Diffusion time estimate (simple model)
Diffusion is the random motion of molecules that leads to net transport from higher concentration to lower
concentration. In biology, diffusion is important for short-distance transport (across membranes, within thin tissues),
but it becomes slow over longer distances.
This calculator uses a simple diffusion timescale approximation that captures the key scaling:
time grows with the square of distance.
Core equation
For a one-dimensional distance \(L\), a common estimate for the characteristic diffusion time is:
\[
t \approx \frac{L^{2}}{2D}
\]
- \(t\): diffusion time (seconds)
- \(L\): diffusion distance (meters)
- \(D\): diffusion coefficient \((\text{m}^{2}/\text{s})\)
The factor “2” corresponds to a 1D mean-square displacement relationship
\(\langle x^{2}\rangle \approx 2Dt\). This calculator treats \(L\) as a characteristic distance in that 1D sense.
Solving for distance
If you know a time \(t\) and want the approximate distance a molecule can diffuse:
\[
L \approx \sqrt{2Dt}
\]
This “distance from time” mode is useful for quick intuition: diffusion distance grows with the square root of time.
Scaling rule (why diffusion becomes limiting)
From \(t \propto L^{2}\):
- If distance doubles (\(L \to 2L\)), time becomes ~4 times larger (\(t \to 4t\)).
- If distance increases 10×, time becomes ~100× larger.
This is why diffusion is excellent at micrometer scales but inefficient across millimeters or centimeters without bulk flow.
Units and conversions used in the calculator
The calculator converts your inputs to SI units for computation, then converts back to your chosen output unit.
Length
\[
1\ \mu\text{m} = 10^{-6}\ \text{m},
\qquad
1\ \text{mm} = 10^{-3}\ \text{m}
\]
Time
\[
1\ \text{min} = 60\ \text{s},
\qquad
1\ \text{h} = 3600\ \text{s}
\]
Diffusion coefficient
\[
D\ (\mu\text{m}^{2}/\text{s}) = D\ (\text{m}^{2}/\text{s}) \cdot 10^{12},
\qquad
D\ (\text{mm}^{2}/\text{s}) = D\ (\text{m}^{2}/\text{s}) \cdot 10^{6}
\]
Because \(D\) is a squared-length per time, its unit conversions are squared as well.
Typical magnitudes of \(D\) (order-of-magnitude intuition)
Diffusion coefficients depend strongly on molecule size, temperature, and the medium (water vs cytoplasm vs gel).
Presets in the calculator provide rough reference values:
- Small molecules in water: on the order of \(10^{-9}\ \text{m}^{2}/\text{s}\)
- Larger molecules: lower \(D\) (often \(10^{-10}\) to \(10^{-11}\ \text{m}^{2}/\text{s}\), sometimes lower)
These are not fixed constants—use them for quick estimates, and enter a custom value if you have one.
Worked example (find time)
Estimate the time for a small molecule to diffuse \(L = 10\ \mu\text{m}\) in water
with \(D = 1\times 10^{-9}\ \text{m}^{2}/\text{s}\).
\[
L = 10\ \mu\text{m} = 10\cdot 10^{-6}\ \text{m} = 1\times 10^{-5}\ \text{m}
\]
\[
t \approx \frac{(1\times 10^{-5})^{2}}{2\cdot (1\times 10^{-9})}
= \frac{1\times 10^{-10}}{2\times 10^{-9}}
= 5\times 10^{-2}\ \text{s}
\]
So the estimate is about \(0.05\ \text{s}\) (50 ms) for ~10 µm.
Worked example (find distance)
With the same \(D = 1\times 10^{-9}\ \text{m}^{2}/\text{s}\), how far in \(t = 1\ \text{s}\)?
\[
L \approx \sqrt{2Dt}
= \sqrt{2\cdot (1\times 10^{-9})\cdot 1}
= \sqrt{2\times 10^{-9}}
\approx 4.47\times 10^{-5}\ \text{m}
\]
\[
4.47\times 10^{-5}\ \text{m}
= 44.7\ \mu\text{m}
\]
In about 1 second, the characteristic diffusion distance is on the order of a few tens of micrometers.
Interpretation and limitations
- 1D timescale: The model is a simplified one-dimensional approximation.
- Not a full concentration profile: It estimates a characteristic time/distance, not the complete diffusion curve.
- Real biology is more complex: Cytoplasm is crowded; membranes and binding sites can slow effective diffusion; active transport and convection can dominate in tissues.
- Use it for intuition: It is excellent for back-of-the-envelope reasoning and comparing scales.
How to use the calculator effectively
- Choose a mode: time from distance or distance from time.
- Select a preset \(D\) to start, then adjust using custom \(D\) if needed.
- Be careful with units (µm vs mm is a 1000× change in distance, which becomes a 1 000 000× change in time through \(L^{2}\)).
- Use batch mode to compare multiple distances/times and see the curve behavior clearly.
