Lecture 6 Molecular motion and Transport p roperties properties

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1 Lecture 6 Molecular motion and Transport properties

2 Molecular motion The Aim: Describe the migration of properties through the matter using simple random motion picture Within this lecture: Transport properties of a substance: ability of transferring matter, energy or other property from one place to another Basis for description of the main transport properties: Diffusion: migration of matter down a concentration gradient Thermal conduction: migration of energy down a temperature t gradient Electric conduction: migration of charge along electric potential Viscosity: migration of linear momentum down a velocity gradient

matter flux, energy e flux Flux of property is usually proportional to the first derivative of some other related property (from

3 Transport properties of a perfect gas Rate of migration of a property is measured by its flux J (quantity of property passing through a unit area per unit time), e.g. matter flux, energy e flux Flux of property is usually proportional to the first derivative of some other related property (from experimental observation), e.g. matter flux~dn/dz (N - number density of particles), energy flux ~dt/dz Fick s first law of diffusion: dn J ( matter ) = D dz Diffusion coefficient, m 2 /s Jenergy ( ) = dt κ dz Coefficient of thermal conductivity

4 Transport properties of a perfect gas Viscosity of fluids (arises due to transfer of momentum) dv J (x-component of momentum) = η dz coefficient of viscosity x

5 Transport properties of a perfect gas Transport properties of gases can be fairly accurately predicted with the kinetic theory gases. Kinetic model assumes that the only contribution to gas energy is a kinetic energy Three main assumptions: Gas consist of molecules of mass m Size of molecules is negligible (d << distances) Molecules interact through elastic collisions (e.g. kinetic energy is conserved at every collision)

6 Transport properties of a perfect gas Application of kinetic theory of gases where λ is a mean free path and c 8RT = vf () v dv = π M is a mean velocity

7 Transport properties of a perfect gas The diffusion coefficient: D 1 = λc 3 1. Mean free path is decreasing when pressure increases, so D decreases with increasing pressure 2. Mean velocity increasing with T, so does the D. 3. Mean free path increases when the collision cross section decreases, so D is larger for small molecules 1 The thermal conductivity: κ = λccvm, [ A] 3 1. Mean free path is decreasing when pressure increases, and therefore inversely proportional to the concentration, so κ is independent on pressure 2. Thermal conductivity is greater for gases with high heat capacity. 1 The viscosity: it η = [ ] 3 M λc A 1. The viscosity is independent on pressure ½ ½ 2. Because the viscosity of gas is increasing with T c T, η T

8 Molecular motion in liquids Experimentally measured in inelastic neutron scattering, as a relaxation time in e.g. NMR and through viscosity Activation origin for viscosity in liquids (i.e. a molecule needs to escape from its neighbors). Thus, it should be inversely proportional to the probability of escape: η exp( E / RT) A Viscosity drops sharply with temperature

9 Conductivity of electrolyte solution Conductance, [S]: G = κ A l conductivity Conductivity depends on a number of ions present: Molar conductivity, typically 10mS m 2 /mol Λ = m κ c concentration The concentration dependence of conductance indicates that there are 2 classes of electrolyte Strong electrolyte: l t molar conductivity it depends d slightly on the molar concentration Weak electrolyte: molar concentration falls sharply as the concentration increases

10 Conductivity of electrolyte solution Strong electrolytes fully ionized in the solution Kohlrausch s law 0 1/2 Λ m =Λm Kc Limiting iti molar conductivity it Law of the independent migration of ions: limiting molar conductivity can be expressed as a sum of ions contribution 0 Λ m = ν + λ+ + ν λ i.e. ions migrate independently in the zero concentration limit For example, limiting molar conductivity of BaCl 2 in water will be: *7.63=27.98 ms m 2 mol -1.

