CHE 318 Lecture 04

Molecular Diffusivity: Theories and Measurement

Author

Dr. Tian Tian

Published

January 12, 2026

Note

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Handwritten notes 👉 Open in pdf

Recap

General solution for diffusion binary mixture gas systems ($N_B = k N_A$)
- $k = -1$ reduces to EMCD solution
- Total flux $N_A$ is EMCD flux times a coefficient

\[\begin{align} N_A = \frac{ c_T D_{AB}}{(z_2 - z_1)} \left(\frac{N_A}{N_A + N_B}\right) \ln\!\left[\dfrac{ \frac{N_A}{N_A + N_B} - x_{A2}} {\frac{N_A}{N_A + N_B} - x_{A1}} \right] \end{align}\]

Brief discussion about diffusivity measurement: two-bulb setup

Demonstration of General Solution

Interaction Time!

participation link

Results and comments to be published after the class

Detailed Explanation for Convection-Driven Flux Change

Note

Advanced topic, not shown in presentation

The ratio between total flux in general solution $N_A(\text{gen})$ and $N_A(\text{EMCD})$ can be written as:

\[\begin{align} \frac{N_A(\text{gen})}{N_A(\text{EMCD})} &= \frac{s \ln\! \left[ \frac{s - x_{A2}}{s - x_{A1}} \right]}{x_{A1} - x_{A2}} \\ &= \frac{s \ln\! \left[ \frac{s - x_{A2}}{s - x_{A1}} \right]}{(s - x_{A2}) - (s - x_{A1})} \\ &= \frac{s}{\text{LM}(s-x_A)} \end{align}\]

where $\text{LM}(u)$ is the log-mean function (similar treatment in Lecture 2), and $s = N_A / (N_A + N_B)$.

There are several limiting cases. Let’s assume $x_{A1} > x_{A2}$ for simplicity, and there is a real solution to the $N_A(\text{gen})$

$s > x_{A1}$:

$s - x_{A1}$ and $s - x_{A2}$ must be both positive to ensure a solution. Therefore $\text{LM}(s-x_A) < s$. We know that $N_A(\text{gen})>N_A(\text{EMCD})$.
- Example: diffusion through stagnant B ($s = 1$)
  
  In this case $\text{LM}(s-x_A) = \text{LM}(x_B)$, we can conclude that the diffusion through stagnant B case always enhances total flux compared with EMCD.
  
  A signature of such transport is the $x_A(z)$ profile is concave, as shown in Figure 1 (a).
$s < 0$:

$s - x_{A2}$ and $s - x_{A1}$ are be both negative. Therefore $|\text{LM}(s-x_A)| > s$. We know that $N_A(\text{gen})<N_A(\text{EMCD})$, but $N_A$ is still positive.
- Example: counter $N_B$ flux ($N_B/N_A < -1$)
  
  The opposite direction of $N_B$ attenuates the molar flux of A. A signature of such transport is the $x_A(z)$ profile is convex, as shown in Figure 1 (b).
$0 < s < x_2$:

$s - x_{A2}$ and $s - x_{A1}$ are be both negative. Since $s >0$, we actually can have $N_A < 0$!

Such situation may be counter-intuitive but the solution is physically valid, as the only solution to satisfy such criteria is $N_B < 0$ and $N_A < 0$ at the same time. As we cannot directly compare $s$ and $\text{LM}(s-x_A)$ in this case, $|N_A(\text{gen})|$ can be either smaller or larger than $N_A(\text{EMCD})$, as shown in Figure 1 (c) and (d).
$x_2 < s < x_1$:

The expression inside the logarithm is negative and we do not have a real solution to the general equation for $N_A$.

Figure 1: Comparison between general solution and EMCD flux. a. Enhance flux via diffusion through stagnant B. b Reduced flux when $s=-1$. c. Enhanced counter flux case d Reduced counter flux case

Learning Outcomes

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

Recall multiple theories of molecular diffusivity
Analyze the diffusivity $D_{AB}$ as function of $(T, P)$
Derive new diffusivity values from standard table
Formulate governing equations for different scenarios measuring the diffusivity

Let’s Look At The General Mass Transfer Equation (Again)

\[ N_A = \frac{D_{AB} c_T}{(z_2 - z_1)} s \ln\!\left[\dfrac{ s - \frac{c_{A2}}{c_T}} {s - \frac{c_{A1}}{c_T}} \right] \quad s = \frac{N_A}{N_A + N_B} \]

What values do we know from the system setup?

