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Atkins & de Paula: Atkins’ Physical Chemistry 9e Chapter 20: Molecules in Motion.

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Presentation on theme: "Atkins & de Paula: Atkins’ Physical Chemistry 9e Chapter 20: Molecules in Motion."— Presentation transcript:

1 Atkins & de Paula: Atkins’ Physical Chemistry 9e Chapter 20: Molecules in Motion

2  transport property, the ability of a substance to transfer matter, energy, or some other property from one place to another.  diffusion, the migration of matter down a concentration gradient.  thermal conduction, the migration of energy down a temperature gradient.  electric conduction, the migration of electric charge along an electrical potential gradient.  viscosity, the migration of linear momentum down a velocity gradient.  effusion, the emergence of a gas from a container through a small hole. MOLECULAR MOTION IN GASES 20.1 The kinetic model of gases  kinetic model, a model of a gas in which the only contribution to the energy is from the kinetic energies of the molecules.  three assumption of kinetic model, The gas consists of molecules of mass m in ceaseless random motion. The size of the molecules is negligible; d << λ Elastic collision, a collision in which the total translational kinetic energy of the molecules is conserved.

3 Chapter 20: Molecules in Motion 20.1(a) Pressure and molecular speeds  pressure of a gas,  root mean square speed, the square root of the mean of the squares of the speeds: c =  v 2  1/2 = (3RT/M) 1/2. # of molecules Momentum change,

4 Chapter 20: Molecules in Motion  distribution of speeds, the function f(v) which, through f(v)dv, gives the fraction of molecules that have speeds in the range v to v + dv.  Maxwell distribution of speeds,

5 Chapter 20: Molecules in Motion

6  mean speed, 475 ms -1 for N 2 in air and 25 o C.  most probable speed,  relative mean speed,

7 Chapter 20: Molecules in Motion 20.1(b) The collision frequency  collision diameter, the distance of approach corresponding to a collision.  collision frequency, z, the number of collisions made by a molecule in an interval divided by the length of the interval; ~5×10 9 s -1 for N 2 at 1 atm and 25 o C.  collision cross-section, σ, σ = πd (c) The mean free path  mean free path, λ, the average distance a molecule travels between collisions; ~70 nm for N 2 at 1 atm=10 3 molecular diameter. σ = πd 2 λ

8 Chapter 20: Molecules in Motion 20.2 Collisions with walls and surfaces  collision flux, Z W, the number of collisions with an area in a given time interval divided by the area and the duration of the interval, ~3×10 23 cm -2 s -1 for O 2 at 1 bar and 300 K.  collision frequency, the collision flux multiplied by the area of the region of interest. # of molecules = N ×volume = # of collisions

9 Chapter 20: Molecules in Motion 20.3 The rate of effusion  effusion, the emergence of a gas from a container through a small hole.  Graham’s law of effusion: the rate of effusion is inversely proportional to the square root of the molar mass.  Knudsen method, a method for the determination of the vapour pressures of liquids and solids. See Example 20.2 Effusion of a gas

10 Chapter 20: Molecules in Motion 20.4 Transport properties of a perfect gas  flux, the quantity of a property passing through a given area in a given time interval divided by the area and the duration of the interval.  matter flux, the flux of matter, J(matter)  d N /dz [m -2 s -1 ].  energy flux, the flux of energy, J(energy)  dT/dz [Jm -2 s -1 ].  Fick’s first law of diffusion: the flux of matter is proportional to the concentration gradient, J(matter) = –Dd N /dz; D: diffusion coefficient. See Further information 20.1  coefficient of thermal conductivity, κ, the coefficient κ in J(energy) = –κdT/dz.

11 Chapter 20: Molecules in Motion  momentum flux, J(momentum)  dv/dz.  Newtonian (laminar) flow, flow that occurs by a series of layers moving past one another.  coefficient of viscosity, η, the coefficient η in J(momentum) = –ηdv x /dz. See Further information 20.1

12 Chapter 20: Molecules in Motion  D; λ  1/p  D  1/p,  T  D  T, λ  1/σ  D  1/molecular dimension  κ; λ  1/p, [A]  p  κ is independent on p, κ  C V,m  η; λ  1/p, [A]  p  η is independent on p,  T  η  T

13 Chapter 20: Molecules in Motion MOLECULAR MOTION IN LIQUIDS 20.5 Experimental results  NMR, EPR, inelastic neutron scattering, viscosity measurements, study on the molecular motion in liquids.  viscosity measurements, η  e E a /RT (mobility of the particles  e -E a /RT )

