CHE 318 Lecture 02

Molecular Diffusion in Gases: Diffusive + Convective Mass Transport

Author

Dr. Tian Tian

Published

January 7, 2026

Note

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Recap

Fick’s 1st law of diffusion \[ J_{Az}^{*} = -D_{AB}\dfrac{d c_A}{dz} \]
Diffusion and convection \[ N_{A} = J_{Az}^{*} + c_A v_m \]
Governing equation for binary mixture system \[ N_{A} = -c_T D_{AB} \dfrac{d x_A}{dz} + x_A(N_A + N_B) \]
We will learn how to solve this equation in this lecture!

Learning Outcomes

After today’s lecture, you will be able to:

Distinguish diffusion vs convection mechanisms
Recall common cases in defining the $N_A$ and $N_B$ relation
Apply constraints to select the right case
Solve:
- Equimolar counter diffusion (EMCD)
- Diffusion through stagnant B
Analyze different solutions for EMCD and stagnant B

Recap: Fick’s 1st Law of Diffusion

Diffusion flux of A in B, z-direction follows:

\[ J^*_{Az} = - D_{AB}\,\frac{d c_A}{d z} \]

*Adolf Fick (1829 - 1901), German physiologist*

Fick was the first to propose the relation between diffusion and concentration gradient driving force
Analog:
- Heat transfer (Fourier’s law)
- Momentum transfer in fluid (Newton equation)
Diffusivity $D_{AB}$ was later linked to molecular Brownian motion by Albert Einstein
$D_{AB}$ has unit of $\mathrm{m^2\,s^{-1}}$

Behaviour of Molecular Diffusion

*Diffusion and Brownian motion of molecules, adapted from Wikipedia*

Brownian motion (Bm) is the inherent motion of molecules
Bm leads to redistribution of molecules until $\dfrac{d c_A}{d z} = 0$
Non-zero flux only when $\dfrac{d c_A}{d z} \neq 0$
Is diffusivity temperature dependent? pressure dependent?
How fast is molecular diffusion?

Typical $D_{AB}$ Range

A Common Misconception

Search for any youtube video with “diffusion experiment dye”

Example Video

Can we verify whether the scenario seen is Ficknian diffusion?

Diffusion vs Convection Length Scale

Length $L$ traveled in time $t$:

Diffusion: $L = 6\sqrt{D_{AB}\,t}$ (Einstein, ~1905)
Convection: $L = v_m\,t$

What is typical $D_{AB}$ in a liquid?
- Often $D_{AB}\sim 10^{-9}$ to $10^{-10}\,\mathrm{m^2/s}$

Assuming $D_{AB} = 10^{-10} \mathrm{m^2/s}$ $v_{m} = 10^{-3} \mathrm{m/s}$

Diffusion vs Convection Summary

Diffusion

Driven by concentration gradient
Associated with Diffusivity $D_{AB}$
Diffusive velocity $v_{Ad}$
Slow at large (industry) length scale

Convection

Driven by bulk motion
Can reinforce or oppose diffusion
Bulk fluid motion $v_m$
Fast at large (industry) length scale

A Note on Setups

Reference frame

$J_{Az}^{*}$ refers to the fluid plane frame (hence the $*$)
$N_{A}$ refers to the stationary frame (lab frame)
We will be dealing with $N_A$ in this course!

Mass balance in stationary frame

\[ [\mathrm{In}] - [\mathrm{Out}] + [\mathrm{Generate}] = [\mathrm{Accumulation}] \]

Steady state (S.S)

$[\mathrm{Generate}] = [\mathrm{Accumulation}] = 0$
$[\mathrm{In}] = [\mathrm{Out}]$
$\dfrac{d N_A}{dz} = 0$ (S.S)

Governing Equation for $N_A$ and $N_B$

The common way of expressing the governing equation is

\[\begin{align} N_{A} &= -c_T D_{AB} \dfrac{d x_A}{dz} + x_A(N_A + N_B) \\ N_{B} &= -c_T D_{BA} \dfrac{d x_B}{dz} + x_B(N_A + N_B) \end{align}\]

How can we solve these?

