Material science

1. DIFFUSION-2
ENG. KAREEM H. MOKHTAR

2. ARRHENIUS EQUATION
• Rate = c e-E/RT
• Rate: rate of diffusion (m2/s
or jump per second)
• E: activation energy
• R: gas constant
• T: temperature
• ln(rate) = ln(c) – E/RT
• Straight line equation
Y= m x + c
Ln (rate)
1/T
Slope= -E/R

3. FICK’S FIRST LAW
• The rate at which atoms, ions, particles or other species diffuse in a material can
be measured by the flux J. Here we are mainly concerned with diffusion of ions or
atoms. The flux J is defined as the number of atoms passing through a plane of
unit area per unit time

4. SHEET 4
• Q6) 0.05 cm layer of magnesium oxide (MgO) is deposited between layers of
nickel (Ni) and tantalum (Ta) to provide a diffusion barrier that prevents reactions
between the two metals. At 1400°C, nickel ions diffuse through the MgO ceramic
to the tantalum. Determine the number of nickel ions that pass through the MgO
per second. At 1400°C, the diffusion coefficient of nickel ions in MgO is 9 * 10-12
cm2/s and the concentration of Ni on Ni/MgO surface equals 8.573 x 1022
atoms/cm3

5. ANSWER
• J =-D
𝑑𝑒𝑙𝑡𝑎 𝐶
𝐷𝑒𝑙𝑡𝑎 𝑥
•
𝑑𝑒𝑙𝑡𝑎 𝐶
𝐷𝑒𝑙𝑡𝑎 𝑥
=
0−8.573 𝑥 1022
0.05
= -1.715 * 1024 atoms/ cm3 .cm
• J= - (9 * 10-12 ) x( -1.715 * 1024 ) = 1.543*1013 𝑁𝑖 𝑎𝑡𝑜𝑚𝑠
𝑐𝑚2
𝑠
• Total Ni passing in one second = J
𝑁𝑖 𝑎𝑡𝑜𝑚𝑠
cm2 𝑠
x area (cm2) = 1.543*1013 x (2cm x
2cm)
• = 6.17 x 1013 Ni atoms/ s

6. SHEET 4
• Q7) An impermeable cylinder 3 cm in diameter and 20 cm long with a membrane
at its half, contains a gas that includes 0.5 * 1020 N atoms per cm3 and 0.5 * 1020 H
atoms per cm3 on one side of an iron membrane . Gas is continuously introduced
to the pipe to ensure a constant concentration of nitrogen and hydrogen. The gas
on the other side of the membrane includes a constant 1 * 1018 N atoms per cm3
and 1 * 1018 H atoms per cm3. The entire system is to operate at 700°C. Design an
iron membrane (BCC) that will

7. SHEET 4 – Q7
• allow no more than 1% of the nitrogen to be lost through the membrane each
hour
• allow 90% of the hydrogen to pass through the membrane per hour.
• Satisfy both later conditions
• given that for N diffusion constant = 0.0047 cm2/s,, and E= 18300 cal/mol
• and for H diffusion constant = 0.0012 cm2/s,, and E= 3600 cal/mol)

8. ANSWER

9. TO ALLOW NO MORE THAN 1% OF THE NITROGEN TO BE
LOST THROUGH THE MEMBRANE EACH HOUR
• The total number of nitrogen atoms in the container is
• 0.5 * 1020 x (pi/4) x 32 cm2 x 10 cm = 35.34 x 1020 N atom
• The maximum number of atoms to be lost = 1% from N per hour
• N atoms lost= 35.34 x 1020 N x 0.01 = 35.34 x 1018 N atoms per hour
J =
35.34 x 1018
3600 ∗32 ∗
pi
4
= 0.00139x 1018 N atoms per cm2 per
second

10. TO ALLOW NO MORE THAN 1% OF THE NITROGEN TO BE
LOST THROUGH THE MEMBRANE EACH HOUR
• Rate of diffusion (D) = c e-E/RT = 0.0047 x e-18300/1.98x973 =3.64x10-7
cm2/s
• dx=
−𝐷 ∗ 𝑑𝑐
𝐽
= −3.64 ∗ 10−7 ∗
1∗1018−0.5∗1020
0.00139 ∗1018 = 0.013 𝑐𝑚

11. ALLOW 90% OF THE HYDROGEN TO PASS
THROUGH THE MEMBRANE PER HOUR.
• H atom loss per hour= (0.90)(35.343 * 1020)= 31.80 * 1020
• J =
31.8 x 1020
3600 ∗32 ∗
pi
4
= 1.2 ∗ 1017 H atoms/ cm2 s
• D = 0.0012 e-3600/(1.98*973) = 1.86 x 10-4
• dx= ( -1.86* 10-4 * -49 * 1018 )/ 1.25 * 1017 = 0.073 cm

12. SATISFY BOTH LATER CONDITIONS
• An iron membrane with a thickness between 0.013 and 0.073 cm will be
satisfactory.

13. BONUS QUESTION
it was found that the rate of diffusion is 5x 106 jumps/s at 500 C and 7x 108 jumps/s at
600 C
a) plot a linear relation that has a slope of -E/R
b) calculate the activation energy needed.

14. BONUS QUESTION SOLUTION
Rate 5000000 700000000
T (k) 773 873
ln R 15.4249485 20.3665909
1/T 0.00129366 0.00114548
Ln(r)= ln (c) – E/RT  plot between Ln r and 1/T

15. 0
5
10
15
20
25
0.00112 0.00114 0.00116 0.00118 0.0012 0.00122 0.00124 0.00126 0.00128 0.0013 0.00132
Ln
R
1/T
Chart Title

16. B) slope= -E/R  -33347.636 = -E/ 8.314= 277252.2 Joules (at 773K)
= - E/1.98= 66028.32 cal (at 873 K)

17. PROJECT DESCRIPTION
• Submit a report and perform a presentation on the topic associated to your
group
• Report
• must be written according technical writing rules
• 7-10 pages without cover page- list of figure – list of tables – references
• Due date Monday 12/12/2022
• Presentation
• Duration: 7 minutes (2 persons) or 10 minutes (3 persons)
• Due date Saturday 17/12/2022

18. ORAL PRESENTATION RUBRIC