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Physical Chemistry

Physical Chemistry. Physical Chemistry A-III Contents Chapter 18 Transport Process & Non-equilibrium Thermodynamics Chapter 19 Fundamental of Chemical Dynamics Chapter 20 Chemical Dynamics in Various Reaction Systems Chapter 21 Rate Theory for Elementary Reaction

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Physical Chemistry

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  1. Physical Chemistry Chemistry Department of Fudan University

  2. Physical Chemistry A-III Contents Chapter 18 Transport Process & Non-equilibrium Thermodynamics Chapter 19 Fundamental of Chemical Dynamics Chapter 20 Chemical Dynamics in Various Reaction Systems Chapter 21 Rate Theory for Elementary Reaction Chapter 22 Molecular Reaction Kinetics Chapter 23 Electrolyte Solutions Chapter 24 Electrochemistry Thermodynamics Chapter 25 Electrochemistry Kinetics & its Applications Chemistry Department of Fudan University

  3. Reference Books • 傅献彩, 沈文霞, 姚天扬,《 物理化学》第四版. 北京高等教育出版社,(1990) • 韩德刚,高执棣,高盘良,《物理化学》,高等教育出版社,(2001) • 胡英等编,《物理化学》第四版,高等教育出版社,(2000) • 江元生,《结构化学》,高等教育出版社,(1997) • 徐光宪,王祥云,《物质结构》第二版,高等教育出版社,(1987) • P. W. Atkins et al, Atkins’《Physical Chemistry》, 7th ed., Oxford University Press,(2002) • Berry, Rice, Ross,《Physical Chemistry》, John Wiley &Sons, (1980) • I. N. Levine《Physical Chemistry》,4th ed., McGraw-Hill, (1995) Chemistry Department of Fudan University

  4. Chapter 18 Transport Process & Non-equilibrium Thermodynamics • §18−1 Fundamentals of Transport Process • (1) Thermal Conduction • (2) Viscosity • (3) Diffusion • §18−2 Non-equilibrium Thermodynamics • (1) Principle for Entropy Production (Increase) • (2) Onsager Reciprocal Relations • (3) Principle for Minimum Entropy Production Chemistry Department of Fudan University

  5. The theory to study the rate and mechanism of transport processes is one branch of kinetics, called physical kinetics Including energy transport (thermal conductivity) and mass transport (flow or diffusion) The theory to study the rate and mechanism of chemical reactions is another branch of kinetics, called chemical kinetics Chemistry Department of Fudan University

  6. Thermal Conductivity:If there is a temperature difference between the system & environment or in the system, namely the system departures from heat equilibrium, the conduction of heat will occur Flow:If there is a non-equilibrium force in the system, a part of the system will move, leading to mass transport Diffusion:If there is a concentration difference in the liquid or gas system, the system will departure from mass equilibrium and mass transport will occur Discussion: Thermal conductivity, flow (viscosity ) and diffusion Chemistry Department of Fudan University

  7. §18−1 Fundamentals of Transport Processes (1) Thermal Conduction Experiments show that the rate of heat flow dq/dt normal to any cross section with area ofA is proportional to the temperature gradient dT/dx, thus substance K isthe substance’sthermal conductivity or coefficient of thermal conduction. Its SI unit is J∙K−1∙m−1∙s−1 reservoir reservoir Adiabatic wall Conduction of heat in a substance Fourier’s law for thermal conduction Chemistry Department of Fudan University

  8. This law is also held when the temperature gradient in the substance is not uniform; in this case, has different values at different places on the x axis, and varies from place to place Although the system in the figure in last slide is not in thermodynamic equilibrium, we assume that any sufficiently small portion of the system can be assigned values of thermodynamic variables, such as T, U, S, and P, and that all the thermodynamic relations between these variables are held in each sub-system. This assumption, called the principle of local state or the hypothesis of local equilibrium, is held well in most (but not all) systems of interest Chemistry Department of Fudan University