For this course we assume $D_{AB}$ and $D_{BA}$ are constants
For industrial purposes we want to know values of $N_A$ and $N_B$
We need relation between $N_A$ and $N_B$ to solve them!
$x_A(z)$ can be solved as a by-product

Three Limiting Cases

Equimolar Counter Diffusion (EMCD)

Assumption

$N_A + N_B = 0$
$v_m = 0$

Example

Two gases exchanging through a thin tube
Idealized membrane diffusion

Diffusion with Stagnant B

Assumption

$N_B = 0$
$N_A \neq 0$

Example

Evaporation of A through non-diffusing air
Gas absorption into a liquid film

General Case

Assumption

$N_A \neq 0$
$N_A = k N_B$ ($k \neq -1$)

Example

Catalyst reaction

Case 1: Equimolar Counter Diffusion (EMCD)

Condition: $N_A + N_B = 0$
No bulk fluid motion

\[\begin{align} N_A &= -N_B \\ &= J_{Az}^{*} \\ &= -c_T D_{AB} \dfrac{d x_A}{d z} \end{align}\]

Relation in EMCD gives:

\[\begin{align} c_T D_{BA} \dfrac{d x_B}{d z} &= -c_T D_{AB} \dfrac{d x_A}{d z} \end{align}\]

Since $x_A + x_B = 1$, we have $D_{AB} = D_{BA}$

Solving EMCD: Flux $N_A$

Assume S.S., constant $D_{AB}$ and integrate from $z_1$ to $z_2$

\[\begin{align} \int_{c_{A1}}^{c_{A2}} d c_A &= -\frac{N_A}{D_{AB}} \int_{z_1}^{z_2} d z \\ c_{A2} - c_{A1} &= -\frac{N_A}{D_{AB}}(z_2 - z_1) \\ \end{align}\] \[\begin{align} \boxed{ N_A = \frac{D_{AB}}{(z_2 - z_1)}(c_{A1} - c_{A2}) } \end{align}\]

For ideal gas:

\[\begin{align} \boxed{ N_A = \frac{D_{AB}}{RT(z_2 - z_1)}(p_{A1} - p_{A2}) } \end{align}\]

Case 2: Diffusion Through Stagnant B

Definition: stagnant species B means zero molar flux in the lab frame: \[ N_B = 0 \]

But diffusion in B phase still occurs!

The governing equation becomes:

\[\begin{align} N_A &= -c_TD_{AB}\dfrac{d x_A}{dz} + x_A N_A \\ N_A(1 - x_A) &= -c_TD_{AB}\dfrac{d x_A}{dz} \end{align}\]

Solving Stagnant B Case

We again do an integration by separating variables:

\[\begin{align} \frac{d x_A}{1-x_A} &= -\frac{N_A}{c_TD_{AB}}\,dz \\ \int_{x_{A1}}^{x_{A2}} \frac{d x_A}{1-x_A} &= -\frac{N_A}{c_TD_{AB}}\int_{z_1}^{z_2} dz \\ \left[-\ln(1-x_A)\right]_{x_{A1}}^{x_{A2}} &= -\frac{N_A}{c_TD_{AB}}(z_2-z_1) \\ \ln\!\left(\frac{1-x_{A1}}{1-x_{A2}}\right) &= -\frac{N_A}{c_TD_{AB}}(z_2-z_1) \end{align}\] \[\begin{align} \boxed{ N_A = \frac{c_TD_{AB}}{(z_2-z_1)} \ln\!\left(\frac{1-x_{A2}}{1-x_{A1}}\right) } \end{align}\]

Various Forms of Stagnant B Solution

In concentration and molar fraction

\[\begin{align} N_A = \frac{c_TD_{AB}}{(z_2-z_1)} \ln\!\left(\frac{1-x_{A2}}{1-x_{A1}}\right) \end{align}\]

