EGEE 510

PHYSICAL CHEMISTRY IN ENERGY, GEO-ENVIRONMENTAL AND MINERAL ENGINEERING

Here is the Fall 2008 syllabus!

In 1927 Heitler and London (Z. Physik, Vol. 44, p. 455) applied quantum mechanics to explain the H-H bonding in H₂ and gave birth to quantum chemistry. Two years later Dirac (Proc. Royal Soc. London, A123, p. 714) proclaimed that the “underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known.” In 1998 the Nobel Prize in Chemistry was awarded to Pople and Kohn for the development of computational methods in quantum chemistry and of the density functional theory. So, in principle (and in most cases still ONLY in principle), chemical kinetics can now be predicted from quantum chemistry (which is introduced in Chs. 8 and 9 of Atkins, PChem8) or from (statistical) thermodynamics (which is discussed in Chs. 2-7 and 16-17). But in practice, 'chemistry' has indeed been reduced to 'physics' during the 20th century! (Hopefully, biology will be reduced to chemistry in the 21st century...)

That's why physical chemistry is today THE most important scientific 'discipline' that forms the basis of ALL (OK... maybe not all, but certainly MOST) engineering applications.

In most practical situations, chemical kinetics (see Chs. 21-24 of Atkins) is still an experimental field, although we can increasingly rely on order-of-magnitude estimations using first principles. In the first part of our course we shall review the highlights of phase and chemical equilibrium which allows us to determine the concentrations of species of interest and the ultimate composition of any reacting mixture. In the second part we shall review the highlights of chemical kinetics -- which allow us to determine more realistic compositions -- and illustrate their relevance and applicability to selected energy, geo-environmental and mineral engineering issues.

The objective of this course, apart from reviewing the applications of thermodynamics and kinetics, is to provide an exercise in targeted and stimulated self-study. In particular, the key to our success will be that you come to 'class' prepared to discuss the topics and issues summarized below (and discussed at length in a physical chemistry textbook).

--OVERVIEW AND APPLICATIONS OF THERMODYNAMICS--

As an introductory exercise, during the first week of class, when we shall have no group meetings (i.e., we shall have no “class”), you are asked to do the following:

(1) Develop a habit of reading and learning with the following tools beside you: (a) Internet (e.g., google.com, Web of Science); (b) Excel, and (c) Mathematica (or any other higher-power math software than Excel, especially for quick visualization of equations).

(2) Retrieve from the electronic library the classical JACS paper by Brunauer, Emmett and Teller where the BET equation has been presented for the first time. (Do you need help to find this paper?)

(3) Which phase (or chemical?!) equilibrium does this equation describe?

(4) If possible, based on the information provided, use Excel to obtain the surface area of any material discussed in the paper. Here is some material for discussion...

(5) Analyze some of the tables and figures as carefully as you can, and try to ‘derive’ (or identify) some of the numbers that appear in some of them, based on the information provided in the text or in (an)other table(s) or figure(s).

(5a) See here a few relevant Mathematica calculations... And here an update... Be prepared to discuss them (or to ask specific questions about them)!

(6) Use the “LeChatelier principle” to justify the adsorption trends shown in Table IV. (Is adsorption always exothermic? Does it matter whether it’s ‘physical’ or ‘chemical’?)

Some of the principal milestones in the development of thermo:

-1760s: Black, heat capacities, latent heats (“calorimetry”)

-1840s: Mayer, interconversion of heat and work (1^st law)

-1840s: Joule, interconversion of heat and work (Who’s done it first... A fascinating story!)

-1850s: Kelvin, absolute temperature and 2^nd law

-1850s: Clausius, entropy and 2^nd law

-1870s: Gibbs, phase rule and chemical potential (“free energy”)

-1880s: Helmholtz, equilibrium and free and bound energy

-1880s: van’t Hoff, equilibrium constant

-1900s: Nernst, 3^rd law

Isn’t it remarkable that in only half a century essentially all the key concepts became rather clear (despite the fact that the relevant issues -- e.g., energy, heat, work -- had been studied for centuries)?!

Key G/S phase equilibrium concepts:

-collision frequency (units?)

