1d:I[72985,["/_next/static/chunks/3th5o5azza5j1.js","/_next/static/chunks/0953gtd9gbswy.js","/_next/static/chunks/01oly6y_17ki0.js","/_next/static/chunks/1xlwfijmtjgqj.js","/_next/static/chunks/2hzca_9a7jrui.js","/_next/static/chunks/0_v6xzhg0htsf.js","/_next/static/chunks/2x0yuymt3nfdv.js","/_next/static/chunks/1hjnhv8wc11is.js"],"default"] 1e:I[63508,["/_next/static/chunks/3th5o5azza5j1.js","/_next/static/chunks/0953gtd9gbswy.js","/_next/static/chunks/01oly6y_17ki0.js","/_next/static/chunks/1xlwfijmtjgqj.js","/_next/static/chunks/2hzca_9a7jrui.js","/_next/static/chunks/0_v6xzhg0htsf.js","/_next/static/chunks/2x0yuymt3nfdv.js","/_next/static/chunks/1hjnhv8wc11is.js"],"default"] 1f:T4094,

In a chemical engineering transport phenomena laboratory experiment, ammonia gas ( $\text{NH}_3$ , Species A) is diffusing in one dimension through a straight cylindrical tube containing stationary nitrogen gas ( $\text{N}_2$ , Species B) under steady-state conditions without bulk convective motion (Stagnant Medium, No Bulk Flow).

The diffusion path length of the tube is $z = 0.15 \text{ m}$ , and the tube maintains a uniform cross-sectional area of $S = 0.005 \text{ m}^2$ . Quantitative boundary concentration measurements show:

The molar concentration of ammonia at the tube inlet ( $z = 0$ ) is maintained constant at $C_{A1} = 0.20 \text{ mol/m}^3$ .
The molar concentration of ammonia at the tube outlet ( $z = 0.15 \text{ m}$ ) is maintained constant at $C_{A2} = 0.05 \text{ mol/m}^3$ .

Under the operating temperature and pressure of this experimental system, the binary mass diffusivity of ammonia in nitrogen is $D_{AB} = 2.30 \times 10^{-5} \text{ m}^2/\text{s}$ .

Determine:

The molar concentration driving force ( $\Delta C_A$ ).
The steady-state molar flux of ammonia ( $J_A$ ) using Fick's First Law.
The total molar transfer rate of ammonia ( $W_A$ ) moving through the cross-sectional area of the tube.

(Note: You can modify the variable values in the input data section below to explore the interactive system response and calculation updates in real time.)

20:Tfe7,

Click to view standard binary gas diffusivity reference table (at 1 atm, 25 °C)

Representative mass diffusivities for binary gas mixtures at atmospheric pressure (1 atm) and 25 °C (298 K) for preliminary chemical engineering design:

Binary Gas Pair (A–B)	Diffusivity (D_AB) (m²/s)	Typical Engineering Application
NH₃ – N₂ (Ammonia in Nitrogen)	2.30 × 10⁻⁵	Gas absorption and fertilizer synthesis
CO₂ – Air (Carbon dioxide in Air)	1.64 × 10⁻⁵	Carbon capture and building ventilation
O₂ – Air (Oxygen in Air)	2.06 × 10⁻⁵	Biochemical aeration and combustion processes
H₂O – Air (Water vapor in Air)	2.56 × 10⁻⁵	Cooling towers, psychrometrics, and drying systems
H₂ – Air (Hydrogen in Air)	6.11 × 10⁻⁵	Hydrogen fuel cell safety and leak detection

21:T7311,

Step 1: Calculate the molar concentration difference ( $\Delta C_A$ ) (Evaluate the magnitude of the chemical driving force by finding the difference between inlet and outlet concentrations)

💡 Gradient Sign Tip: In one-dimensional diffusion, the actual physical gradient is negative: $\frac{dC_A}{dz} = \frac{C_{A2} - C_{A1}}{z} = \frac{0.05 - 0.20}{0.15} = -1.00 \text{ mol/m}^4$ , reflecting that molecules migrate down the concentration gradient from high to low. The negative sign in Fick's First Law ( $J_A = -D_{AB} \frac{dC_A}{dz}$ ) cancels this out, ensuring the net transport flux $J_A$ is positive in the $+z$ direction.

