B T / q, with the copper electron mobility example returning about 3000 mm^2/(V s) at room temperature.

According to NIST CODATA, Boltzmann constant, the Boltzmann constant is fixed at 1.380649 x 10^-23 J/K by the 2019 redefinition of the SI, which is the value used in the Einstein-Smoluchowski relation.

When the project needs the matching conductivity sigma = n q mu after the mu value comes back, conductivity to resistivity calculator converts between S/m and ohm-metres at the same temperature so the answer can be paired with the conductivity value the circuit actually uses.

Four small concepts explain why the mu value carries through to conductivity, drift velocity, and the Hall coefficient.

Once you know mu, three other transport quantities fall out: conductivity sigma = n q mu, drift velocity u = mu * E, and the Hall coefficient R_H = 1 / (n q).

When the carrier is a single bound electron and the mu value has to be paired with the orbital radius and the energy levels of the same atom, Bohr model calculator returns the Bohr radius, energy, and orbital speed so the answer lines up with the underlying atomic model.

Six steps take you from a measured diffusion constant to a mu value plus the matching drift velocity on one screen.

1. 1 Pick the Carrier Preset: Choose the carrier that matches the diffusion constant. The preset fills the charge field with the right signed multiple of e.
2. 2 Enter the Diffusion Constant: Type D in m^2/s. For textbook values use 7.577e-5 m^2/s for copper electrons at 293.15 K or 1.31e-9 m^2/s for Na+ in water.
3. 3 Confirm the Carrier Charge: Check the charge field shows the right signed multiple of e. Edit freely for a custom carrier the preset does not cover.
4. 4 Set the Temperature: Type T in kelvin. 293.15 K is 20 C; 298.15 K is 25 C; cryogenic carriers use 4.2 K, 77 K, or the actual bath temperature.
5. 5 Add an Electric Field for Drift Velocity: Optional. Type the electric field in V/m to read u = mu * E. Leave at 0 to skip the drift velocity output.
6. 6 Read the Mobility and Drift Velocity: The results panel shows mu in m^2/(V s), cm^2/(V s), and mm^2/(V s), then drift velocity in m/s. Match the prefix to the table being checked.

Picture a copper Hall bar measured at 20 C. Pick the Electron preset, type 7.577e-5 m^2/s for D, leave q at 1.602176634e-19 C, type 293.15 K, and enter 100 V/m. The calculator returns mu = 2.999e-3 m^2/(V s) = 30.0 cm^2/(V s) = 2999 mm^2/(V s) and a drift velocity of 0.30 m/s.

When the drift velocity u = mu * E needs a velocity, distance, or acceleration check against the same carrier over a known gap, kinematics and motion calculator runs the kinematic formulas for constant acceleration so the result can be cross-checked against a different motion model.

A small dedicated mobility calculator saves the unit-conversion steps and gives the same answer in every unit a transport table could ask for.

• • Three Mobility Units at Once: Returns mu in m^2/(V s), cm^2/(V s), and mm^2/(V s), so one input covers every reference unit.
• • Carrier Preset Saves a Lookup: Fills the signed multiple of e for electron, proton, Na+, K+, and Cl-, so the user does not have to remember the sign.
• • Drift Velocity Built In: Adds u = mu * E when an electric field is supplied, so the same screen gives the drift velocity for J = n q u.
• • Uses Exact NIST Constants: Reads k_B and e at the 2019 SI values from NIST CODATA.
• • Covers Standard and Custom Carriers: Switches to a free-text charge field for exotic ions.

The biggest practical benefit is that the carrier preset removes the most common mistake: forgetting the sign of the carrier charge when D was measured for a negative carrier like an electron or Cl-.

When the carrier is an ion in an electrolyte and the mu value has to be paired with the equilibrium potential across the same cell, Nernst equation calculator returns the cell voltage from concentration, temperature, and electron count so the answer lines up with the electrochemistry result.

Five physical and measurement factors shift the computed mu up or down from the textbook value.

• • The Einstein relation assumes the carrier population is in thermal equilibrium with the host. Out-of-equilibrium carriers from laser excitation break the relation until the gas thermalizes.
• • The formula gives a single mu per carrier per temperature. Real materials have a distribution across grains and defects; the calculator returns the average.

Practical measurements depend on how D was measured. A Hall bar gives the Hall mobility, which differs from the drift mobility when more than one band is occupied; NMR gives the tracer diffusion, which is what the Einstein relation expects.

According to Wikipedia, Electrical mobility, electrical mobility is the ratio of the drift velocity to the magnitude of the electric field, and the Einstein relation mu = q D / (k T) is exact in both the gas phase and the liquid phase.

When the carrier population sits in a thermal distribution and the mu value has to be paired with the relative probability of the same energy level, Boltzmann factor calculator returns exp(-E / k_B T) so the answer can be read against the same thermal weight.