Ideal gas with partial pressure

\[\begin{align} N_A = \frac{p_TD_{AB}}{RT(z_2-z_1)} \ln\!\left(\frac{p_T-p_{A2}}{p_T-p_{A1}}\right) \end{align}\]

Log-mean Form

Historically, engineers liked to write the stagnant B equation using linear terms.

\[\begin{align} \ln\!\left(\frac{p_T-p_{A2}}{p_T-p_{A1}}\right) &= \ln\!\left(\frac{p_{B2}}{p_{B1}}\right) \\ (p_{B2} - p_{B1})\ln\!\left(\frac{p_T-p_{A2}}{p_T-p_{A1}}\right) &= \ln\!\left(\frac{p_{B2}}{p_{B1}}\right) (p_{A1} - p_{A2}) \\ \ln\!\left(\frac{p_T-p_{A2}}{p_T-p_{A1}}\right) &= \ln\!\left(\frac{p_{B2}}{p_{B1}}\right) \left( \frac{p_{A1} - p_{A2}}{p_{B2} - p_{B1}} \right) \end{align}\]

Define $p_{Bm} = \dfrac{p_{B2} - p_{B1}}{\ln\!(p_{B2}/p_{B1})}$

We get

\[\begin{align} N_A = \frac{D_{AB}}{RT(z_2-z_1)} \frac{p_T}{p_{Bm}}(p_{A1}-p_{A2}) \end{align}\]

Solving Concentration Profiles in Stagnant B

We will utilize the fact $d N_A / dz = 0$ (S.S).
Do we have $d c_A / dz = 0$ like in EMCD?

\[\begin{align} d N_A / dz &= -cD_{AB} \frac{d}{dx} \left[\frac{1}{1 - x_A}\frac{d x_A}{dz}\right] \\ &= 0 \\ \frac{1}{1 - x_A}\frac{d x_A}{dz} &= \mathrm{Const} \\ \int_{x_{A1}}^{x} \frac{1}{1 - x_A^{'}}d x_A^{'} &= \int_{z_1}^{z} \mathrm{Const}\, dz^{'} \end{align}\]

Can you solve the profile for $x_A(z)$?

Wrap-up

“Case selection”: choosing the right constraint
Different constraints lead to different solvable cases
Equimolar counter diffusion (EMCD): linear concentration profiles
Stagnant B: non-linear profiles and logarithmic driving force
Stagnant ≠ no diffusion: 👉 $N_B = 0$

