Lab 102-C — Diffusion Analogue: Dye in Agar or Gelatin

Goes with: Unit 7. Time: 1 h setup + readings over 2–4 days. Cost: about $15–$30 (estimate).

Goal

Measure how far food dye diffuses into a gel over time at three temperatures; show that depth grows as $\sqrt{t}$; extract a diffusion coefficient $D$ at each temperature; and fit an Arrhenius activation energy. This mirrors, safely, the mathematics of dopant diffusion (POCl₃/BBr₃ diffusion itself is lecture-only).

Shopping list

Item	Spec / example	Qty	Approx. cost (estimate)	Where
Agar powder	culinary agar-agar (sets firm, melts ~85 °C — usable at 40 °C)	25–50 g	$6–10	grocery (Asian section), Amazon
Alternative unflavoured gelatin	e.g., Knox (melts ~30–35 °C, so only for ≤ 20 °C runs)	1 box	$4	grocery
Food dye	liquid, dark (blue or red)	1 set	$4	grocery
Clear test tubes or straight-sided glass vials (or clear drinking straws plugged at one end)	10–15 cm long	9+	$8–12	Amazon
Ruler with mm scale, phone camera	—	—	—	household
Thermometer	—	1	$8	—
Warm-water bath for 40 °C	slow cooker on "warm", sous-vide stick, or insulated cooler topped up with warm water	1	$0–60	household

Safety

Food-safe materials. Keep hot water and electrical heaters apart; don't leave heaters unattended.

Procedure

Make 1.5–2 % agar (1.5–2 g per 100 mL water): boil while stirring until clear, then pour into the tubes to ~2 cm below the top. Let set at room temperature (1 h), capped to prevent drying.
Prepare a dye solution (e.g., 10 drops dye in 10 mL water). Add 1 cm of dye solution on top of the gel in each tube at time $t = 0$. Cap.
Put three tubes in each environment: fridge (~5 °C), room (~20–22 °C), warm bath (~40 °C). Record actual temperatures.
Photograph each tube against a mm scale at ~1, 3, 6, 12, 24, 48 and 72 h. Keep tubes upright and still.
Measure the depth $x_{1/2}$ where the colour is visibly half as intense as at the top (or use a consistent visual threshold; better, extract colour intensity along the tube from photos with Python).

Analysis

The 1-D constant-source solution (Unit 7) is $C = C_s\,\text{erfc}(x/2\sqrt{Dt})$. Where $C = C_s/2$, $x/2\sqrt{Dt} = 0.477$, so

\[x_{1/2} = 0.954\sqrt{Dt} \quad\Rightarrow\quad D = \frac{x_{1/2}^2}{0.910\,t}\]

For each temperature, plot $x_{1/2}^2$ vs $t$: a straight line through the origin confirms $\sqrt{t}$ behaviour. Slope $= 0.910\,D$.

Make an Arrhenius plot: $\ln D$ vs $1/T$ (kelvin). Slope $= -E_a/k_B$.

import numpy as np
T = np.array([278, 295, 313.])          # your temperatures (K)
D = np.array([2.5e-6, 4e-6, 6.5e-6])    # your fitted D (cm^2/s) -- replace!
m, b = np.polyfit(1/T, np.log(D), 1)
print("Ea =", -m*8.617e-5, "eV")

(The numbers in the code are placeholders, not results.)

Expected results

$x_{1/2}^2$ grows linearly with time (depth ∝ √t): it takes ~4× as long to go twice as deep.
Small dye molecules in dilute gels have $D$ of order 10⁻⁶–10⁻⁵ cm²/s at room temperature (estimate) → a front of order 0.5 cm after 1 day.
$D$ increases with temperature by tens of per cent per 10 °C; fitted $E_a$ of order 0.15–0.25 eV (estimate; similar to water's viscous flow).

How to interpret

Same equations, very different numbers: dye in gel has $E_a \approx 0.2$ eV and diffuses millimetres per day near room temperature; boron in silicon has $E_a \approx 3.5$ eV and needs ~1000 °C to move ~0.1 µm per hour. The much larger activation energy is why dopants stay put at room temperature (devices are stable for decades) yet move readily in furnaces — and why furnace temperature control (± a few °C) matters so much for junction depth.

Sources

Crank, The Mathematics of Diffusion (erfc solution).
Unit 7 references for silicon values.

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