Free Electrical Mobility Calculator

Understanding the Einstein-Smoluchowski Relation

This electrical mobility calculator is built around the Einstein-Smoluchowski relation (commonly called the Einstein relation), a key equation that connects the random thermal motion of charge carriers to the directed flow they experience under an applied electric field. Whether you are studying semiconductors, electrolytes, or metals, this tool acts as both an electron mobility calculator and a diffusion constant calculator, allowing you to compute one transport parameter from the others.

The Diffusion Constant and Random Carrier Motion

Charge carriers inside a conductor are never truly stationary; they undergo continuous thermal agitation. If carriers are initially confined to a small region, their random displacements cause them to spread throughout the bulk material. The diffusion constant $D$ quantifies how rapidly this spreading occurs. Its units are area per time (for instance, $\text{m}^2/\text{s}$). A helpful way to picture it: if the carriers occupy a certain area at one instant, $D$ represents the speed at which that area expands over time.

Drift Velocity and Electrical Mobility

When a voltage difference is placed across a conductor, an electric field sets the carriers in motion. The carriers accelerate under the field but also undergo collisions with one another and with the lattice. The resulting steady average velocity is called the drift velocity $u$. The electrical mobility $\mu$ is defined as the ratio of the drift velocity to the magnitude of the electric field $E$:

$$\mu = \frac{u}{E}$$

Here $E = \Delta V / L$ (voltage divided by conductor length). Mobility is a material‑specific quantity that expresses how easily a carrier moves in a given medium under an electric field.

The Einstein‑Smoluchowski Formula

The central relation implemented in this calculator links the diffusion constant to the electrical mobility through the Boltzmann constant, temperature, and carrier charge:

$$D = \frac{\mu \, k_B \, T}{q}$$

Variables and units:

• $D$ – diffusion constant ($\text{m}^2/\text{s}$)
• $\mu$ – electrical mobility ($\text{m}^2/(\text{V} \cdot \text{s})$)
• $k_B$ – Boltzmann constant ($1.380649 \times 10^{-23}\ \text{J/K}$)
• $T$ – absolute temperature ($\text{K}$)
• $q$ – charge of the mobile carrier ($\text{C}$)

The formula shows that the diffusion constant scales linearly with both temperature and mobility, and inversely with the carrier charge. It applies to any charged species undergoing thermally activated diffusion, including electrons, holes, and ions.

Worked Examples

Electrons in copper (room temperature)
In a typical copper wire the charge carriers are electrons, so $q = 1.602 \times 10^{-19}\ \text{C}$. The electron mobility in copper at about 300 K is roughly $\mu = 3000\ \text{mm}^2/(\text{V} \cdot \text{s})$, which is $3.0 \times 10^{-3}\ \text{m}^2/(\text{V} \cdot \text{s})$. Plugging into the relation:

$$D = \frac{(3.0 \times 10^{-3}) \cdot (1.38 \times 10^{-23}) \cdot 300}{1.602 \times 10^{-19}} \approx 77.08\ \text{m}^2/\text{s}$$

Sodium ions in water
For Na⁺ ions dissolved in water, the mobility is much lower: $\mu = 0.0519\ \text{mm}^2/(\text{V} \cdot \text{s})$ (which equals $5.19 \times 10^{-8}\ \text{m}^2/(\text{V} \cdot \text{s})$). With $q = 1.602 \times 10^{-19}\ \text{C}$ and the same temperature, the diffusion constant becomes:

$$D = \frac{(5.19 \times 10^{-8}) \cdot (1.38 \times 10^{-23}) \cdot 300}{1.602 \times 10^{-19}} \approx 1.333 \times 10^{-9}\ \text{m}^2/\text{s}$$

which is often expressed as $0.001333\ \text{mm}^2/\text{s}$. These examples highlight how drastically the medium and carrier type influence transport properties.

Using the Calculator

To obtain any unknown parameter — diffusion constant, electrical mobility, temperature, or carrier charge — simply enter the known values into this Einstein‑Smoluchowski relation calculator. The tool accepts common non‑SI units (like $\text{mm}^2/(\text{V} \cdot \text{s})$ for mobility) and converts them automatically before performing the calculation. You can switch between the roles of $D$ and $\mu$ depending on what you are solving for, making it a flexible diffusion constant calculator and electron mobility calculator in one.