-surface coverage (monolayer, multilayer)

-intermolecular interactions

-adsorption isotherm (Langmuir, BET, Freundlich, Dubinin, etc.)

-others?

HW1 (accepted until 9/17, preferably in appropriate Angel dropbox), Atkins8: D7.1-7.5, E7.1, E7.9, E7.12, P7.2, P7.4, A7.36, D25.3, D25.5-25.7, E25.1, E25.4, E25.8, E25.13, P25.2, P25.6, A25.33. (Note: Parts of these problems will be solved during our class discussions or as ‘hints’, to be posted here in due course.)

Criteria for selection of a ‘good’ paper for (exam) analysis:

-Any research paper (or thesis, or report) must contain figures and/or tables that present one or all of the following types of results:

(a) ‘Raw’ data: e.g., temperature or pressure vs. composition.

(b) Data analysis: in our case, this is typically the determination of equilibrium parameters (e.g., properties such as equilibrium constant or partition coefficient) for a system, reaction, or process.

(c) Correlations: establishment (e.g., discovery, confirmation, rediscovery) of relationships between structure and properties (in a ‘scientific’ study) or between properties and behavior (in an ‘engineering’ study).

-A ‘good’ paper for analysis will contain information in each one, or at least two, of these categories.

HW1 hints/solutions:

-In order to use Sandler’s chemeq.bas program, do you agree that the following line should be added for CuSO₄ to the react.dta file? (And for CuSO₄*5H₂O?)

“CuSO4”,-661.8,-771.4,0.414,-26.7,-22.6,72.2,0.

-Here is a partial solution to A7.36. I suggest you use it as a ‘template’ for the solution of the other problems, in terms of (a) results provided, (b) assumptions made and (c) commentary offered. Be sure to explicitly address ALL the questions posed in the statement of the problem.

-E25.1(a): Hydrogen(i)=1.08x10²⁵ m^-2 s^-1; Hydrogen(ii)=1.44x10¹⁸ m^-2 s^-1; Propane(i)=2.3x10²⁴ m^-2 s^-1, Propane(ii)=3.1x10¹⁷ m^-2 s^-1.

-E25.4(a): 12.7 m²/g.

-E25.8(a): (i) 0.21 kPa; (ii) 22 kPa.

-E25.13(a): -12.3 kJ/mol.

-P25.2: (a) ca. 2x10⁸; (b) ca. 2x10³.

-P25.6: Assuming 0.157 nm² per H₂ molecule (reasonable?), the Langmuir eqn gives a surface area of ca. 6 m²/g.

-A25.33: (a) R²=0.988 for a, R²=0.932 for b (good enough?); (b) k_a=3.65x10^-3 kg/kg, k_b=2.7x10^-5 ppm^-1, DH_a=-8.69 kJ/mol, DH_b=-15.6 kJ/mol; (c) Does the cited paper shed any light on this issue? (See also this summary and this analysis of the Langmuir equation.)

-E7.1(a): 2.85x10^-6, 240 kJ/mol, 0 kJ/mol.

-E7.9(a): y_B=0.90, y_IB=0.10. In E7.9(b) note the error regarding the number of moles of N2, so the correct solution is y_NO=0.0176.

-E7.12(a): Above approx. 1150 K. For E7.12(b) the calculation shown assumes that C_p is independent of T; it’s not easy to find Cp=f(T) for the hydrate, but if the 298 K value is used, the calculated decomposition temperature (ca. 400 K) is very close to that shown in the solution manual (397 K).

-P7.2: (a) 1.24x10^-9; (b) 1.29x10^-8; (c) 1.8x10^-4.

-P7.4: (a) K = 1.22x10^-6 (1395 K), 2.80x10^-6 (1443 K), 7.23x10^-6 (1498 K). This is in reasonable agreement with Chemeq: 9.3x10^-7, 2.1x10^-6, 4.9x10^-6. It’s much better to check the reliability of Chemeq, and then use it to determine the other thermochemical parameters, than to make all these calculations by hand.