11 Conductivity of electrolyte solution Weak electrolytes: not fully ionized in the solution + HA( aq) + H O H O ( aq) + A ( aq) K = 2 3 If we define degree of deprotonation: HO + 3 = αc A = αc HA = (1 α ) c [ ] a a H O 3 a a + HA A K a m 2 1/2 α c K 4 = 1 1 a c α = + 1 α 2c K a Λ = αλ 0 m Ostwald s dilution law 1 1 αc K α = + 0 ( 0 ) 2 m m K a 1 1 cλm Λ = Λ + Λ a m

12 Mobilities of ions We need to know how ions move to understand e.g. Kohlrausch law Let s consider uniform electric field between the electrodes in solution: Δφ E = l zeδ φ F = zee = l Ffric 6 πη rs = (for ion with radius r and velocity v) s = zee 6πηr s = μe μ ze 6πη r = hydrodynamic radius, might be different from the ionic radius, small ions are more solvated than the bulk ones Example: Cs + ion: z=1, r=170pm, η=1*10-3 kg m -1 s -1. Than μ=5*10-8 m 2 /Vs, i.e. with 1V applied across 1cm the drift speed is 5um/s.

Mobilities of ions Mobility and conductivity λ = zμf F = N e

electrolyte: sδta ν cn A Jions ( ) = = scn ν A AΔt J ( charge )

zμνcfa l Δφ Δφ I G φ κa R l = = Δ = κ = zμν cf In solution Λ = +

13 Mobilities of ions Mobility and conductivity λ = zμf F = N e Faraday constant, A Let s consider a fully dissociated electrolyte: sδta ν cn A Jions ( ) = = scn ν A AΔt J ( charge ) = ze s ν cn A = zs ν cf = z μ E ν cf Δφ I = J A = zμeνcfa = zμνcfa l Δφ Δφ I G φ κa R l = = Δ = κ = zμν cf In solution Λ = + 0 ( ) m z+ μν + + z μν F Example: if μ=5*10-8 m 2 /Vs and z=1, Λ=10mS m 2 mol -1.

14 Mobilities of ions Transport numbers fraction of total current carried by ions of specific type I± t =, t = t + t I ± + Limiting iti transport t numbers defined d in the limit it of zero concentration ti of electrolyte solution t 0 z± μ± ν ± ± = z+ μν z μν as z ν = zν t = ± μ± μ + μ + or, using λ=zμf 0 ν λ ν λ t± = = ν λ ± ± ± ± ν λ Λ m as we can measure transport number of individual ions, the ionic conductivity and ionic mobility can be determined as well

15 Moving boundary technique Measuring transport numbers Moving boundary technique zclaf t = + + IΔt leading solution (less dense + higher mobility) indicator solution (denser + low mobility)

Conductivities and ion-ion interactions Reduction

atmosphere is called relaxation effect Enhanced

opposite direction is called electrophoretic

atmosphere Th t ff t t k i t t l d t ½ Those two

16 Conductivities and ion-ion interactions Reduction of ion mobility due to retardation of ionic atmosphere is called relaxation effect Enhanced viscous drag due to ionic atmosphere moving in opposite direction is called electrophoretic effect still ion moving ion incompletely formed atmosphere Th t ff t t k i t t l d t ½ Those two effect taken into account lead to c ½ dependence in Kolrausch law

and close in response to some stimuli (e.

17 Ion channels and Ion pumps passive ion transport: ions transport t down the concentration ti and membrane potential gradient active transport: driven by an exergonic process transport mechanisms: by carrier molecule through a channel former Channel formers: ion channels: highly selective proteins that can open and close in response to some stimuli (e.g. potential built-up or effector molecule) ion pumps: proteins actively transporting particular ions pipette size: 1um typ current: 1pA

18 Ion channels and Ion pumps K + ion channel: 1. K + ion stripped of the hydration shell 2. gripped by carbonyl group in the selectivity filter region 3. K + ion is driven out by electrostatic interaction with the next ion selectivity filter H + ATP i th t l H + -ATPase ion pump that couples proton flow to synthesis of ATP

19 Diffusion Thermodynamic force If the chemical potential depends on position, the maximum nonexpansion work μ dw = dμ = dx x pt, Comparing with dw = Fdx w = Δμ μ μ + Δμ F μ = x pt, Thermodynamic force

20 Diffusion Thermodynamic force of concentration gradient μ = μ0 + RTln a F For ideal solution ln a RT c = RT F = x c x p, pt, p T Fick s law of diffusion: Particles flux: J ~ drift velocity ~ F ~ c xx