  9. According to the theory of gas kinetics, for the ideal gas with molar concentration [A] the coefficient of thermal conduction k can beexpressed as  is the mean free path is average velocity of gas molecules is the heat capacity for the system with a constant volume   1/p and  1/[A] At higher pressure,   1/p , k has no relationship with p At lower pressure, k p , has no relationship with p Chemistry Department of Fudan University

  10. Most probable velocity Mean velocity Root mean square velocity Mean free path  Relative mean velocity <vAB> = (8RT/ ) ½  = [(MA + MB)/MAMB] 1/2 Total number of collision between A and B per unit time is ZAB =  dAB2 <vAB> (NB/V) (NA/V) =  dAB2 ( 8 RT/  ) 1/2 N2 [A][B] For like molecules ZAA = (0.5) 1/2  dA2 (8 RT/MA)1/2N2 [A]2/2 NA <vA> = 2ZAAV = V/ (0.5) 1/2  dA2NA Chemistry Department of Fudan University

  11. (2) Viscosity Viscosityis the property to characterize a fluid’s resistance to flow Let Fy be the frictional force exerted by the slower-moving fluid on one side of the surface (side 1 in the figure) on the faster-moving fluid (side 2). Experiments on fluid flow show that Fy is proportional to the surface area of contact and to the gradient dvy/dxof flowspeed. The proportionality constant is the fluid’s viscosity  plane surface area A The unit for viscosity  iskg∙m−1∙s−1 this is Newton’s law of viscosity A fluid flowing between two planar plates Chemistry Department of Fudan University

  12. The flow rate is not high Obey Newton’s law of viscosity Laminar flow Turbulent flow High flow rates Does not obey Newton’s law of viscosity Newtonian fluid is one for which  is independent of Gases and most pure non-polymeric liquids are Newtonian For a non-Newtonian fluid,  changes as changes Polymer solution, liquid polymers, and colloidal suspensions are non-Newtonian Chemistry Department of Fudan University

  13. Since Newton’s law of viscosity can be written as where is the changing rate of the y direction component of the momentum against time caused by the interaction of the fluid on one side of a surface with the fluid on the other side The molecular explanation of viscosity is that it is due to the transport of momentum across planes perpendicular to the x axis The momentum transfer between layers occurs mainly by collisions between molecules in adjacent layers (for most liquids) or by transfer of molecules between layers (for most gases) Chemistry Department of Fudan University

  14. According to the kinetic theory of gas, the viscosity for ideal gas is Since the pressure is inversely proportional to , and is proportional to concentration, the viscosity is independent of pressure. Because of ,so ,namely the viscosity of gases increases with the temperature increment Since A and , we obtain The viscosity of gases is related to molecular weight, diameter and temperature Chemistry Department of Fudan University

  15. (3) Diffusion Removable partition Right figure When the partition is removed, the two phases are in contact, and the random molecular motion of i and j molecules will reduce and ultimately eliminate the concentration difference between the two solutions Area Plane P phase phase constant-temperature bath When the partition is removed, diffusion occurs This spontaneous decrease in concentration difference is diffusion Chemistry Department of Fudan University

  16. Experiments show that the following equations are obeyed indiffusion Above equation isFick’s first law of diffusion In the formula, dni /dtis the net rate of flow of j (in moles per unit time) across a plane P of area A perpendicular to the x axis; dci /dx is the concentration gradient at plane P with respect to the x coordinate; and Dijis called the (mutual) diffusion coefficient, its unit ism2∙s−1 Thediffusion coefficient Dij is a function of the local state of the system and therefore depends on T, P and the local composition of the solution. If solutions A and B mix with no volume change, then Chemistry Department of Fudan University