Discussion questions:

7.1: How does the mixing of reactants and products affect the position of chemical equilibrium? (Does the equilibrium constant depend on the concentrations of the species involved in the reaction?)

7.2: What are the various ways in which K can be expressed as a ratio of ‘concentrations’?

7.3: Effects of pressure (for deltaN≠0) and T?

7.4: On the van’t Hoff plot, sketch a typical result for an endothermic and an exothermic reaction. What are the ‘molecular’ implications (in terms of favoring reactants or products)? (Hint: A quick and convenient use of Sandler’s chemeq.bas should be useful here.)

-Here is a comprehensive van’t Hoff plot available from the literature. Can you reproduce some of these graphs?

Summary of key issues in V/L equilibrium:

-see handout “Summary of VLE criteria”

-solubility (Henry’s law): G --> L

-vapor pressure: L --> G

-single component: Antoine eqn, etc.

-solutions (mixtures): see Figure 6.8, Atkins8

-enthalpy of vaporization (Clausius-Clapeyron eqn)

-phase rule and phase diagrams

-fugacity [f = (fugacity coefficient)(pressure)] and activity [a = (activity coefficient)(mole fraction in liquid)]

-standard states (‘solvent’: Raoult’s law; ‘solute’: Henry’s law)

-ideal solutions (DH_mix=0, H^E=0)

-regular solutions (DH_mix≠0, H^E≠0, DS_mix=0)

-calculations using equation of state (e.g., vdW)

-here is a Mathematica file that allows easy preparation of the p vs. composition diagram for VLE

-for analysis of combined phase and chemical equilibrium, see handout “G/L Rxn Equilibrium”, as well as the accompanying Mathematica file.

HW2 (due 10/1), Atkins8: D1.3-1.6, E1.13, E1.22, A1.25, A2.45, D3.6-3.8, P3.18, A3.39, E4.8, P4.8, A4.22, A4.23, E5.3, E5.10, P5.9, P5.12, A5.25, D6.1, E6.2, E6.5, P6.3.

-the importance of having some math software is illustrated here for convenient differentiation (E5.10) and here for a quick solution of integrals in eqn 3.60 (P3.18)

-E1.22a: b=4.6x10^-5 m³/mol; Z= 0.661 (Makes sense? Why?)

-P3.18: Using a fourth-order polynomial (Why? Necessary?), f = 0.81, f = 81 atm.

-E4.8a: DH_vap=49.1 kJ/mol; T_b=215 ^oC; DS_vap=101 J/mol/K (Can check these values?)

-P4.8: 83.6 ^oC; DH_vap=38.0 kJ/mol (Can check these values?)

-A4.22: T_b=111.5 ^oC; DH_vap=7.9 kJ/mol (Can check these values?)

-E5.10a: n6/n7=1.0; m6/m7=0.86.

-P5.12: Check the Francesconi et al. (1996) reference!

-E6.2a: P=58.6 kPa, xA=0.27.

-E6.5a: (i) yM = 0.36; (ii) yM = 0.82.

-P6.3: Is Raoult's law (y_iP=x_iP_i,sat) mentioned? Applicable? (Here is another example of the significance of Raoult's law and deviations from it.)

Note: Our next group meeting will be on 9/29. In the meantime, be sure to verify the partial solutions and/or hints for the HW problems that will be posted here. Also, see whether you can find a suitable research paper that is of interest to you and contains figures and/or tables that illustrate phase and/or chemical equilibrium issues. Upon mutual agreement, such a paper will probably serve as the basis for your Exam1. Of course, for help with any of these activities, as well as further clarification of the material posted on the class web site, be sure to contact me by e-mail 24/7.

Similar tools and concepts can be extended, in a more or less straightforward way, to understand L/S phase and chemical equilibria... but this must be left for your own study (when needed). HW3 illustrates a few examples. It is also meant to cover, albeit very briefly, the increasingly important field of statistical thermodynamics (Chs. 16 and 17 in Atkins8), which, among other virtues, forms the basis for quantum thermochemistry calculations -- e.g., vibrational frequencies and energy changes between reactants, products and their transitions states (see, for example, Section 11.8 in Atkins8) -- and thus provides a natural link to our next topic of discussion, chemical kinetics.

