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 (5 o C) steps. Small molecules in nonviscous fluid typically jump through about 1 radian (57 o C). 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∞ e Ea/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, Λ m, is defined as: c is the molar concentration Λ m 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.

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: Λ 0 m 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; MgCl 2, v + = 1 and v - = 2)

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, [H 3 O + ] = αc, [A - ] = αc, [HA] = c –αc Since only fraction, α, of electrolyte is actually presents as ions, the measure conductivity Λ m, is given by: Λ m = αΛ 0 m

Ostwald’s dilution law

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) F fric = (6πηa)s, a is the hydrodynamic radius Mobility of an ion: u is called the mobility of the ion

Mobility and conductivity λ = z u F ( λ is an ion’s molar conductivity) For the solution: Λ 0 m = (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, t 0 ±, is defined for the limit of zero concentration of the electrolyte solution.

The measurement of transport numbers Moving boundary method Indicator solution Leading solution

Conductivities and ion-ion interactions To explain the c 1/2 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.

Since μ = μ ө + RTlnα One get Using concentrations to replace the activity: Connections between the thermodynamic force and the concentration gradient

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 ( d c/ d x) Express d c/ d x 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 ion λ = zuF = (z 2 DF 2 )/(RT) For electrolyte Λ m = (v + Z + 2 D + + v - Z - 2 D - )F 2 /(RT)