Molecular motion in liquids

21.5 Experimental results

• Measuring techniques: NMR, ESR, inelastic neutron scattering, etc.

• Big molecules in viscous fluids typically rotate in a series of small (5oC) steps.

• Small molecules in nonviscous fluid typically jump through about 1 radian (57oC).

• For a molecule to move in liquid, it must acquire at least a minimum energy to escape from its neighbors.

• The probability that a molecule has at least an energy Ea is proportional to e-Ea/RT.

• Viscosity, η, is inversely proportional to the mobility of the particles, η∞ eEa/RT

Temperature dependence of the viscosity of water

24.6 The conductivities of electrolyte solutions

• Conductance (G, siemens) of a solution sample decreases with its length l and increases with its cross-sectional area A:

k is the conductivity (Sm-1).

• Molar conductivity, Λm, is defined as:

c is the molar concentration

• Λm varies with the concentration due to two reasons:

• Based on the concentration dependence of molar conductivities, electrolytes can be classified into two categories:

1. Strong electrolyte: its molar conductivity depends only slightly on the molar concentration.

2. Weak electrolyte: its molar conductivity is normal at diluted environment, but falls sharply as the concentration increases.

l

kAG

c

km

Strong electrolyte

• Strong electrolyte is virtually fully ionized in solution, such as ionic solid, strong acids and bases.

• According to Kohlrausch’s law, the molar conductivity of strong electrolyte varies linearly with the square root of the concentration:

• Λ0m can be expressed as the sum of contributions from its individual ions:

where v+ and v- are the numbers of cations and anions per formula unit. (For example: HCl: v+ = 1 and v- = 1; MgCl2, v+ = 1 and v- = 2)

210 /cmm

vvm0

Weak electrolyte• Weak electrolytes are not fully ionized in solution, such as weak

acids and bases.

• Degree of ionization (α): defined as the ratio of the amount of ions being formed in the solution and the amount of electrolyte added to the solution.

• For the acid HA at a molar concentration c,

[H3O+] = αc, [A-] = αc , [HA] = c –αc

• Since only fraction, α, of electrolyte is actually presents as ions, the measure conductivity Λm, is given by:

Λm = αΛ0m

14

12

21 /

a

a

K

c

c

Ka

Ostwald’s dilution law

200

11

ma

m

mm K

c

24.7 The mobility of ions

• Drift speed (s): the terminal speed reached when the accelerating force is balanced by the viscous drag.

• Accelerating force induced by a uniform electric field (E = Δø/l): F = z e E = z e Δø/l

• Friction force (Stokes formula) Ffric = (6πηa)s, a is the hydrodynamic radius

• Mobility of an ion:

• u is called the mobility of the ion

uEsora

zeEs

6

a

zeu

6

Mobility and conductivity

• λ = z u F ( λ is an ion’s molar conductivity)

• For the solution:

Λ0m = (z+u+v+ + z-u-v-) F

Transport numbers

• The fraction of total current carried by the ions of a specified type.

• The limiting transport number, t0±, is defined for the limit of zero

concentration of the electrolyte solution.

I

It

I

It

uvzuvz

uvzt0

vv

vt 0

The measurement of transport numbers

• Moving boundary method

• Indicator solution

• Leading solution

tI

clAFzt

Conductivities and ion-ion interactions

• To explain the c1/2 dependence in the Kohlrausch law.

Hückel-Onsager Theory

21.9 The thermodynamic view of diffusion

• The maximum amount of Non-Expansion work can be done when moving a substance from local x to x+dx is:

• When expressed with an opposite force:

dw = - F dx

Then one gets:

Therefore: The slope of the chemical potential can be interpreted as an effect force, thermodynamic force. This force represents the spontaneous tendency of the molecules to disperse.

dxx

ddwTp,

TpxF

,

• Since μ = μө + RTlnα

• One get

• Using concentrations to replace the activity:

TpTp x

aRT

x

aRTuF

,,

ln})ln(

{

Connections between the thermodynamic force and the

concentration gradient

Tpx

c

c

RTF

,

Fick’s first law of diffusion revisit

• Fick’s law of diffusion discussed earlier was developed from the kinetic theory of gases.

• The flux of diffusing particles is due to a thermodynamic force arising from concentration gradient (i.e. the thermodynamic force is proportional to the concentration gradient).

• The drift speed is proportional to the thermodynamic force.

• The particle flux, J, is proportional to the drift speed.

• The chain of proportionalities (J ~ s, s ~ F, F ~ dc/dx) implies that J is proportional to concentration gradient.

The Einstein relation

• The flux is related to the drift speed by J = sc

• Comparing the above equation with the Fick’s law, one gets sc = -D (dc/dx)

• Express dc/dx in terms of F, one gets s = (DF)/(RT)

• The drift speed of an ion equals s = u E

• Therefore, u E = (DF)/(RT) = (zFED)/(RT)

• Reorganizing the above equation to D = (uRT)/(zF) (Einstein relation between the diffusion coefficient and the ionic mobility, F is

the Faraday constant)

The Nernst – Einstein Equation

• Provides a link between the molar conductivity of an electrolyte and the diffusion coefficients.

• Can be applied to determine the ionic diffusion coefficients from conductivity measurement.

• For each type of ionλ = zuF = (z2DF2)/(RT)

• For electrolyte

Λm = (v+Z+2D+ + v-Z-

2D-)F2/(RT)