Step 2: Calculate the steady-state molar flux ( $J_A$ ) (Compute the molar transfer rate per unit cross-sectional area per unit time according to Fick's First Law)

💡 Dimensional Verification: Evaluating the units: $\left(\text{m}^2/\text{s}\right) \times \left(\frac{\text{mol/m}^3}{\text{m}}\right) = \left(\text{m}^2/\text{s}\right) \times \left(\text{mol/m}^4\right) = \text{mol/(m}^2\cdot\text{s)}$ . Notice that because the geometric ratio $\frac{\Delta C_A}{z} = \frac{0.15}{0.15} = 1.00 \text{ m}^{-1}$ , the numerical value of the molar flux is identical to the diffusivity $D_{AB}$ in this benchmark case.

Step 3: Calculate the total molar transfer rate ( $W_A$ ) (Determine the total moles of species A crossing the entire cross-sectional area $S$ per unit time)

22:T86b5,

From the calculations above, the numerical results directly connect to core chemical engineering principles and transport phenomena across four key aspects:

Driving Force and Linear Concentration Profile ( $\Delta C_A$ and Concentration Gradient): The calculated concentration driving force of mol/m³ over the conduit length of m establishes a uniform concentration gradient magnitude of $\left|\frac{dC_A}{dz}\right| = \frac{\Delta C_A}{z} = 1.00 \text{ mol/m}^4$ . This gradient serves as the primary driving force for molecular mass transfer, directly analogous to the temperature gradient ( $\frac{dT}{dx}$ ) in Fourier's law of heat conduction and the electric potential gradient ( $\frac{dV}{dx}$ ) in Ohm's law.
Molar Flux and Molecular Scale Dynamics ( $J_A$ ): The resulting molar flux is mol/(m²·s), representing the number of moles of ammonia molecules traversing a $1 \text{ m}^2$ cross-section each second. In gases, large mean free paths between collisions produce diffusivities of order $10^{-5} \text{ m}^2/\text{s}$ , approximately four orders of magnitude faster than in liquids ( $\approx 10^{-9} \text{ m}^2/\text{s}$ ) and up to nine orders of magnitude faster than in solids ( $\approx 10^{-14} \text{ m}^2/\text{s}$ ).
Total Molar Rate and Mass Flow Conversion ( $W_A$ ): Across the actual conduit cross-section of m², the total molar flow is mol/s (or $0.115\ \mu\text{mol/s}$ ). Converting this to an equivalent mass flow rate ( $\dot{m}_A = W_A \times M_A$ ) using ammonia's molecular weight ( $M_{\text{NH}_3} \approx 17.03 \text{ g/mol}$ ): $\dot{m}_A = (1.15 \times 10^{-7} \text{ mol/s}) \times (17.03 \text{ g/mol}) \approx 1.96 \times 10^{-6} \text{ g/s} \approx 7.05 \text{ mg/hr}$ This quantitative result illustrates a fundamental chemical engineering reality: pure molecular diffusion through stagnant media is an inherently slow transport mechanism.
Implications for Industrial Process Design (Convective Scale-Up): Because purely molecular diffusion yields milligrams per hour at laboratory scales, commercial separation units—such as gas absorbers, strippers, distillation columns, and chemical reactors—cannot rely solely on stagnant diffusion. Chemical engineers actively induce convective mass transfer through mechanical agitation, high-Reynolds turbulent flow, and structured packing. These methods drastically compress the boundary layer thickness ( $\delta$ ), elevating the local gradient $\frac{\Delta C_A}{\delta}$ by orders of magnitude and accelerating industrial mass throughput tens of thousands of times over stagnant diffusion.

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This example demonstrates how to analyze and calculate steady-state one-dimensional molecular diffusion according to Fick's First Law, determining both the molar flux and the total molar transfer rate of a gaseous species through a stagnant medium.

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