Geometry: $z_1$, $z_2$, $c_{A1}$, $c_{A2}$, $c_T$
Chemical reaction stoichiometry: $k$

What else value(s) do we need to solve $N_A$?

$D_{AB}$: generally $D_{AB}(z) = \text{Const}$
But $D_{AB}=f(T, P, \cdots)$

Motivation to Have Theory About Diffusivity

Solving the general solution for gas mass transfer requires parameter $D_{AB}$ (and $D_{BA}$)!
Measuring every $D_{AB}$ pair for gases is tedious
$D_{AB}$ is dependent on conditions $T, P$
We need to have theories that can predict diffusivity $D_{AB}$ without doing all pair experiments and all $(T, P)$ conditions!
In the lease case, the theory should allow extrapolating a measured $D_{AB}(T_1, P_1)$ to $D_{AB}(T_2, P_2)$

Molecular Diffusivity Theory – Kinetic Theory

In dilute gas regime ($p_T \approx \text{atm}$), the kinetic behaviour of gas molecules can be described using kinetic theory. A few assumptions are made:

Interdiffusion between A and B are dominated by collision between molecules (low $p_T$ 👉 only 2 molecules can collide at the same time)
The molecules are rigid spheres having particle masses $m_A$, $m_B$ and diameters $d_A$ and $d_B$

Kinetic Theory – Key Results

There are some results from the kinetic theory: ¹

Mean molecular speed $\overline{u}$ \[ \overline{u} = \sqrt{\frac{8 k_B T}{\pi \overline{m_{AB}}}} \]
Mean free path (between collisions): $\lambda_{A,B}$ \[ \lambda_{AB}=\frac{1}{\sqrt{2} \pi d_{AB}^2 c_T} \]

Frequency of molecule A colliding with wall ($Z$, unit $\text{m}^{-2}\cdot \text{s}^{-1}$): \[ Z_A = \frac{1}{4} c_A \overline{u} \]

where $d_{AB} = \frac{d_A + d_B}{2}$, $m_{AB}^{-1} = m_{A}^{-1} + m_B^{-1}$, $k_B$ is the Boltzmann constant

We also have the average distance $a$ to one plane from last collision: \[ a = \frac{2}{3}\lambda_{AB} \]

Kinetic Theory – Mass Transfer

When there is only molecular diffusion (no convection), at plane $z=0$, the flux can be described by:

\[\begin{align} J_{Az}^*\vert_{z=0} &= \text{[In]}\vert_{z=-a} - \text{[Out]}\vert_{z=+a} \\ &= \frac{1}{4} c_T [x_A \overline{u}]\vert_{z=-a} \\ &- \frac{1}{4} [c_T x_A \overline{u}]\vert_{z=+a}\\ &= -\frac{1}{4} c_T \cdot 2a \cdot \overline{u} \frac{d x_A}{dz} \\ &= -D_{AB} c_T \frac{dx_A}{dz} \end{align}\]

We can now link $D_{AB}$ to microscopic properties!

Kinetic Theory of Diffusivity – Conclusion

Results:

\[ D_{AB} = \frac{1}{3} \overline{u} \lambda_{A,B} \]

$\overline{u}$: average molecular velocity
$\lambda$: mean free path between collisions

Assumptions:

Rigid sphere type molecules (what are they?)
No interaction upon collisions
Collisions are elastic (momentum conserved)
Good for dilute gases (low $P$)
Not accurate otherwise

Molecular Diffusivity – Chapman-Enskog Theory

We can take real interaction between molecules into the kinetic theory!

\[\begin{align} D_{AB} &= \frac{1.8583\times{}10^{-7} T^{3/2}} {p_T \sigma_{AB}^2 \Omega_{D, AB}} \left(\frac{1}{m_A} + \frac{1}{m_B} \right)^{\frac{1}{2}} \\ &\propto \frac{T^{\frac{3}{2}}}{p_T} \left(\frac{1}{m_A} + \frac{1}{m_B} \right)^{\frac{1}{2}} \end{align}\]

$m_A$, $m_B$, molecular weight
$\sigma_{AB}$: average collision radius between A and B
$\Omega_{D, AB}$: Lennard-Jones collision integral
- $\Omega_{D, AB}=1$ for elastic collision
- Chapman-Enskog theory accounts for non-elastic collision using $\Omega_{D, AB}\neq1$