  17. According to the theory of molecular motion, the motion of a diffusing molecule is random, x, the net displacement in the x direction that occurs in time t, is likely to be positive as negative, so theaverage value x is zero. Therefore we consider the average of the square of the displacement x should not be zero. In 1905, Einstein proved that where D is the diffusion coefficient. The formula is called the Einstein-Smoluchowskiequation. The quantity of the root-mean-square net displacement of a diffusing molecule in the x direction in time t is Taking t to be 60 s and D to be 10-1, 10-5, and 10-20 cm2/s, we find the typical (x )rmsof molecules in 1 min at room temperature and 1 atm to be only 3 cm for gases, 0.3 cm for liquids, and less than 1 Å for solids Chemistry Department of Fudan University

  18. The continuous random motion of the colloidal particles in a suspension caused by the thermal motion of the mediummolecules is called Brownian motion. The particle’s average square displacement in the x direction increases with time according to If the colloidal particles are spheres with radiusr, then Stoke’s law gives and the friction coefficient is , above formula becomes If we assume that the i molecules are spherical with radius ri and assume that Stokes’ law can be applied to the motion of i molecules through the solvent B, then , the diffusion coefficient can be written as for ri > rB, liquid solution; ri = rB,64; ri < rB, 6 less than 4 The above formula is the Stokes-Einstein equation Chemistry Department of Fudan University

  19. 1.Avogadro's number(5pts) Spherical water droplets are dispersed in argon gas. At 27oC, each droplet is 1.0 micrometer in diameter and undergoes collisions with argon. Assume that inter-droplet collisions do not occur. The root-mean-square speed of these droplets was determined to be 0.50 cm/s at 27oC. The density of a water droplet is 1.0 g/cm3. 1-1. Calculate the average kinetic energy (mv2/2) of this droplet at 27oC. The volume of a sphere is given by (4/3) π r3 where r is the radius. If the temperature is changed, then droplet size and speed of the droplet will also change. The average kinetic energy of a droplet between 0oC and 100oC as a function of temperature is found to be linear. Assume that it remains linear below 0oC. At thermal equilibrium, the average kinetic energy is the same irrespective of particle masses (equipartition theorem). The specific heat capacity, at constant volume, of argon (atomic weight, 40) gas is 0.31 J g-1 K-1. 1-2.Calculate Avogadro's number without using the ideal gas law, the gas constant, Boltzmann’s constant). Chemistry Department of Fudan University

  20. 1-1. answer The mass of a water droplet: m = Vρ = [(4/3) r3]  = (4/3)  (0.510-6 m)3 (1.0 g/cm3) = 5.210-16 kg Average kinetic energy at 27oC: KE = mv2/2 = (5.210-16 kg) (0.5110-2 m/s)2/2 = 6.910-21 kg m2/s2 = 6.910-21 J Chemistry Department of Fudan University

  21. 1-2. The average kinetic energy of an argon atom is the same as that of a water droplet. KE becomes zero at –273oC. From the linear relationship in the figure, KE = aT (absolute temperature) where a is the increase in kinetic energy of an argon atom per degree. a = KE/T = 6.910-21 J/(27+273 K) = 2.310-23 J/K S: specific heat of argon; N: number of atoms in 1 g of argon S = 0.31 J/g K = aN; N = S/a = (0.31 J/g K) / (2.310-23 J/K)= 1.41022 Avogadro’s number (NA) : Number of argon atoms in 40 g of argon NA = (40)(1.41022)= 5.61023 Chemistry Department of Fudan University

  22. Assignments 18 - 1,2,3, 4 Chemistry Department of Fudan University

  23. §18−2 Non-equilibrium Thermodynamics At invariable external restriction conditions, such as fixed boundary condition or restricted concentration condition, the system in which the macroscopic properties vary is in a non-equilibrium state After a certain time span the system, in which its macroscopic properties do not vary with time but the macroscopic processes are still going on, reaches a non-equilibrium steady state, for short we call it as steady state If there is no external restriction condition, the system definitely reaches a steady state in which there is no macroscopic process. This special steady state is the equilibrium state Under non-equilibrium states including the non-equilibrium steady state, the definitions of classical temperature, pressure, thermodynamic functions such as Gibbs function are invalid. So classical thermodynamics is not applicable to life systems, also not applicable to the universe Chemistry Department of Fudan University