Questions to be explored whenever you read a paper that is important for your research: (And for your EGEE 510 exams?)

(i) Which equation(s) in this paper are identical, or similar, to equation(s) in Atkins? Does the paper analyze phase equilibrium or chemical equilibrium, or both?

(ii) Which graph(s) in this paper are identical, or similar, to graph(s) in Atkins?

(iii) Does the information provided in the paper (e.g., tables) allow you to verify (and confirm?) some of the statements made in the paper?

(iv) Does the information provided in the paper (e.g., tables) allow you to reproduce the trend(s) shown in some of the figures?

(v) Can you make reasonable assumptions that will allow you to reproduce the trend(s) shown in some of the figures?

(vi) Do(es) the conclusion(s) of the authors follow directly from the results and discussion presented, and do(es) it/they make thermodynamic sense?

Summary of key concepts/issues in electrochemical thermodynamics:

-electric (electrostatic) potential ("electric pressure"): electric energy/electric charge (f = E_e/Q; 1 V = 1 J/1 C).

-electrochemical potential: m'_i= m_i + zFf (For z=0, uncharged species, it reduces to the chemical potential.)

-electromotive force (E): It's NOT a force! But it IS a driving force, because it is a potential difference... See Illustration 7.9 in Atkins8.)

-S/L adsorption equilibrium can be severely affected by the development of a double layer at the interface: in addition to dispersive attraction, need to take into account the electrostatic repulsion or attraction between the charged (electrode) surface and anions/cations in solution.

-standard cell potential (standard emf): E^o = -DG^o/zF.

-galvanic cell: 'produces' electric energy (i.e., converts chemical energy to electricity).

-electrolytic cell: 'consumes' electric energy (i.e., converts electric energy to, say, chemical energy).

-Nernst equation (for a cell at equilibrium): ln K = zFE^o/RT. (If cell is NOT at equilibrium, then eq 7.29 in Atkins8 applies, where instead of K need to use the ratio of actual activities, and not their equilibrium values.)

-Debye-Hückel equation (limiting law): -log g = Abs(z₊z_-)AI^0.5 (i.e., it allows us to determine activity coefficients, and thus activities, of ions in dilute solutions of known ionic strength I; A=0.509 at 25 ^oC in aqueous solution).

-Extended D-H eq: introduces additional temperature-dependent parameter(s), see eq 5.72 in Atkins8, as well as the effective ion diameter.

-electrochemical (redox) series of metals: Au > Pt > Ag > ... > Ni > Fe > Zn > ... > Na > Ca > K (e.g., Zn + AgO --> ZnO + Ag, E^o(Ag) > E^o(Zn) => "low reduces high".

Here is Table 7.2 from Atkins8. (Consistent with electrochemical redox series of metals? Should it be?)

HW3 (due midnight 10/19, in Angel dropbox): E4.5, E4.9, P4.6, E5.18, E5.20, E5.21, P5.6, P5.16, E6.11, E6.14b, E7.14, E7.16, P7.12, P7.17, D16.1-16.3, D16.5, D16.6, E16.2, E16.8, P16.12, E17.1, E17.3, E17.10, P17.10.

-E4.5a: T_fr = 281.7 K. (Of course, as P increases, so does the freezing point... But is this what makes ice skating possible?)

-E4.5b: Do we need to know the molecular weight of “a certain liquid”?

-E4.9a: T_fus = 272.7 K. (As expected?)

-P4.6: T_fr = 234.4 K. (As expected?)

-P5.6: At low solute concentrations its apparent MolWt is roughly half of the actual (e.g., 26.8 g/mol at a molality of 0.037), so KF is totally dissociated; at the higher concentrations, dissociation is NOT complete because the apparent MolWt of KF is higher, 35-48 g/mol, and the ratio of molalities is increasingly less than two. (Verified the Emsley paper?)

-P5.16: Is your graph similar to this one?

-E5.20: Why is the "mean ionic activity coefficient" apparently 'based' (see Solution Manual) on the component present in smaller amount?