Chapman-Enskog Theory: How to Use

\[\begin{align} D_{AB} = \frac{1.8583\times{}10^{-7} T^{3/2}} {p_T \sigma_{AB}^2 \Omega_{D, AB}} \left(\frac{1}{m_A} + \frac{1}{m_B} \right)^{\frac{1}{2}} \end{align}\]

The Chapman-Enskog theory formula in Geankoplis book need to use the following units:

$T$: absolute temperature in $\text{K}$
$m_A$, $m_B$: molecular weight in $\text{kg} \cdot (\text{kg mol})^{-1}$ or $\text{g} \cdot (\text{g mol})^{-1}$
$p_T$: absolute pressure in $\text{atm}$
$\sigma_{AB}$: average collision diameter in Å
$\Omega_{D, AB}$: collision integral (dimensionless)

Resulting $D_{AB}$ is in $\text{m}^2 \cdot \text{s}^{-1}$

Chapman-Enskog Theory & Lennard Jones Potential

Parameters $\sigma_{AB}$ and $\Omega_{D, AB}$ can be derived using Lennard-Jones (LJ) potential
LJ potential described the interaction energy $U_{AB}$ between 2 molecules at distance $r$ follows:

\[\begin{align} U_{AB}(r) = 4\epsilon_{AB}\left[ (\frac{\sigma_{AB}}{r})^{12} - (\frac{\sigma_{AB}}{r})^{6} \right] \end{align}\]

How to get the parameters from table:

$\sigma_{AB} = (\sigma_{A} + \sigma_{B})/2$
$\epsilon_{AB} = \sqrt{\epsilon_A \epsilon_B}$
$\Omega_{D, AB}$: get interpolated table values as function of $T^*=k_B T / \epsilon_{AB}$

Note

Check online course materials for the $\Omega_{D, AB}$ table!

Molecular Diffusivity Theory – Fuller Method

The Chapman Enskog theory can be difficult to use, engineers need some simplied empirical rules.

\[\begin{align} D_{AB} &= \frac{1.0\times{}10^{-7} T^{1.75} }{p_T \left[ (\sum \nu_A)^{1/3} + (\sum \nu_B)^{1/3}\right]^2} \left(\frac{1}{m_A} + \frac{1}{m_B} \right)^{\frac{1}{2}} \\ &\propto \frac{T^{1.75}}{p_T} \end{align}\]

Mainly to overcome the complicated estimation of $\Sigma_{D, AB}$ in Chapman-Enskog Theory
$\sum \nu_i$ sum of structural volume increments
$\sum \nu_i$ can be estimated from individual atoms
Easier to use than the Chapman-Enskog Theory, but less accurate

Fuller Method: How to Use

\[\begin{align} D_{AB} = \frac{1.0\times{}10^{-7} T^{1.75} }{p_T \left[ (\sum \nu_A)^{1/3} + (\sum \nu_B)^{1/3}\right]^2} \left(\frac{1}{m_A} + \frac{1}{m_B} \right)^{\frac{1}{2}} \end{align}\]

The Fuller method formula in Geankoplis book need to use the following units:

$T$: absolute temperature in $\text{K}$
$m_A$, $m_B$: molecular weight in $\text{kg} \cdot (\text{kg mol})^{-1}$ or $\text{g} \cdot (\text{g mol})^{-1}$
$p_T$: absolute pressure in $\text{atm}$
$\nu_{A}$, $\nu_{B}$: structural volume increments (dimensionless)

Resulting $D_{AB}$ is in $\text{m}^2 \cdot \text{s}^{-1}$

How to Use the Fuller Method From a Table

We need to determine $\sum \nu_A$ and $\sum \nu_B$, Check Table 6.2-2 of Geankoplis 4th ed.

$\sum \nu_i$ are dimensionless numbers
For known gases in the table, use its value in the table
- Air: $\nu = 20.1$
- O$_2$: $\nu = 16.6$
For an unknown gas, use its chemical composition
- For chemical formula $\text{C}_x \text{H}_y \text{O}_z$, \[ \sum \nu(\text{C}_x \text{H}_y \text{O}_z) = \nu(\text{C}) \cdot x + \nu(\text{H}) \cdot y + \nu(\text{O}) \cdot z \]

Estimating $D_{AB}$ At Different $(T, P)$

For the same pair of (A, B), both the Chapman-Enskog and Fuller methods have the same form \[ D_{AB} \propto \frac{T^n}{P} \]

This means we can find one existing $D_{AB}$ value from table and extrapolate: \[ \frac{D_{AB}\vert_1}{D_{AB}\vert_2} = \left(\frac{T_1}{T_2}\right)^{n} \left( \frac{P_2}{P_1} \right) \]

Fuller method is often used in chemical engineering, where $n=1.75$.