  24. 均匀温度梯度的介质中间一点温度是多少? 均匀介质 300 K 400 K 外层绝热 Basic problem: Can thermodynamic state functions describe the properties of non-equilibrium states? What is the temperature at a certain point in the medium with uniform temperature gradient? Uniform medium External adiabatic wall The zeroth law of thermodynamics requires the thermometer and the system reaching thermodynamic equilibrium firstly when we measure the temperature of the system. In a non-equilibrium system constituted by a lot of particles the equilibrium itself is not reached So it is meaningless to discuss temperature for such system Chemistry Department of Fudan University

  25. Postulations of Local Equilibrium 1. Divide the system studied into subunits with small volume, which is microscopically large enough that thermodynamic variables such as T, P, and have well-defined local values, but macroscopically small enough that these local values are the same everywhere within the subunit 2. At the time scale, dt is small enough. The thermodynamic properties of the subunit isolated fromthe surroundingenvironment at time t can be represented by the thermodynamic properties of the subunit reaching equilibrium at time t + dt 3. Further assume that these local thermodynamic quantities satisfy the same thermodynamic equations that we have derived from equilibrium systems Applicable area: the gradient should not be too large within the mean free path l of the rarefied gases Chemistry Department of Fudan University

  26. (1) Principle for Entropy Production (Increase) Entropy is always produced in a spontaneous process > 0 for the spontaneous process in an isolated system = 0 for the reversible process in an isolated system For other types of systems, it is convenient to view the entire entropy change as consisting of two parts Where dSprod is the part created by any spontaneous process. This quantity is always positive; dSexch is the part resulting from heat exchange with the surroundings. This quantity is given by , which can be positive, negative, or zero Chemistry Department of Fudan University

  27. The entire entropy change of the system in the figure is The part resulting from the exchange with the surroundings is Reservoir 1 Reservoir 2 The part created by the spontaneous process is A two-compartment system with each compartment in contact with an heat reservoir. The two parts are separated by a rigid, heat-conducting wall The non-equilibrium thermodynamicsis based upon the fact thatdSprod > 0 for any spontaneous process Chemistry Department of Fudan University

  28. Total entropy of the system is Because each compartment is in equilibrium individually, we can write (i=1, 2) The system is isolated In the isolated system The first term is S caused by energy exchange The second one is S caused by volume change The third one is S caused by matter exchange Rigid, adiabatic wall Diathermal, flexible, permeable wall An isolated two-compartment system in which the two parts are separated by a diathermal (heat-conducting), flexible, permeable wall. Each part is equilibrated individually, but the two parts are not in equilibrium with each other Chemistry Department of Fudan University

  29. If the two compartments are separated by a rigid, diathermal, permeable wall so or = 0 for the equilibrium system > 0for the non-equilibrium system The flux of energy The flux of matter and are called thermodynamic forces The non-zero values of the thermodynamic forces are responsible for the fluxes of energy and mass Note that the fluxes are the time derivatives of extensive thermodynamic properties and that the forces are differences in intensive thermodynamic properties Chemistry Department of Fudan University

  30. (2)Onsager Reciprocal Relations Many flux-force relations are linear If there are gradients for more than one physical quantities in the system, it turns out experimentally that where the Ls are called phenomenological coefficients, which are determined experimentally. The physical meaning of the diagonal phenomenological coefficients (Lnn and LUU) is thatthey relate a flux to its “own” force. Lnn and LUU are positive. The off-diagonal phenomenological coefficients are called coupling or cross coefficients and relate an indirect dependence of a flux on the “direct” force of some other flux. They can be positive or negative Chemistry Department of Fudan University

