ChE 505 – Chapter 7N: Appendix Updated 02-08/05 Chapter 7: Appendix Diffusion: A General Treatment of Molecules and Ions We have seen that diffusion may be regarded as the net movement of solutes in solution (or in gas phase) due to a concentration gradient. We have also seen that ions move due to potential gradients. Hence, the unified treatment asserts that solutes in solution will move with the velocity proportional to the electrochemical potential gradient. This can be expressed as follows:

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) ( φµ ∇+∇−= ezuv iiii (1) (ionic species i velocity) = (ionic mobility) (chemical forces + electrical forces) where iµ is the molecular chemical potential of i ( Gibbs free energy of i not accounting for the electrostatic contribution is aii NG µ= ), z i is the charge number on the ion, e is the Faraday constant (or charge on electrons) and φ is the electrostatic potential. u i - ion mobility is the physical property of the ion which must be measured. Often it is taken to be given by Einstein's equation µµπ where6/1 ir is solvent viscosity and r i is the effective ion radius (which we still do not know for most of the situations). Hence, u i is a parameter to be determined from experiments. The c g s. units on u i are (s/g) Recall that the chemical potential of species i (2) iB

oii anTk l+=µµ

so that iBi anTk l∇=∇µ (3)

where k B is the Boltzman constant. We will assume an ideal solution, then ⎟⎟⎠

⎞⎜⎜⎝

⎛=

MCa i

i 1 so that

ii

Bi C

CTk∇=∇µ (4)

where C i is the concentration of ionic species i (in mole i per unit volume). The molar flux of species i is given by J i (in mol per unit area and second) (5) iii CvJ = Substitution of eq (4) and eq (1) into eq (5) yields upon reorganization:

[ ] ⎟⎟⎠

⎞⎜⎜⎝

⎛∇+∇−= φ

TkCeZCTkuJ

B

iiiBii (6)

By comparison with the situation when 0=∇φ we recognize that diffusivity D i is given by

(7) TkuD Bii= If we take k B = 1.381 x 10-16 erg K-1, T in (oK), and u i (s/g) we have D i (cm2/s). If we take k B = 1.381 x 10-23 JK-1, we must take u i (s/kg) to get D i (m2/s). It is customary to rewrite eq (6) as

⎟⎠⎞

⎜⎝⎛ ∇+∇−= φi

iiii C

RTFZCDJ (8)

where F is the Faraday constant (F = 9.6485 x 104 C mol-1), R is the ideal gas constant. This is often called the Nernst-Planck equation. Strong electrolytes of charges Z 1 and Z2 Since the condition of electro-neutrality has to be satisfied it can be shown that eq (8) can be rewritten for strong electrolytes as (9) TT CDJ ∇−= where

1

2

2

1

ZJ

ZJ

JT == and C T is the equivalent concentration

1

2

2

1

ZC

ZC

CT == (10)

and D is the composite diffusivity

2

1

1

2

21

DZ

DZ

ZZD

+

+= (11)

Weak 1-1 electrolytes These often rapidly dimerize in solution. For example an aqueous-solution of acetic acid will contain hydrated protons [H3O+], acetate ions [CH3COO-], and acetic acid molecules [CH3COOH]. The cations and anions of this weak electrolyte diffuse at the same rate because of electrostatic interactions i.e they act as monomers of the same species. The electrolyte molecules i.e acid molecules, roughly equivalent to a dimer, diffuse at a different rate. As diffusion proceeds, the concentrations change and so do the fraction of monomer and dimer. Hence we have a concentration dependent diffusion process. We assume the total solute concentration expressed in equivalents is: C T = C 1 + 2 C 2 (12) Furthermore the dimerization is much more rapid than diffusion so that

2

(13) 212 CKC =

where K is the equilibrium constant identical with the association constant for a weak electrolyte. Then one can show that

[ ]

TTT

TT CDC

CKCKDD

J ∇−=∇⎟⎟⎠

⎞⎜⎜⎝

⎛

+

−++−=

8118121 (14)

