Atkins & de Paula: Atkins’ Physical Chemistry 9e

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

2. Chapter 20: Molecules in Motion • 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 = v21/2 = (3RT/M)1/2. Momentum change, # of molecules

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. Chapter 20: Molecules in Motion • mean speed,475 ms-1 for N2 in air and 25oC. • 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; ~5×109 s-1 for N2 at 1 atm and 25oC. • collision cross-section, σ, σ = πd2 . • 20.1(c) The mean free path • mean free path, λ, the average distance a molecule travels between collisions; ~70 nm for N2 at 1 atm=103 molecular diameter. λ σ = πd2

8. Chapter 20: Molecules in Motion 20.2 Collisions with walls and surfaces collision flux, ZW, the number of collisions with an area in a given time interval divided by the area and the duration of the interval, ~3×1023 cm-2 s-1 for O2 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) dN/dz [m-2s-1]. • energy flux, the flux of energy, J(energy) dT/dz [Jm-2s-1]. • Fick’s first law of diffusion: the flux of matter is proportional to the concentration gradient, J(matter) = –DdN/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) = –ηdvx/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 CV,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eEa/RT (mobility of the particles  e-Ea/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, Λm = 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, Λm = Λm – Kc1/2. • limiting molar conductivity, Λm, the molar conductivity at zero concentration. • law of the independent migration of ions, Λm = 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, CuSO4, v+ = 1 and v– = 2 for MgCl2).

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 = uE; 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 H2O molecules transfer a proton. Table 20.5 1.5 ps Grotthuss mechanism; high u of H+

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, Λm = Λm –Kc1/2 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 + BΛm. No E Λm = Λm – Kc1/2 retardation of an ion’s mobility E

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. K+ channel patch clamp technique

19. Chapter 20: Molecules in Motion • DIFFUSION • 20.8 The thermodynamic view The diffusion equation • thermodynamic force, dw =dμ = (μ/x)p,Tdx, dw = -Fdx F = –(μ/x)p,T. • Fick’s first law of diffusion • μ = μo+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=DF /RT • For electrolyte solutions; s=uE, F=NAezE= zFE  uE= zFE 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.

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2c/x2. 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 20.10 Diffusion probabilities average distance travelled in diffusion, x = 2(Dt/π)1/2. root mean square distance travelled in diffusion, x21/2 = (2Dt)1/2. # of particles=cANAdx Probability that any of the N0=n0NAparticle in the slab =cANAdx/N0 Adx dx dx 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.