-E6.11a: T_m is approx. 180 ^oC and x_B is approx. 0.26. The lower-right region is solid B+D, the middle region is D+L (with D being the decomposing product of composition AB_x, with x=??), and so on.

-E6.14b: Surely we can find the UF₄-ZrF₄ phase diagram in the literature, and verify all the relevant statements... Right?

-E7.16a: Do you agree that the activity coefficients of KBr and Cd(NO₃)₂ are 0.77 and 0.67, and that therefore the cell potential is -0.62 V?

-P7.12: Can you derive the expression DH = 2.3RT? And then, do you agree that the values for H₂ and CO are 14.7 and 18.8 kJ/mol? (Do these values make sense?)

-E16.2a: At 300 K it’s 2.57x10²⁷ and at 600 K it’s 7.26x10²⁷ (units?).

-E16.8a: Molar energy and entropy at 10 K are 22 J/mol and 4.8 J/mol/K.

-P16.12: (a) (Incalculable?) exp(1.25x10²⁴); (b) (Incalculable?) exp(1.31x10²⁴); (c) If W in (b) is greater than W in (a), then...

-E17.1a: For I₂ it’s 3.5R, for methane 3R and for benzene 7R... Close enough to experimental values?

-E17.3a: At 25 ^oC it’s 19.6 and at 250 ^oC it’s 34.3.

-E17.10a: rotational = -13.8 kJ/mol; vibrational = -0.20 kJ/mol.

-P17.10: By analogy with P17.11 (see Solutions Manual), the ratios of translational and rotational partition functions are, respectively, 0.964 and 6.24. After some (manageable?) math, at 27 ^oC K=945; and at 827 ^oC K=37. Obviously, you want to do this math “by hand” only once, to see clearly how it works; subsequently, you want to use high-power software such as Gaussian or Gamess (see below) or, at least, a template such as this Mathematica file provided by Prof. Metiu with his recent PChem textbook.

Note: To find the (necessary?) values of rotational constants for, e.g., methane, use Web of Science with “methane and rotational constant*” to locate, for example, the 1994 paper by Hollenstein et al. (see Table IV). Is there also a compilatory monograph of spectroscopic (including rotational) constants of substances?

Note: Developing a feel for RELATIVE values of fugacity coefficients and activity coefficients is among the MOST valuable claims to fame of an 'expert'. Contrary to "popular belief", engineers, certainly -- and scientists to a large extent -- are increasingly being paid to make reasonable estimates and assumptions, and not so much to calculate...

Here is an illustration of this process for activity coefficients. (Can you solve the problem?) How would you update your information on the activity coefficients of electrolytes?

Pending conceptual questions:

-Is the characteristic shape of the fugacity coefficient vs. pressure, or activity coefficient vs. concentration, related in any way to (e.g., conditioned by) the intermolecular potential vs. distance relationship (e.g., Lennard-Jones 6-12 curve)?

-Why is molality (mol/kg solvent) a more convenient measure of concentration than molarity (mol/L solution), or mole fraction, when dealing with electrolyte solutions? (Does the addition of a second solute change the molality of the first? Is molality temperature-dependent? Is density information needed?)

-Can you confirm the value of A=0.509 kg^0.5/mol^0.5 in the Debye-Hückel eq?

-Can you confirm the validity of this table that conveniently summarizes the values of the mean ionic molality for common types of electrolytes?

Exam #1: Analyze the following paper: Anal Chem 2006, 78, 4642-53; see questions (i)-(vi) above. Due in Angel dropbox midnight 10/15.

-What is the meaning of a linear van't Hoff plot? (For a typical homogeneous reaction -- e.g., CO+H₂O=CO₂+H₂ -- is the van't Hoff plot linear over a wide temperature range?)

-Can you 'reproduce' Figures 2 and 3 using the information provided elsewhere in the paper? (Table(s)?)

-Which van't Hoff plots make more sense, those in Figure 5 or those in Figure 7?

-The authors do not seem to be sufficiently familiar with the Le Chatelier principle!? Are you surprised to read the following: "It is well known that retention factors almost always decrease with increasing temperature. The questions that we want to answer are, Why? What does cause this decrease? How do the saturation capacities and the equilibrium constants of adsorbates change with increasing temperature?" Aren't the answers to at least two of these questions also well known?