The quantity in parentheses is the apparent diffusivity of the dimerizing solute. At very low C T this is equal to monomer diffusivity D 1. At high C T it equals to the diffusivity of the dimer D 2.. For the values of D/ D 1 see Figure 1. ____________________________________________________________________________ Example: The p K a = 4.756 for acetic acid. The diffusion coefficient for the acid molecules is D2 = 1.80 x 10-5 cm2/s at 25oC. What is D for a 1 M solution? Recall

[ ][ ][ ]HA

AHpKa

−+

−= log

so that [ ]

[ ][ ]apK

AHHA 10=−+

but since [H +] = [A - ] we get 41070.510 xK apK == Due to the high total concentration and the fact that K is so large, equation (14) reveals that D≈D2 so that

MCT 1≈

s

cmxDDCDJ TT

25

22 1080.1 −==∇−=

Micelle Forming Solutions In micelle formation, typical of detergent solutions, n monomers combine to make an n-mer. No other sizes are present. The diffusion coefficient in such detergent solutions represents an average over monomer and micelle present in solution. The mass balance requires (15) mT CnCC += 1

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and rapid equilibration is assumed (16) n

m CKC 1= Now we have (17) mmT CDnCDJ ∇−∇−= 11

In order to get a workable equation one must experimentally determine the critical micelle concentration CCMC at which the properties of the solution change dramatically. Approximately so1CCCMC≈

( )CMCTm CCm

C −=1

and

( ) nCMCT CC

nKC /1

11

−=

Finally, we get

( )

( ) TCMCT

n

mTT CCCn

nKDDCDJ ∇⎥⎦

⎤⎢⎣

⎡

−+−=∇−=

− /11 (18)

This works very well for non-ionic detergents. It does not work for ionic detergents at low ionic strength. Isodesmic Association Here aggregation occurs 1 molecule at a time (19) 11CCKC ii −= where K is equilibrium constant assumed independent of size

( )21

1

1 1 CKCCiC

iiT −==∑

∞

=

(20)

This yields

(21)

( )( ) T

T

T

T

TT C

DDDDCK

DDDCK

DDKCD

CDJ ∇⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

+++−−

+−

+−−

−=∇−=

....)167211256(

92415

44

432133

32122

211

mmT CDnCDJ ∇+∇−= 11

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FIGURE 7A-1: The diffusion coefficient of a dimerizing solute. As a solute dimerizes, its average diffusion coefficient changes from that of the monomer to that of the dimer. The concentration cT at which this occurs is roughly the reciprocal of the association constant K.

TABLE 7-1: Ionic diffusion coefficients in water at 25°C.

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FIGURE 7A-2: Micelle formulation and isodesmic aggregation. In the type of micelle formation discussed here, n monomers combine to form an n-mer. No other sizes are present. In isodesmic association, monomers add with equal facility to monomers or aggregates of any size.

FIGURE 7A-3: Types of solute aggregation. The detergent sodium dodecylsulfate aggregates

abruptly to form micelles, and the dye Orange II has isodesmic aggregates (see Fig. 6.2-4). The bile salt sodium taurodeoxycholate falls between these two limits. These results were obtained using ion-selective electrodes. [Data from Kale et al. (1980).]

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FIGURE 7A.4: Diffusion of the detergent Triton X-100 at 25ºC. The variation with concentration is predicted by Eq. ????. The intercept is the micelle’s diffusion coefficient, and the slope is related to the monomer’s diffusion coefficient. [From Weinheimer et al., (1981), with permission]

.

FIGURE 7A.5: Diffusion of sodium dodecylsufate (SDS) at 25°C. The diffusion coefficients in

this case increase as SDS concentration and solution viscosity rise. This increase is the result of aggregation and electrostatic interaction. [Data from Weinheimer et al. (1981)].

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TABLE 7.1.2: Lennard-Jones potential parameters found from viscosities

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TABLE 7.1.3: The collision integral Ω

TABLE 7.1.4: Atomic diffusion volumes for use in Eq. ???

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TABLE 7.2.1: Diffusion coefficients at infinite dilution in water at 25°C

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TABLE 7.2.2: Diffusion coefficients at infinite dilution in nonaqueous liquids

FIGURE 7A.5: Alternatives to Stokes-Einstein equation for diffusion in liquids a,b

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