  31. Lars Onsager (1903-76), Norwegian-American chemist and physicist. Born (1903-11-27) in Oslo, Ch. E. degree from Norges Tekniske Høiskole, Trondheim, in 1925. Emigrated to the USA in 1928 and became an American citizen in 1945. Got his Ph. D. degree in 1935. J. Willard Gibbs Professor at Yale University in 1945. Received the 1968 Nobel Prize for Chemistry for work done in 1931 on irreversible thermodynamics. Theoretical professor in Miami university in 1972. Won 1968 Nobel Prize Chemistry Department of Fudan University

  32. For any isolated two-compartment system, we define a set of thermodynamic forces and a set of corresponding fluxes, the rate of entropy production is given by We then express the fluxes as linear combination of the forces according to The formulism is called linear non-equilibrium thermodynamics because the flux-force relations are restricted to being linear. Onsager discovered that the phenomenological coefficients satisfy the condition They are called the Onsager reciprocal relations which give us relations between various properties of non-equilibrium systems Chemistry Department of Fudan University

  33. If we assume N = 2 and use Onsager reciprocal relations, then From the above formula, we can prove that The above results are easy to generalize. If we simply take Then we obtain the general results The above equation means that the off-diagonal terms cannot dominate the flux-force relations Chemistry Department of Fudan University

  34. For a discontinuous system, the entropy production is For a continuous system only including one compartment, or a non-dispersed system, we can subdivide the system into slides of thickness x, the volume of which is Ax. According to the Postulations of Local Equilibrium, we now apply above equation to any sub-unit and divide by V= Ax to write The differential form is Among them, the flux of energy is , the flux of matter is Chemistry Department of Fudan University

  35. (3) Principle for Minimum Entropy Production For a discontinuous non-equilibrium system In the above formula Using Onsager reciprocal relations,we obtain An equilibrium state, in which all properties of the system are equivalent everywhere and all fluxes and forces are zero, is a state of zero entropy production, namely A steady state, in which all fluxes and forces of the system do not change with time, is a state of entropy production, namely Chemistry Department of Fudan University

  36. Let’s see how varies with Xn whilekeeping XU fixed. To do this mathematically we already have We differentiate given by above equation with respect to Xn, and use the fact that at the steady state to obtain We consider the case in which only XUis held constant using one compartment in contact with a heat bath at temperature T1 and the other compartment in contact with a heat bath at temperature T2. Then letXnchangeuntilthe systemcomes to a steadystate in which the flux of matter is equal to zero, namely Jn= 0 and Xn 0 at the steady state. We derive this result in terms of equations as the following At the steady state the rate of the entropy production does reach extremum Chemistry Department of Fudan University

  37. A second derivative of with respect to Xn gives So we can see that the rate of entropy production is a minimum at the steady state, namely at the steady state, the unrestrained force will adjust itself so that the rate of entropy production is a minimum. This is the principle of Minimum Entropy Production An equilibrium state is a state of zero entropy production; a steady state is a state of minimum entropy production. In a sense, a steady state plays the role in non-equilibrium thermodynamics as an equilibrium state plays in classical (equilibrium) thermodynamics The Principle of Minimum Entropy Production is limitedto the cases in which all the fluxes are linearly related to the forces, namely Onsager reciprocal relationsis held and phenomenological coefficients do not change with time Chemistry Department of Fudan University

  38. Ilya Prigogine was born in Moscow, Russia on 1917-1-25. He obtained both his undergraduate and graduate education in chemistry at the Universite Libre de Bruxelles. He was Regental Professor and Ashbel Smith Professor of Physics and Chemical Engineering at the University of Texas at Austin. In 1967, he founded the Center for Statistical Mechanics, later renamed the Ilya Prigogine Center for Studies in Statistical Mechanics and Complex Systems. Since 1959, he was the director of the International Solvay Institutes in Brussels, Belgium. Ilya Prigogine was awarded the Nobel Prize in chemistry in 1977 for his contributions to non-equilibrium thermodynamics, particularly the theory of dissipative structures Chemistry Department of Fudan University

  39. Assignments 18 - 5,6,7 Chemistry Department of Fudan University

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