-Do the authors provide any justification for the existence of only two or three types of sites on the heterogeneous adsorbent surface? Why not four or five? Or a continuous distribution? With two or three, and without a physical interpretation, aren't they just adding an excesive number of adjustable (and meaningless?!) parameters to their model?

-Does the annotated paper (see above) help to ease its 'digestion' and (critical) analysis?

-Do some of your van't Hoff plots look like this one?

-Is this Mathematica template helpful for your nonlinear regression analysis? (Is a convenient Excel template available as well? Such as this one?)

Summary of key concepts in statistical thermodynamics:

-quantum mechanics <--> relativity: large Hadron collider ("standard model" OK?)

-quantum mechanics <--> ('macrosopic') thermodynamics: statistical ('microscopic') thermodynamics

-Boltzmann distribution (e.g., barometric pressure P=P^oexp[-gMh/RT], or Maxwell's distribution of molecular speeds)

-partition function (e.g., at T=0 only the ground state is accessible to a molecule, whereas at high T virtually all energy levels are accessible)

-distinguishable particles: Q=q^N; indistinguishable particles: Q=q^N/N! (Stirling: lnX! = X lnX - X)

-etc.

Here is a nice example of the (fundamental and elegant!) link between quantum chemistry, (statistical) thermodynamics and chemical kinetics.

--OVERVIEW AND APPLICATIONS OF (CHEMICAL) KINETICS--

‘Mystery’ graphs: A picture is worth “a thousand words”. A graph is worth much more, because it summarizes and illustrates many scientific or engineering concepts; you ought to think about a graph every time you walk into a shower… (What else are you going to think about while you are there?!)

Below is a summary of key issues/equations in relevant chapters in Atkins. (Be sure to work through the units, verify their consistency, and develop a feel for the relevant orders of magnitude. Here is a Mathematica program which should help to give us that ‘feel’ for the key parameters in the various equations.)

Key concepts:

-reaction rate, molecularity vs. order

-mean free path, collision diameter, collision frequency, c vs. c_rel (kinetic theory of gases)

-collision theory

-Arrhenius equation (frequency factor, activation energy)

-transport coefficients (‘physical’ kinetics)

-rate-determining step (RDS)

-steady-state approximation (SSA)

-transition-state theory

-liquid-phase concentrations

-any missing (that are important for your research?)

HW4 (due in Angel dropbox by midnight 11/9; apart from these Atkins8 problems, see also Exercises 1-7 below):

-E21.2b: 475 m/s, 4.5x10⁴ m, 1.1x10^-2 1/s.

-E21.3a: Assuming lambda=0.1 m, 8.0x10^-7 atm.

-E21.7a: 0.009.

-E21.14a: 16.9 J/s, 17 W (assuming 100% efficiency… Good assumption? And if window is a fancy one, argon-filled, instead of air?)

-E21.22a: 7.6x10^-3 S m² mol.

-E21.29a: 0.42 nm.

-P21.16: Li, 0.237 nm (vs. 0.059); Na, 0.184 (vs. 0.10); K, 0.125 (vs. 0.14); Rb, 0.120 (vs. 0.15)… Comments? (These ‘subtle’ differences between Na⁺ and K⁺ in particular – with and without their hydration shells – are important, of course, not only for EGEE, but for life itself, because of “ion channels” in cell membranes. As Eric Kandel discusses in In Search of Memory, 2006, p. 89, “the ionic hypothesis … did for the cell biology of neurons what the structure of DNA did for the rest of biology.”)

-P21.22: 0.83 nm (with u=1.1x10^-4 cm²/s/V).

-E24.2b:

-E24.4a: 1.03x10^-5 m³/mol/s.

-E24.7a: 7.4x10⁶ m³/mol/s, 1.3x10^-7 s.

-E24.14a: -45.8 J/mol/K, 5.0 kJ/mol, 18.7 kJ/mol.

-E24.15a: 20.9 L²/mol²/min.