Example 1: Chapman-Enskog Theory vs Fuller Method

Question: calculate the diffusivity $D_{AB}$ for methan–ethan system at 313 K and 1 atm.

Chapman-Enskog Theory

$m_A=16.04$, $m_B=30.07$
$\rho_A (Å)=3.822$, $\rho_B (Å)=4.418$
$\epsilon_A = k_B * (137\ \text{K})$, $\epsilon_B = k_B * (230\ \text{K})$
$\Omega_{D, AB} = 1.125$ in this case

Fuller Method

$m_A=16.04$, $m_B=30.07$
$\nu(\text{C})=16.5$, $\nu(\text{H})=1.98$

The experimental value at this condition is $D_{AB}=1.84\times 10^{-5}\ \text{m}^2\cdot s^{-1}$. Compare the percent errors in both methods.

Solution to Example 1:

Chapman-Enskog Theory:

Diffusivity (m^2/s) = 1.666e-05
Percentage error: 9.45%

Fuller Method:

Diffusivity (m^2/s) = 1.728e-05
Percentage error: 6.07%

Summary

Diffusivity $D_{AB}$ value is the key to solve the transport problem in gases
$D_{AB}$ can be measured from experiments, but not exhaustively
Several theories predict the value for $D_{AB}$ with certain accuracy
Chapman-Enkog theory and Fuller method both have $D_{AB}\propto T^n/P$ form
Fuller method more often used as it can approximate $D_{AB}$ solely by taking values from table