Next lecture: we remove simplifying constraints and discuss the general case

--- title: "CHE 318 Lecture 02" subtitle: "Molecular Diffusion in Gases: Diffusive + Convective Mass Transport" author: "Dr. Tian Tian" date: "2026-01-07" format: html: {} revealjs: output-file: slides.html pdf: output-file: L02.pdf --- ::: {.content-visible when-format="html" unless-format="revealjs"} ::: {.callout-note} - Slides 👉 [Open presentation🗒️](./slides.html) - PDF version of course note 👉 [Open in pdf](./L02.pdf) - Handwritten notes 👉 [Open in pdf](./public/L02_annotated.pdf) ::: ::: ## Recap - Fick's 1st law of diffusion $$ J_{Az}^{*} = -D_{AB}\dfrac{d c_A}{dz} $$ - Diffusion and convection $$ N_{A} = J_{Az}^{*} + c_A v_m $$ - **Governing equation** for binary mixture system $$ N_{A} = -c_T D_{AB} \dfrac{d x_A}{dz} + x_A(N_A + N_B) $$ - We will learn how to solve this equation in this lecture! --- ## Learning Outcomes After today's lecture, you will be able to: - **Distinguish** diffusion vs convection mechanisms - **Recall** common cases in defining the $N_A$ and $N_B$ relation - **Apply** constraints to select the right case - **Solve**: - Equimolar counter diffusion (EMCD) - Diffusion through stagnant B - **Analyze** different solutions for EMCD and stagnant B --- ## Recap: Fick's 1st Law of Diffusion Diffusion flux of A in B, z-direction follows: $$ J^*_{Az} = - D_{AB}\,\frac{d c_A}{d z} $$ ::: {.columns} ::: {.column width="20%"} ![*Adolf Fick (1829 - 1901), German physiologist*](./public/500px-Adolf_Fick_8bit_korr_klein1.jpg){fig-alt="Adolf Fick"} ::: ::: {.column width="80%"} - Fick was the first to propose the relation between diffusion and concentration gradient driving force - Analog: - Heat transfer (**Fourier's law**) - Momentum transfer in fluid (**Newton equation**) - Diffusivity $D_{AB}$ was later linked to molecular Brownian motion by Albert Einstein - $D_{AB}$ has unit of $\mathrm{m^2\,s^{-1}}$ ::: ::: --- ## Behaviour of Molecular Diffusion :::: {.columns} ::: {.column width="40%"} ::: {.content-visible when-format="html"} ![*Diffusion and Brownian motion of molecules, adapted from Wikipedia*](./public/DiffusionMicroMacro.gif){width="95%"} ::: ::: ::: {.column width="60%"} - Brownian motion (Bm) is the inherent motion of molecules - Bm leads to redistribution of molecules until $\dfrac{d c_A}{d z} = 0$ - Non-zero flux only when $\dfrac{d c_A}{d z} \neq 0$ - Is diffusivity temperature dependent? pressure dependent? - How fast is molecular diffusion? ::: :::: --- ## Typical $D_{AB}$ Range ```{=html} <iframe width="100%" height="800" src="../../scripts/L02_diffusivity_demo.html" title="Webpage example"></iframe> ``` ## A Common Misconception Search for any youtube video with *"diffusion experiment dye"* [Example Video](https://www.youtube.com/embed/STLAJH7_zkY){preview-link="true"} Can we **verify** whether the scenario seen is Ficknian diffusion? --- ## Diffusion vs Convection Length Scale :::: {.columns} ::: {.column width="45%"} Length $L$ traveled in time $t$: - **Diffusion:** $L = 6\sqrt{D_{AB}\,t}$ *(Einstein, ~1905)* - **Convection:** $L = v_m\,t$ **What is typical $D_{AB}$ in a liquid?