Gas Diffusivity Explorer (embed)

Gas Diffusivity Explorer

Start here: choose gases A and B, set T and P, then press Remove membrane and watch the mixing index.

Binary gas pair

Species A

Formula (or air)

Rings

M_A (g/mol)

Σv_A

Species B

Formula (or air)

Rings

M_B (g/mol)

Σv_B

Temperature T (K)

Pressure P (atm)

101.325 kPa

Playback speed

Demo, wrong on purpose: read Table 5-11 literally (√M_AB in the numerator)

Compare with a measured value rescaled by T^1.75/P

Animated schematic

Calculated binary diffusivity

D_AB (cm²/s)

–

D_AB (m²/s)

–

D_ABP (cm²·atm/s)

–

Mixing in the box

Mixing index

Simulated t (s)

0.00

Time to 90% (s)

not yet

Temperature effect at constant pressure

Pressure effect at constant temperature

Equations, assumptions and references

Governing equations

Fuller method (binary pair A + B)

Perry’s 9th ed., Eq. 5-202; Table 5-11

D_{AB} = \frac{0.001 T^{1.75} \sqrt{\frac{1}{M_{A}} + \frac{1}{M_{B}}}}{P {[{(\sum v_{A})}^{1 / 3} + {(\sum v_{B})}^{1 / 3}]}^{2}}

D_AB in cm²/s, T in K, P in atm, M in g/mol, Σv from Table 5-13. 1 cm²/s = 10⁻⁴ m²/s; 1 atm = 101.325 kPa.

Perry’s 9th ed., Eq. 5-210: reduced molar mass

M_{AB} = {(\frac{1}{M_{A}} + \frac{1}{M_{B}})}^{- 1}

\sqrt{\frac{1}{M_{A}} + \frac{1}{M_{B}}} = \frac{1}{\sqrt{M_{AB}}}

The square-root factor is 1/√M_AB. Reading Table 5-11 literally (√M_AB in the numerator) is the “wrong on purpose” demo.

Perry’s 9th ed., Table 5-13: diffusion volumes

\sum v = \sum_{k} n_{k} Δ v_{k} - 20.2 N_{ring}

Formula builder: C, H, O, N, Cl and S, plus preset values for He, Ar and air (example: C₆H₅Cl with 1 ring). Each aromatic or heterocyclic ring subtracts 20.2. M and Σv can also be typed directly.

Perry’s 9th ed., Table 2-141: optional measured reference

D = D_{ref} {(\frac{T}{T_{ref}})}^{1.75} (\frac{P_{ref}}{P})

Enter a measured binary D at T_ref and P_ref; it is moved to the current T and P with the same T^1.75/P scaling. Away from the measurement conditions both numbers are estimates.

Limits. Low-pressure gas correlation. D_ABP is constant at low pressure but falls as pressure rises (Perry’s p. 5-49), so the region above 10 atm is shaded as a caution zone. The exponent 1.75 is approximate. Check that each species is still a gas at the chosen T and P.

Reading the results. For fixed species D_AB ∝ T^1.75/P, so D_ABP is the same at every pressure. Fuller diffusion volumes are empirical parameters, not bulk molar volumes.

Plots. Temperature plot: both axes logarithmic, so the Fuller line has slope 1.75. Pressure plot: logarithmic P axis; left axis D_AB (log), right axis D_ABP (linear). Comparison bars: the Fuller value against the measured value moved to the same T and P; the gap shows method spread, not which one is right.

References

Perry’s Chemical Engineers’ Handbook, 9th ed., Section 5 (Eqs. 5-202, 5-210; Tables 5-11, 5-13) and Table 2-141.
E. N. Fuller, P. D. Schettler and J. C. Giddings, Ind. Eng. Chem. 58(5), 18 (1966).
R. B. Bird, W. E. Stewart and E. N. Lightfoot, Transport Phenomena, 2nd ed., Wiley (2002), Ch. 17 (kinetic theory of gases).

Model self-tests

Each check recomputes a published or analytic value with the same functions the simulator uses (0.6% tolerance). None of these is a homework answer.

