NPTEL – Chemical – Mass Transfer Operation 1
MODULE 2: DIFFUSION
LECTURE NO. 4
2.4
DIFFUSION
COEFFICIENT:
MEASUREMENT
AND
PREDICTION
The proportionality factor of Fick’s law is called diffusivity or diffusion coefficient
which can be defined as the ratio of the flux to its concentration gradient and its
unit is m2/s. It is a function of the temperature, pressure, nature and
concentration of other constituents. Diffusivity decreases with increase in
pressure ( DAB  1 / p for moderate ranges of pressures, upto 25 atm) because
number of collisions between species is less at lower pressure. But the diffusivity
is hardly dependent on pressure in case of liquid. The diffusivity increases with
increase in temperature ( D AB  T 1.5 ) because random thermal movement of
molecules increases with increase in temperature. The diffusivity is generally
higher for gases (in the range of 0.5×10–5 to 1.0 × 10-5 m2/s) than for liquids (in
the range of 10–10 to 10-9 m2/s). The diffusivity value reported for solids is higher
in the range of 10
–13
to 10-5 m2/s. Diffusion is almost impossible in solids
because the particles are too closely packed and strongly held together with no
‘empty space’ for particles to move through. Solids diffuse much slower than
liquids because intermolecular forces in solid are stronger enough to hold the
solid molecules together.
Joint initiative of IITs and IISc – Funded by MHRD
Page 1 of 5
NPTEL – Chemical – Mass Transfer Operation 1
2.4.1 Measurement of gas-phase diffusion coefficient
There are several methods of experimental determination of gas-phase diffusion
coefficient. Two methods are (a) Twin-bulb method and (b) Stefan tube method.
Predictive Equations are sometimes used to determine diffusivity. These may be
empirical, theoretical or semi-empirical.
(a) Twin-bulb method
Two large bulbs are connected by a narrow tube. The schematic representation
is shown in Figure 2.6. In the beginning two bulbs are evacuated and all the three
valves [V1, V2 and V3] are kept closed. Then V2 is opened and bulb 1 is filled
with pure A at a pressure P. After that V3 is opened and bulb 2 is filled with pure
B at the same pressure P. At steady state
aN A 
aDAB ( p A1  p A2 )
 aN B
RTl
(2.37)
where, a is cross sectional area of the connecting tube of length l. If pA1 and pA2
are partial pressures of A in two bulbs at any time,

V 1 dp A1
 aN A
RT dt
and
V 2 dp A2
 aN A
RT dt
(2.38)
(2.39)
From Equations (2.38) and (2.39) we have

d ( p A1  p A2 )
1 
 1
 aN A RT  

dt
 V1 V 2 
(2.40)

d ( p A1  p A2 ) aDAB ( p A1  p A2 )  1
1 

 

dt
l
 V1 V 2 
(2.41)
Boundary conditions:
t=0; (pA1-pA2) = (P-0)=P
t=t/, (pA1-pA2) = ( p A/ 1  p A/ 2 )
Joint initiative of IITs and IISc – Funded by MHRD
Page 2 of 5
NPTEL – Chemical – Mass Transfer Operation 1
applying the above boundary conditions, Equation (2.41) is integrated to obtain
the expression of DAB as follows:
ln
P
aDAB  1
1 /

 
t
/
( p  p A2 )
l  V1 V 2 
(2.42)
/
A1
V2
V3
1
t=0, pA=P, pB=0
t=t/ , p A  p A/ 1 ,
2
V1
t=0, pA=0, pB=P
t=t/ , p A  p A/ 2 ,
pB  pB/ 1
pB  pB/ 2
p A/ 1  pB/ 1  P
p A/ 2  pB/ 2  P
l
Figure 2.6: A schematic of the twin bulb apparatus
(b) Stefan tube method
Stefan tube consists of a T tube made of glass, placed in a constant temperature
water bath. Air pump is used to supply the air, passed through the T tube as
shown in Figure 2.7. Volatile component is filled in the T tube and air passed
over it by the pump and change in the level is observed by the sliding
microscope. Let, at any time t, partial pressure of A at the Z distance from the top
of the vertical tube is pA1 and that at the top it is pA2. The diffusional flux of A is
given as:
NA 
DAB P ( p A1  p A2 )
RT Z
pB , lm
(2.43)
The rate of evaporation is given by
NA 
 A dZ
M A dt

DAB P ( p A1  p A2 )
RT Z
pB , lm
(2.44)
Boundary conditions:
t=0; Z=Z0
Joint initiative of IITs and IISc – Funded by MHRD
Page 3 of 5
NPTEL – Chemical – Mass Transfer Operation 1
t=t/; Z=Z/
Integration of Equation (2.44) using the above boundary conditions gives,
RTpB , lm ( Z /  Z 02 )
2
DAB 
(2.45)
2 PM A ( p A1  p A2 )t /
where, partial pressure of a at liquid surface, pA1 is equal to vapor pressure at the
same temperature. The partial pressure of A at the top of the vertical tube, pA2 is
zero due to high flow rate of B.
Gas B
A+ B
Z
pA2=0
pA1=vapor pressure of liquid A
dZ
Volatile liquid A
Figure 2.7: A representation of the Stafan tube
(c)Predictisve Equations:
(I) Empirical: Fuller, Schettler and Giddings
D AB 
 1
1 



2
1
M
MB 
 ( ) B3   A

1.0133  10 7 T 1.75
1
P ( ) A3

1
2
(2.46)
where,
T is temperature in K
MA, MB are molecular weights of A and B
P is total pressure in bar
νA, νA are atomic diffusion volume in m3.
(II) Theoretical: Chapman-Enskog Equation
Joint initiative of IITs and IISc – Funded by MHRD
Page 4 of 5
NPTEL – Chemical – Mass Transfer Operation 1
DAB
1.858  107 T 1.5

2
P AB
D
 1
1 
M  M 
B
 A
1
2
(2.47)
where, σAB is characteristic length parameter of binary mixture in Å, ΩD is
collision integral=f(kT/εAB)
 AB 
( A   B )
and εAB = ( εA +εB)0.5
2
Joint initiative of IITs and IISc – Funded by MHRD
(2.48)
Page 5 of 5