14 Chapter 20: Molecules in Motion 20.6 The conductivities of electrolyte solutions  conductance, G, the inverse of resistance; [G]=Ω -1 or S.  conductivity, the constant κ in G = κA/l; [κ]=Sm -1.  molar conductivity, Λ m = κ/c.  strong electrolyte, an electrolyte with a molar conductivity that varies only slightly with concentration.  weak electrolyte, an electrolyte with a molar conductivity that is normal at concentrations close to zero, but falls sharply to low values as the concentration increases.  Kohlrausch’s law, for the concentration dependence of the molar conductivity of a strong electrolyte, Λ m = Λ m  – K c 1/2.  limiting molar conductivity, Λ m , the molar conductivity at zero concentration.  law of the independent migration of ions, Λ m  = v + λ + + v – λ – ; λ + and λ – are the limiting molar conductivity of cations and anions, respectively, v + and v – are the numbers of cations and anions per formula unit of electrolyte (v + = v – = 1 for HCl, CuSO 4, v + = 1 and v – = 2 for MgCl 2 ).

15 Chapter 20: Molecules in Motion 20.7 The mobilities of ions 20.7(a) The drift speed  drift speed, s, the terminal speed when an accelerating force is balanced by the viscous drag.  mobility of an ion, the coefficient u in the expression s = u E ; u = ze/6πηa.  hydrodynamic radius (Stokes radius), the effective radius of a particle in solution.  Grotthuss mechanism, a mechanism for the conduction of protons in solution in which neighbouring H 2 O molecules transfer a proton. Grotthuss mechanism; high u of H + Table ps

16 Chapter 20: Molecules in Motion 20.7(b) Mobility and conductivity  ionic conductivity, the contribution of ions of one type to the molar conductivity: λ = zuF.  Kohlrausch’s law, Λ m = Λ m  – K c 1/2 ion–ion interactions

17 Chapter 20: Molecules in Motion 20.7(c) ion–ion interactions  relaxation effect, the reduction of an ion’s mobility due to distortion of the ionic atmosphere.  electrophoretic effect, the enhanced viscous drag due to the counter current of oppositely charged ions.  Debye–Hückel–Onsager theory, a theory of the concentration dependence of the molar conductivity of a strong electrolyte, K = A + BΛ m . retardation of an ion’s mobility No E E Λ m = Λ m  – K c 1/2

18 Chapter 20: Molecules in Motion I20.2 Ion channel  passive transport, the tendency for a species to move spontaneously down a concentration or potential gradient.  active transport, transport that must be driven by an exergonic process.  channel former, a protein that creates a hydrophilic pore in a membrane.  ion channel, a protein that effects the movement of a specific ion down a potential gradient.  ion pump, proteins that effect the active transport of ions.  patch clamp technique, for studying ion transport across biological membranes. patch clamp technique K + channel

19 Chapter 20: Molecules in Motion DIFFUSION 20.8 The thermodynamic view The diffusion equation  thermodynamic force, dw = dμ = (  μ/  x) p,T dx, dw = - F dx  F = –(  μ/  x) p,T.  Fick’s first law of diffusion μ = μ o +RTlna  F =-RT(  lna/  x) p,T, for ideal solution F =-RT/c(  c/  x) p,T  Einstein relation, D = uRT/zF. J=-Ddc/dx, J=sc  sc=-Ddc/dx  s=-D/c dc/dx=D F /RT For electrolyte solutions; s=u E, F= N A ez E= zF E  u E= zF E D/RT  D = uRT/zF  Stokes–Einstein equation, u=ez/f  D = kT/f = kT/6πηa. No charge term!!  It can apply to neutral molecules in solution. No charge term!!  It can apply to neutral molecules in solution.

20 Chapter 20: Molecules in Motion 20.9 The diffusion equation  diffusion equation (Fick’s second law of diffusion), the relation between the rate of change of concentration at a point and the spatial variation of the concentration at that point:  c/  t = D  2 c/  x 2. Nature abhors a wrinkle!!

21 Chapter 20: Molecules in Motion 20.9(a) Diffusion with convection  convection, the transport of particles arising from the motion of a streaming fluid.  convective flux, the amount of substance passing through an area in a given interval by convection divided by the area and the length of the interval; J = cv.  generalized diffusion equation, the diffusion equation including convection 20.9(b) Solutions of the diffusion equation

22 Chapter 20: Molecules in Motion Diffusion probabilities  average distance travelled in diffusion,  x  = 2(Dt/π) 1/2.  root mean square distance travelled in diffusion,  x 2  1/2 = (2Dt) 1/2. dx Adx dx # of particles=cAN A dx Probability that any of the N 0 =n 0 N A particle in the slab =cAN A dx/N 0 Diffusion is a very slow process!

23 Chapter 20: Molecules in Motion Impact on Nanotechnology; DLA  diffusion limited aggregation (DLA)  Formation of nanoporous membranes through DLA process S. W. Han et al., J. Mater. Chem. 2008, 18, 2208.


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