Show computed and expected values

Not run.

How the animated schematic is scaled

Fuller-calibrated stochastic diffusion model (random flight), not molecular dynamics. The box holds only the two species. A theoretical membrane at the midline keeps A (red filled dots) on the left and B (blue rings) on the right: molecules move thermally but bounce off it, so nothing mixes until Remove membrane is pressed. Insert membrane traps every molecule on the side where it is. Reset run, or the Reset button at the top, puts A and B back in their halves with the membrane in.

Changing T, P or a species during a run updates speeds and free paths at once without restarting, so you can watch the rate change; the run is then flagged “inputs changed”, because its times mix two conditions. For a clean comparison press Reset run, then Remove membrane.

Molecule speeds

Each molecule flies straight at its kinetic-theory mean speed:

v = S_{v} \sqrt{\frac{T}{M}} \propto \sqrt{\frac{8 R T}{π M}}

S_v is one fixed constant (air at 298 K moves at 150 box units per second; the box is 300 units wide and is stretched to fit the screen). Doubling T raises every speed by √2; H₂ moves √(28.01/2.016) = 3.73 times faster than N₂ at the same T.

Collisions

After an exponentially distributed free path the molecule is scattered into a random direction. The mean free path of each species is set so that this random flight diffuses at exactly the Fuller value:

λ_{i} = \frac{2 D_{px}}{v_{i}}, D_{px} = S_{D} D_{AB} (i = A, B)

A random flight in two dimensions has diffusivity vλ/2, so both species interdiffuse with D_AB (for a binary mixture D_AB = D_BA). One fixed factor S_D = 11 500 converts cm²/s to (box units)²/s; D_px is in those units. Molecules reflect at the walls, and at the membrane while it is in. When the free path is far shorter than one frame of motion, the equivalent Gaussian step of variance 2D_pxΔt per axis is used.

Spreading of A

Along the box, the spread of A obeys

⟨ x^{2} ⟩ = 2 D_{AB} t

so A mixes faster at higher T (D_AB ∝ T^1.75) and lower P (D_AB ∝ 1/P), and more slowly for heavier or bulkier molecules. The model self-tests check this variance against 2D_ABt within 10%, and that doubling T or halving P changes the measured spread rate by 2^1.75 and 2.

Mixing index

The box is cut into 8 equal vertical slices. x_A,i is the mole fraction of A among the n_i molecules in slice i (the strip under the box), and σ₀ = 1/2 is the spread for the fully separated start:

I = 1 - \frac{σ}{σ_{0}}, σ^{2} = \frac{1}{8} \sum_{i} {(x_{A, i} - \frac{1}{2})}^{2} - \frac{1}{8} \sum_{i} \frac{x_{A, i} (1 - x_{A, i})}{n_{i} - 1}

The second sum removes the scatter expected from counting a finite number of molecules. I runs from 0% (separated) to 100% (uniform). It uses all 8,000 simulated molecules (4,000 A and 4,000 B); one in 40 is drawn so that single paths stay visible. Even a perfectly mixed box never counts out exactly even, so I keeps flickering by a few percent around 99%. To keep that flicker from deciding the result, Well mixed is shown once I averaged over the last 1 s of model time reaches 95%, and it stays on until the next reset. Time to 90% records when I first reaches 90%; this flicker moves it by well under 1%.

Time base

Elapsed time is model time and runs only while the membrane is out. Because D_px = S_DD_AB, the 300-unit box behaves like a cell L = 300/√11 500 ≈ 2.8 cm long, so mixing times are true diffusion times for that length and scale as L²/D_AB. Molecule sizes, spacings, speeds and free paths are not to scale.

Binary gas pair

Calculated binary diffusivity

Mixing in the box

Temperature effect at constant pressure

Pressure effect at constant temperature

Two estimates at the same T and P

Equations, assumptions and references

Governing equations

Fuller method (binary pair A + B)

References

Model self-tests

How the animated schematic is scaled

Molecule speeds

Collisions

Spreading of A

Mixing index

Time base