** - Often $D_{AB}\sim 10^{-9}$ to $10^{-10}\,\mathrm{m^2/s}$ ::: ::: {.column width="55%"} Assuming $D_{AB} = 10^{-10} \mathrm{m^2/s}$ $v_{m} = 10^{-3} \mathrm{m/s}$ ```{=html} <iframe width="100%" height="800" src="../../scripts/L02_diffusion_length.html" title="Webpage example"></iframe> ``` ::: :::: --- ## Diffusion vs Convection Summary ::: columns ::: column **Diffusion** - Driven by concentration gradient - Associated with Diffusivity $D_{AB}$ - Diffusive velocity $v_{Ad}$ - **Slow** at large (industry) length scale ::: ::: column **Convection** - Driven by bulk motion - Can reinforce or oppose diffusion - Bulk fluid motion $v_m$ - **Fast** at large (industry) length scale ::: ::: --- ## A Note on Setups ##### Reference frame - $J_{Az}^{*}$ refers to the fluid plane frame (hence the $*$) - $N_{A}$ refers to the stationary frame (lab frame) - We will be dealing with $N_A$ in this course! ##### Mass balance in stationary frame $$ [\mathrm{In}] - [\mathrm{Out}] + [\mathrm{Generate}] = [\mathrm{Accumulation}] $$ ##### Steady state (S.S) - $[\mathrm{Generate}] = [\mathrm{Accumulation}] = 0$ - $[\mathrm{In}] = [\mathrm{Out}]$ - $\dfrac{d N_A}{dz} = 0$ (**S.S**) --- ## Governing Equation for $N_A$ and $N_B$ The common way of expressing the governing equation is ```{=tex} \begin{align} N_{A} &= -c_T D_{AB} \dfrac{d x_A}{dz} + x_A(N_A + N_B) \\ N_{B} &= -c_T D_{BA} \dfrac{d x_B}{dz} + x_B(N_A + N_B) \end{align} ``` How can we solve these? - For this course we assume $D_{AB}$ and $D_{BA}$ are constants - For industrial purposes we want to know values of $N_A$ and $N_B$ - We need relation between $N_A$ and $N_B$ to solve them! - $x_A(z)$ can be solved as a by-product --- ## Three Limiting Cases :::: {.columns} ::: {.column width="40%"} Equimolar Counter Diffusion (EMCD) **Assumption** - $N_A + N_B = 0$ - $v_m = 0$ **Example** - Two gases exchanging through a thin tube - Idealized membrane diffusion ::: ::: {.column width="30%"} Diffusion with Stagnant B **Assumption** - $N_B = 0$ - $N_A \neq 0$ **Example** - Evaporation of A through non-diffusing air - Gas absorption into a liquid film ::: ::: {.column width="30%"} General Case **Assumption** - $N_A \neq 0$ - $N_A = k N_B$ ($k \neq -1$) **Example** - Catalyst reaction ::: :::: --- ## Case 1: Equimolar Counter Diffusion (EMCD) - Condition: $N_A + N_B = 0$ - No bulk fluid motion ```{=tex} \begin{align} N_A &= -N_B \\ &= J_{Az}^{*} \\ &= -c_T D_{AB} \dfrac{d x_A}{d z} \end{align} ``` Relation in EMCD gives: ```{=tex} \begin{align} c_T D_{BA} \dfrac{d x_B}{d z} &= -c_T D_{AB} \dfrac{d x_A}{d z} \end{align} ``` Since $x_A + x_B = 1$, we have $D_{AB} = D_{BA}$ --- ## Solving EMCD: Flux $N_A$ Assume S.S., constant $D_{AB}$ and integrate from $z_1$ to $z_2$ ```{=tex} \begin{align} \int_{c_{A1}}^{c_{A2}} d c_A &= -\frac{N_A}{D_{AB}} \int_{z_1}^{z_2} d z \\ c_{A2} - c_{A1} &= -\frac{N_A}{D_{AB}}(z_2 - z_1) \\ \end{align} ``` ```{=tex} \begin{align} \boxed{ N_A = \frac{D_{AB}}{(z_2 - z_1)}(c_{A1} - c_{A2}) } \end{align} ``` For ideal gas: ```{=tex} \begin{align} \boxed{ N_A = \frac{D_{AB}}{RT(z_2 - z_1)}(p_{A1} - p_{A2}) } \end{align} ``` ## Case 2: Diffusion Through Stagnant B Definition: **stagnant** species B means zero molar flux in the lab frame: $$ N_B = 0 $$ But diffusion in B phase **still occurs**! The governing equation becomes: ```{=tex} \begin{align} N_A &= -c_TD_{AB}\dfrac{d x_A}{dz} + x_A N_A \\ N_A(1 - x_A) &= -c_TD_{AB}\dfrac{d x_A}{dz} \end{align} ``` ## Solving Stagnant B Case We again do an integration by separating variables: ```{=tex} \begin{align} \frac{d x_A}{1-x_A} &= -\frac{N_A}{c_TD_{AB}}\,dz \\ \int_{x_{A1}}^{x_{A2}} \frac{d x_A}{1-x_A} &= -\frac{N_A}{c_TD_{AB}}\int_{z_1}^{z_2} dz \\ \left[-\ln(1-x_A)\right]_{x_{A1}}^{x_{A2}} &= -\frac{N_A}{c_TD_{AB}}(z_2-z_1) \\ \ln\!