Footnotes

adapted from BSL Transport Phenomena ch 17.3↩︎

--- title: "CHE 318 Lecture 04" subtitle: "Molecular Diffusivity: Theories and Measurement" author: "Dr. Tian Tian" date: "2026-01-12" format: html: {} revealjs: output-file: slides.html pdf: output-file: L04.pdf --- ::: {.content-visible when-format="html" unless-format="revealjs"} ::: {.callout-note} - Slides 👉 [Open presentation🗒️](./slides.html) - PDF version of course note 👉 [Open in pdf](./L04.pdf) - Handwritten notes 👉 [Open in pdf](./public/L04_annotated.pdf) ::: ::: ::: {.content-visible when-format="revealjs"} ::: {.callout-note} - **Course Website Updated!** [https://tiangroup-uofa.github.io/che318-mass-transfer/](https://tiangroup-uofa.github.io/che318-mass-transfer/) <div style="width: 100%; text-align: center;"> <img src="https://quickchart.io/qr?text=https://tiangroup-uofa.github.io/che318-mass-transfer/&size=250" alt="QR Code" style="width: 150px; height: 150px; display: block; margin: 0 auto;" /> </div> ::: ::: ## Recap - **General solution** for diffusion binary mixture gas systems ($N_B = k N_A$) - $k = -1$ reduces to EMCD solution - Total flux $N_A$ is EMCD flux times a coefficient ```{=tex} \begin{align} N_A = \frac{ c_T D_{AB}}{(z_2 - z_1)} \left(\frac{N_A}{N_A + N_B}\right) \ln\!\left[\dfrac{ \frac{N_A}{N_A + N_B} - x_{A2}} {\frac{N_A}{N_A + N_B} - x_{A1}} \right] \end{align} ``` - Brief discussion about diffusivity measurement: two-bulb setup ## Demonstration of General Solution ```{=html} <iframe width="100%" height="800" src="../../scripts/L03_xa_profile.html" title="Webpage example"></iframe> ``` ## Interaction Time! [participation link](https://app.wooclap.com/318L04?from=instruction-slide) *Results and comments to be published after the class* ::: {.content-visible when-format="revealjs"} --- ## Convection-Driven Flux Change ![](./public/NA_comparison_plots.svg){#fig-na-comp width="100%"} ::: ::: {.content-visible when-format="html" unless-format="revealjs"} --- ## Detailed Explanation for Convection-Driven Flux Change ::: {.callout-note} Advanced topic, not shown in presentation ::: The ratio between total flux in general solution $N_A(\text{gen})$ and $N_A(\text{EMCD})$ can be written as: ```{=tex} \begin{align} \frac{N_A(\text{gen})}{N_A(\text{EMCD})} &= \frac{s \ln\! \left[ \frac{s - x_{A2}}{s - x_{A1}} \right]}{x_{A1} - x_{A2}} \\ &= \frac{s \ln\! \left[ \frac{s - x_{A2}}{s - x_{A1}} \right]}{(s - x_{A2}) - (s - x_{A1})} \\ &= \frac{s}{\text{LM}(s-x_A)} \end{align} ``` where $\text{LM}(u)$ is the log-mean function (similar treatment in [Lecture 2](../L02)), and $s = N_A / (N_A + N_B)$. There are several limiting cases. Let's assume $x_{A1} > x_{A2}$ for simplicity, and there is a real solution to the $N_A(\text{gen})$ 1. $s > x_{A1}$: $s - x_{A1}$ and $s - x_{A2}$ must be both positive to ensure a solution. Therefore $\text{LM}(s-x_A) < s$. We know that $N_A(\text{gen})>N_A(\text{EMCD})$. - Example: diffusion through stagnant B ($s = 1$) In this case $\text{LM}(s-x_A) = \text{LM}(x_B)$, we can conclude that the diffusion through stagnant B case **always enhances** total flux compared with EMCD. A signature of such transport is the $x_A(z)$ profile is **concave**, as shown in @fig-na-comp (a). 2. $s < 0$: $s - x_{A2}$ and $s - x_{A1}$ are be both negative. Therefore $|\text{LM}(s-x_A)| > s$. We know that $N_A(\text{gen})<N_A(\text{EMCD})$, but $N_A$ is still positive. - Example: counter $N_B$ flux ($N_B/N_A < -1$) The opposite direction of $N_B$ **attenuates** the molar flux of A. A signature of such transport is the $x_A(z)$ profile is **convex**, as shown in @fig-na-comp (b). 3. $0 < s < x_2$: $s - x_{A2}$ and $s - x_{A1}$ are be both negative. Since $s >0$, we actually can have $N_A < 0$! Such situation may be counter-intuitive but the solution is physically valid, as the only solution to satisfy such criteria is $N_B < 0$ and $N_A < 0$ at the same time. As we cannot directly compare $s$ and $\text{LM}(s-x_A)$ in this case, $|N_A(\text{gen})|$ can be either smaller or larger than $N_A(\text{EMCD})$, as shown in @fig-na-comp (c) and (d). 