21 The Einstein relation Diffusion c s Δ J = D J = Atc = sc x AΔt sc c c = D x s D c DF = = c x RT We know drift speed vs force relation for ion mobility, so we can deduce diffusion constant s = μe = DzeN AE RT D = μrt z F μ = 510 i m / sv we find = 110 i m / For example: for 510 / D m s

22 Diffusion The Nernst-Einstein equation 2 2 zdf Molar conductivity it of ions in the solution λ = zμf= RT F + z + D+ ν z D Λ m = ( ν z D + z D ) RT The Stokes-Einstein equation Frictional force μe= E eze f D = μrt zf D zert zekn AT kt = = = fzf fzen f A Using Stokes s s law D kt 6πηa = No charge involved -> applicable to all molecules

Diffusion equation How concentration distribution evolves

J Adt J J = = t Aldt Aldt l c J J = cv cv = c c+ l v= vl

23 Diffusion equation How concentration distribution evolves with time due to diffusion c JAdt J Adt J J = = t Aldt Aldt l J J = D + D = D + D c+ l = Dl 2 x x x x x x 2 c c = D 2 t x 2 c c c c c Diffusion with convection c c JAdt J Adt J J = = t Aldt Aldt l c J J = cv cv = c c+ l v= vl x c c = v due to convection only t x c x 2 c c c = D v 2 t x x

24 Solution of diffusion equation 2 c c c = D v 2 t x x 2 nd order differential equation: two boundary condition are required for spatial dependence and single for time dependence Example 1: one surface of a container (x=0) is coated with N 0 molecules at time t=0 cxt (, ) = n 0 A( π Dt) 1/2 e x 2 /4 Dt Example 2: dissolution infinitely small solid containing N 0 molecules at time t=0 in 3D solvent n0 crt (,) = 8( π Dt) 3/2 e r 2 /4Dt

25 Diffusion probabilities Probability to find a particle at a given slab of thickness dx is proportional to the concentration there: The mean distance traveled by the particles: p( x) = c( x) AN dx/ N A 0 cxandx ( ) A 1 x /4Dt x xe dx 1/2 N 0 0 ( πdt) 0 2 Dt = = = 2 π 1/2 x 2 1/2 1/2 = (2 Dt ) Diffusion in unstirred solution, D=5x10-10 m 2 /S

26 Random Walk Apparently diffusion can be modeled as a random walk, where particle is jumping distance λ in a time τ. Direction of the jump is chosen randomly One dimensional walk: p 1/2 2τ 2 2 x τ /2tλ = πt Comparing with the solution of diffusion i equation e D λ 2 = Einstein-Smoluchowski equation 2τ Connection between microscopic and macroscopic parameters.

27 Problems Atkins 21.24a What fraction of the total current is carried by Li+ ions when current flows through h an aqueous solution of LBr at 25 C. Atkins 21.25a25a The limiting molar conductivities of KCl, KNO 3, and AgNO 3 are ms m 2 mol -1, ms m 2 mol -1, and ms m 2 mol -1, respectively (all at 25 C). What is the limiting iti molar conductivity it of AgCl at tthis temperature? t Atkins P The diffusion coefficient of particular RNA molecule is 1.0x10-11 m 2 /s. Estimate time required for a molecule to diffuse 1 um from nucleus to the cell wall

28 Assignment VI P21.8 Conductivities are often measured by comparing the resistance of a cell filled with some standard solutions. The conductivity of water is 76 ms/m at 25ºC and 0.1M KCl(aq) is S/m. A cell has resistance of Ohm when filled with 0.1M KCl(aq) and Ohm when filled with 0.1M CH 3 COOH. What is the molar conductivity of the acetic acid at that concentration and temperature P21.16 Estimate the diffusion coefficient and the effective hydrodynamic radii of the alkali metal cations at 25ºC. Estimate the approximate number of water molecules dragged along by the cations. Take ionic i radii from the Table 20.3

CHAPTER 19 Molecules in Motion. 1. Matter transport is driven by concentration gradient.

CHAPTER 19 Molecules in Motion I. Transport in Gases. A. Fick s Law of Diffusion. 1. Matter transport is driven by concentration gradient. Flux (matter) = J m = D dn dz N = number density of particles