\left(\frac{1-x_{A1}}{1-x_{A2}}\right) &= -\frac{N_A}{c_TD_{AB}}(z_2-z_1) \end{align} ``` ```{=tex} \begin{align} \boxed{ N_A = \frac{c_TD_{AB}}{(z_2-z_1)} \ln\!\left(\frac{1-x_{A2}}{1-x_{A1}}\right) } \end{align} ``` --- ## Various Forms of Stagnant B Solution - In concentration and molar fraction ```{=tex} \begin{align} N_A = \frac{c_TD_{AB}}{(z_2-z_1)} \ln\!\left(\frac{1-x_{A2}}{1-x_{A1}}\right) \end{align} ``` - Ideal gas with partial pressure ```{=tex} \begin{align} N_A = \frac{p_TD_{AB}}{RT(z_2-z_1)} \ln\!\left(\frac{p_T-p_{A2}}{p_T-p_{A1}}\right) \end{align} ``` --- ## Log-mean Form Historically, engineers liked to write the stagnant B equation using linear terms. ```{=tex} \begin{align} \ln\!\left(\frac{p_T-p_{A2}}{p_T-p_{A1}}\right) &= \ln\!\left(\frac{p_{B2}}{p_{B1}}\right) \\ (p_{B2} - p_{B1})\ln\!\left(\frac{p_T-p_{A2}}{p_T-p_{A1}}\right) &= \ln\!\left(\frac{p_{B2}}{p_{B1}}\right) (p_{A1} - p_{A2}) \\ \ln\!\left(\frac{p_T-p_{A2}}{p_T-p_{A1}}\right) &= \ln\!\left(\frac{p_{B2}}{p_{B1}}\right) \left( \frac{p_{A1} - p_{A2}}{p_{B2} - p_{B1}} \right) \end{align} ``` Define $p_{Bm} = \dfrac{p_{B2} - p_{B1}}{\ln\!(p_{B2}/p_{B1})}$ We get ```{=tex} \begin{align} N_A = \frac{D_{AB}}{RT(z_2-z_1)} \frac{p_T}{p_{Bm}}(p_{A1}-p_{A2}) \end{align} ``` --- ## Solving Concentration Profiles in Stagnant B - We will utilize the fact $d N_A / dz = 0$ (S.S). - Do we have $d c_A / dz = 0$ like in EMCD? ```{=tex} \begin{align} d N_A / dz &= -cD_{AB} \frac{d}{dx} \left[\frac{1}{1 - x_A}\frac{d x_A}{dz}\right] \\ &= 0 \\ \frac{1}{1 - x_A}\frac{d x_A}{dz} &= \mathrm{Const} \\ \int_{x_{A1}}^{x} \frac{1}{1 - x_A^{'}}d x_A^{'} &= \int_{z_1}^{z} \mathrm{Const}\, dz^{'} \end{align} ``` *Can you solve the profile for $x_A(z)$*? --- ## Wrap-up - “Case selection”: choosing the right constraint - Different **constraints** lead to different solvable cases - **Equimolar counter diffusion (EMCD):** linear concentration profiles - **Stagnant B:** non-linear profiles and logarithmic driving force - **Stagnant ≠ no diffusion:** 👉 $N_B = 0$ **Next lecture:** we remove simplifying constraints and discuss the **general case**

CHE 318 Lecture 02

Recap

Learning Outcomes

Recap: Fick’s 1st Law of Diffusion

Behaviour of Molecular Diffusion

Typical \(D_{AB}\) Range

A Common Misconception

Diffusion vs Convection Length Scale

Diffusion vs Convection Summary

A Note on Setups

Reference frame

Mass balance in stationary frame

Steady state (S.S)

Governing Equation for \(N_A\) and \(N_B\)

Three Limiting Cases

Case 1: Equimolar Counter Diffusion (EMCD)

Solving EMCD: Flux \(N_A\)

Case 2: Diffusion Through Stagnant B

Solving Stagnant B Case

Various Forms of Stagnant B Solution

Log-mean Form

Solving Concentration Profiles in Stagnant B

Wrap-up