4. $x_2 < s < x_1$: The expression inside the logarithm is negative and we do not have a real solution to the general equation for $N_A$. ![Comparison between general solution and EMCD flux. **a**. Enhance flux via diffusion through stagnant B. **b** Reduced flux when $s=-1$. **c**. Enhanced counter flux case **d** Reduced counter flux case](./public/NA_comparison_plots.svg){#fig-na-comp width="100%"} ::: --- ## Learning Outcomes {.center} After today's lecture, you will be able to: - **Recall** multiple theories of molecular diffusivity - **Analyze** the diffusivity $D_{AB}$ as function of $(T, P)$ - **Derive** new diffusivity values from standard table - **Formulate** governing equations for different scenarios measuring the diffusivity --- ## Let's Look At The General Mass Transfer Equation (Again) $$ N_A = \frac{D_{AB} c_T}{(z_2 - z_1)} s \ln\!\left[\dfrac{ s - \frac{c_{A2}}{c_T}} {s - \frac{c_{A1}}{c_T}} \right] \quad s = \frac{N_A}{N_A + N_B} $$ What values do we know from the system setup? - Geometry: $z_1$, $z_2$, $c_{A1}$, $c_{A2}$, $c_T$ - Chemical reaction stoichiometry: $k$ What else value(s) do we need to solve $N_A$? - $D_{AB}$: generally $D_{AB}(z) = \text{Const}$ - But $D_{AB}=f(T, P, \cdots)$ --- ## Motivation to Have Theory About Diffusivity {.center} - Solving the general solution for gas mass transfer requires parameter $D_{AB}$ (and $D_{BA}$)! - Measuring every $D_{AB}$ pair for gases is tedious - $D_{AB}$ is dependent on conditions $T, P$ - We need to have theories that can **predict** diffusivity $D_{AB}$ **without** doing all pair experiments and all $(T, P)$ conditions! - In the lease case, the theory should allow extrapolating a measured $D_{AB}(T_1, P_1)$ to $D_{AB}(T_2, P_2)$ --- ## Molecular Diffusivity Theory -- Kinetic Theory In dilute gas regime ($p_T \approx \text{atm}$), the kinetic behaviour of gas molecules can be described using kinetic theory. A few assumptions are made: 1. Interdiffusion between A and B are dominated by **collision** between molecules (low $p_T$ 👉 **only 2** molecules can collide at the same time) 2. The molecules are rigid spheres having particle masses $m_A$, $m_B$ and diameters $d_A$ and $d_B$ ![Scheme of gas kinetic theory](./public/kinetic-theory-gas.png){width="80%"} ## Kinetic Theory -- Key Results There are some results from the kinetic theory: ^[adapted from *BSL Transport Phenomena ch 17.3*] ::: {.columns} ::: {.column width="50%"} 1. Mean molecular speed $\overline{u}$ $$ \overline{u} = \sqrt{\frac{8 k_B T}{\pi \overline{m_{AB}}}} $$ 2. Mean free path (between collisions): $\lambda_{A,B}$ $$ \lambda_{AB}=\frac{1}{\sqrt{2} \pi d_{AB}^2 c_T} $$ ::: ::: {.column width="50%"} 3. Frequency of molecule A colliding with wall ($Z$, unit $\text{m}^{-2}\cdot \text{s}^{-1}$): $$ Z_A = \frac{1}{4} c_A \overline{u} $$ where $d_{AB} = \frac{d_A + d_B}{2}$, $m_{AB}^{-1} = m_{A}^{-1} + m_B^{-1}$, $k_B$ is the Boltzmann constant 4. We also have the average distance $a$ to one plane from last collision: $$ a = \frac{2}{3}\lambda_{AB} $$ ::: ::: --- ## Kinetic Theory -- Mass Transfer When there is only molecular diffusion (no convection), at plane $z=0$, the flux can be described by: ::: {.columns} ::: {.column width="60&"} ```{=tex} \begin{align} J_{Az}^*\vert_{z=0} &= \text{[In]}\vert_{z=-a} - \text{[Out]}\vert_{z=+a} \\ &= \frac{1}{4} c_T [x_A \overline{u}]\vert_{z=-a} \\ &- \frac{1}{4} [c_T x_A \overline{u}]\vert_{z=+a}\\ &= -\frac{1}{4} c_T \cdot 2a \cdot \overline{u} \frac{d x_A}{dz} \\ &= -D_{AB} c_T \frac{dx_A}{dz} \end{align} ``` We can now link $D_{AB}$ to microscopic properties! ::: ::: {.column width="40%"} ![Mass balance in gas kinetic theory](./public/kt-mass-balance.png){width="80%"} ::: ::: --- ## Kinetic Theory of Diffusivity -- Conclusion ::: {.columns} ::: {.column width="50%"} Results: $$ D_{AB} = \frac{1}{3} \overline{u} \lambda_{A,B} $$ - $\overline{u}$: average molecular velocity - $\lambda$: mean free path between collisions ::: :::{.column width="50%"} Assumptions: - Rigid sphere type molecules (what are they?) - No interaction upon collisions - Collisions are elastic (momentum conserved) - Good for dilute gases (low $P$) - Not accurate otherwise ::: ::: --- ## Molecular Diffusivity -- Chapman-Enskog Theory We can take real interaction between molecules into the kinetic theory! ```{=tex} \begin{align} D_{AB} &= \frac{1.8583\times{}10^{-7} T^{3/2}} {p_T \sigma_{AB}^2 \Omega_{D, AB}} \left(\frac{1}{m_A} + \frac{1}{m_B} \right)^{\frac{1}{2}} \\ &\propto \frac{T^{\frac{3}{2}}}{p_T} \left(\frac{1}{m_A} + \frac{1}{m_B} \right)^{\frac{1}{2}} \end{align} ``` - $m_A$, $m_B$, molecular weight - $\sigma_{AB}$: average collision radius between A and B - $\Omega_{D, AB}$: Lennard-Jones collision integral - $\Omega_{D, AB}=1$ for elastic collision - Chapman-Enskog theory accounts for non-elastic collision using $\Omega_{D, AB}\neq1$ --- ## Chapman-Enskog Theory: How to Use ```{=tex} \begin{align} D_{AB} = \frac{1.8583\times{}10^{-7} T^{3/2}} {p_T \sigma_{AB}^2 \Omega_{D, AB}} \left(\frac{1}{m_A} + \frac{1}{m_B} \right)^{\frac{1}{2}} \end{align} ``` The Chapman-Enskog theory formula in *Geankoplis book* need to use the following units: - $T$: absolute temperature in $\text{K}$ - $m_A$, $m_B$: molecular weight in $\text{kg} \cdot (\text{kg mol})^{-1}$ or $\text{g} \cdot (\text{g mol})^{-1}$ - $p_T$: absolute pressure in $\text{atm}$ - $\sigma_{AB}$: average collision diameter in Å - $\Omega_{D, AB}$: collision integral (dimensionless) Resulting $D_{AB}$ is in $\text{m}^2 \cdot \text{s}^{-1}$ --- ## Chapman-Enskog Theory & Lennard Jones Potential - Parameters $\sigma_{AB}$ and $\Omega_{D, AB}$ can be derived using Lennard-Jones (LJ) potential - LJ potential described the interaction energy $U_{AB}$ between 2 molecules at distance $r$ follows: ```{=tex} \begin{align} U_{AB}(r) = 4\epsilon_{AB}\left[ (\frac{\sigma_{AB}}{r})^{12} - (\frac{\sigma_{AB}}{r})^{6} \right] \end{align} ``` How to get the parameters from table: - $\sigma_{AB} = (\sigma_{A} + \sigma_{B})/2$ - $\epsilon_{AB} = \sqrt{\epsilon_A \epsilon_B}$ - $\Omega_{D, AB}$: get interpolated table values as function of $T^*=k_B T / \epsilon_{AB}$ ::: {.callout-note} Check online [course materials]() for the $\Omega_{D, AB}$ table! ::: --- ## Molecular Diffusivity Theory -- Fuller Method The Chapman Enskog theory can be difficult to use, engineers need some simplied empirical rules. ```{=tex} \begin{align} D_{AB} &= \frac{1.0\times{}10^{-7} T^{1.75} }{p_T \left[ (\sum \nu_A)^{1/3} + (\sum \nu_B)^{1/3}\right]^2} \left(\frac{1}{m_A} + \frac{1}{m_B} \right)^{\frac{1}{2}} \\ &\propto \frac{T^{1.75}}{p_T} \end{align} ``` - Mainly to overcome the complicated estimation of $\Sigma_{D, AB}$ in Chapman-Enskog Theory - $\sum \nu_i$ sum of structural volume increments - $\sum \nu_i$ can be estimated from individual atoms - Easier to use than the Chapman-Enskog Theory, but less accurate --- ## Fuller Method: How to Use ```{=tex} \begin{align} D_{AB} = \frac{1.0\times{}10^{-7} T^{1.75} }{p_T \left[ (\sum \nu_A)^{1/3} + (\sum \nu_B)^{1/3}\right]^2} \left(\frac{1}{m_A} + \frac{1}{m_B} \right)^{\frac{1}{2}} \end{align} ``` The Fuller method formula in *Geankoplis book* need to use the following units: - $T$: absolute temperature in $\text{K}$ - $m_A$, $m_B$: molecular weight in $\text{kg} \cdot (\text{kg mol})^{-1}$ or $\text{g} \cdot (\text{g mol})^{-1}$ - $p_T$: absolute pressure in $\text{atm}$ - $\nu_{A}$, $\nu_{B}$: structural volume increments (dimensionless) Resulting $D_{AB}$ is in $\text{m}^2 \cdot \text{s}^{-1}$ ## How to Use the Fuller Method From a Table We need to determine $\sum \nu_A$ and $\sum \nu_B$, *Check Table 6.2-2 of Geankoplis 4th ed*. - $\sum \nu_i$ are **dimensionless** numbers - For known gases in the table, use its value in the table - Air: $\nu = 20.1$ - O$_2$: $\nu = 16.6$ - For an unknown gas, use its chemical composition - For chemical formula $\text{C}_x \text{H}_y \text{O}_z$, $$ \sum \nu(\text{C}_x \text{H}_y \text{O}_z) = \nu(\text{C}) \cdot x + \nu(\text{H}) \cdot y + \nu(\text{O}) \cdot z $$ ## Estimating $D_{AB}$ At Different $(T, P)$ For the same pair of (A, B), both the Chapman-Enskog and Fuller methods have the same form $$ D_{AB} \propto \frac{T^n}{P} $$ This means we can find one existing $D_{AB}$ value from table and extrapolate: $$ \frac{D_{AB}\vert_1}{D_{AB}\vert_2} = \left(\frac{T_1}{T_2}\right)^{n} \left( \frac{P_2}{P_1} \right) $$ Fuller method is often used in chemical engineering, where $n=1.75$. --- ## Example 1: Chapman-Enskog Theory vs Fuller Method Question: calculate the diffusivity $D_{AB}$ for methan--ethan system at 313 K and 1 atm. ::: {.columns} ::: {.column width="50%"} #### Chapman-Enskog Theory - $m_A=16.04$, $m_B=30.07$ - $\rho_A (Å)=3.822$, $\rho_B (Å)=4.418$ - $\epsilon_A = k_B * (137\ \text{K})$, $\epsilon_B = k_B * (230\ \text{K})$ - $\Omega_{D, AB} = 1.125$ in this case ::: ::: {.column width="50%"} #### Fuller Method - $m_A=16.04$, $m_B=30.07$ - $\nu(\text{C})=16.5$, $\nu(\text{H})=1.98$ ::: ::: The experimental value at this condition is $D_{AB}=1.84\times 10^{-5}\ \text{m}^2\cdot s^{-1}$. Compare the percent errors in both methods. ::: {.content-visible when-format="html" unless-format="revealjs"} --- ## Solution to Example 1: ```{python} #| echo: false import marimo as mo def chapman(ma, mb, T, p, sig_a, sig_b, Omega): coeff = 1.8583e-7 numerator = coeff * T ** 1.5 denominator = p * ((sig_a + sig_b) / 2) ** 2 * Omega mass_inv = (1/ma + 1/mb) ** 0.5 return numerator / denominator * mass_inv def fuller(ma, mb, T, p, nu_a, nu_b): coeff = 1.0e-7 numerator = coeff * T ** 1.75 denominator = p * (nu_a ** (1/3) + nu_b ** (1/3)) ** 2 mass_inv = (1/ma + 1/mb) ** 0.5 return numerator / denominator * mass_inv def nu_ch(x_c, x_h): nu_c = 16.5 nu_h = 1.98 return nu_c * x_c + nu_h * x_h def calculate(): ma = 16.04 mb = 30.07 symb_a = (1, 4) symb_b = (2, 6) _T = 313 _p = 1.0 sig_a = 3.822 sig_b = 4.418 Omega = 1.125 v_chap = chapman(ma, mb, _T, _p, sig_a, sig_b, Omega) v_fuller = fuller(ma, mb, _T, _p, nu_ch(*symb_a), nu_ch(*symb_b)) v_real = 1.84e-5 output = mo.vstack([ mo.md(f"""**Chapman-Enskog Theory**: - Diffusivity (m^2/s) = {v_chap:.3e} - Percentage error: {abs((v_chap - v_real) / v_real) * 100:.2f}% """), mo.md(f""" **Fuller Method**: - Diffusivity (m^2/s) = {v_fuller:.3e} - Percentage error: {abs((v_fuller - v_real) / v_real) * 100:.2f}% """ )]) return output calculate() ``` ::: --- ## Summary - Diffusivity $D_{AB}$ value is the key to solve the transport problem in gases - $D_{AB}$ can be measured from experiments, but not exhaustively - Several theories predict the value for $D_{AB}$ with certain accuracy - Chapman-Enkog theory and Fuller method both have $D_{AB}\propto T^n/P$ form - Fuller method more often used as it can approximate $D_{AB}$ solely by taking values from table

Recap

Demonstration of General Solution

Interaction Time!

Detailed Explanation for Convection-Driven Flux Change

Learning Outcomes

Let’s Look At The General Mass Transfer Equation (Again)

Motivation to Have Theory About Diffusivity

Molecular Diffusivity Theory – Kinetic Theory

Kinetic Theory – Key Results

Kinetic Theory – Mass Transfer

Kinetic Theory of Diffusivity – Conclusion

Molecular Diffusivity – Chapman-Enskog Theory

Chapman-Enskog Theory: How to Use

Chapman-Enskog Theory & Lennard Jones Potential

Molecular Diffusivity Theory – Fuller Method

Fuller Method: How to Use

How to Use the Fuller Method From a Table

Estimating \(D_{AB}\) At Different \((T, P)\)

Example 1: Chapman-Enskog Theory vs Fuller Method

Chapman-Enskog Theory

Fuller Method

Solution to Example 1:

Summary

Footnotes