-P24.2: P=0.0065, and the reactive cross-section is 3.9x10^-21 m². (Temperature-dependent?)

-P24.4: The second point in the table should be -20.93 ^oC, instead of -20.73… Right? Therefore, with E_a=86 kJ/mol and A=1.26x10¹⁴ s^-1, DH=84 kJ/mol (makes sense?), DS=18 J/mol/K (makes sense for a thermolysis reaction?), DU=84 kJ/mol, and DG=79 kJ/mol.

-P24.10: DH=60 kJ/mol, DU=63 kJ/mol, DS=-181 J/mol/K, DG=115 kJ/mol. (See Note 4 below!)

-P22.2: After 43.8 h the concentration drops to ca. 0.97 mol/m³.

-P22.10: Do you agree that k=7.6x10^-4 1/s?

-P22.16: Do your results confirm those of the original paper (3.98 kcal/mol and 11.1x10⁹ L/mol/s)?

-P23.1: The appropriate time interval, for the concentrations and k values as shown, is NOT 0-10 ns (as in the problem statement) but 0-10 ms (as shown in the solution manual). Do you agree? (Note that, as expected for a consecutive reaction scheme, there is a more or less monotonic decrease in the concentration of H and NO₂, O and OH are intermediate products whose concentration therefore goes through a maximum, whereas the appearance of the final product O₂ is ‘delayed’.

-P23.2: The most appropriate kinetics is pseudo-first-order… Right? (But does it reproduce properly the experimental [O] vs. t behavior?)

Notes:

-if you do not agree with the answers to the HW questions that will eventually be posted here (before the due date of the HW), let’s discuss them. (The procedures used are, of course, more important than the exact numbers obtained…)

-in E21.14 it is asumed that the RDS is heat transfer through the gas space between the panes, and NOT through either of the two glass panes; is this a reasonable assumption? (Are you familiar with the concept of R-value of insulating materials? See what you can find about it on wikipedia or using google.com. Also, when you walk toward campus from the east-side corner of College and Burrowes, be sure to read this!)

-in analyzing the units of the various quantities involved in the motion of ions (in liquids), remember that 1 volt is 1 joule per coulomb (1 V = 1 J/C).

-see here a clarification of the relationship between the observable (experimental) activation energy and the internal energy increase required for the formation of an activated compex (transition state): in analogous fashion, one can derive the relevant expressions for a unimolecular reaction, and then conclude that the entropy of activation in E24.14b should be -32.4 and not -24.1 J/mol/K. (Do you agree?)

Applications where these issues are important are, for example:

-combustion of gases and liquids (homogeneous kinetics)

-atmospheric reactions (e.g., soot + ozone = CO + O₂)

-aqueous-phase reactions (e.g., SO₂(aq) -> SO₃(aq) -> H₂SO₄(aq))

-thermal decomposition of organic compounds (e.g., pyrolysis of oil shale, biomass and coal, decomposition of soil constituents)

-gasification and combustion of solids (heterogeneous kinetics, catalysis)

-others? (Here list some of YOUR topics of interest…)

One of the most important skills to be developed is the “feel” for orders of magnitude (see class handout). Toward this goal, it’s always instructive to put some important markers on the scale of chemical reaction rates:

(a) A very fast rate: Show (using a class handout) that a typical coal (carbon) combustion rate is of the order of 1 mol/cm³/s. (Gas-phase combustion reactions, as we saw, are some three orders of magnitude faster than this.)

(b) At the other extreme, let’s show that the rate of the metabolic process(es) in our body is some 10 orders of magnitude less than that.

Ch. 21 (Molecules in motion) and Ch. 24 (Molecular reaction dynamics):

The “bottom line” of this section, for our purposes, is the following:

-using the concepts of mean free path and mean molecular speed, developed from the Maxwell distribution function, can quantify collisions between molecules, or of molecules with surfaces, and thus estimate the preexponential term in the chemical kinetic rate expression

-using the same concepts, can estimate transport coefficients

-in conjunction with the Stokes-Einstein and Einstein-Smoluchowski equations, establish a link between microscopic parameters of particle motion (e.g., mean free path, random walk step) and macroscopic (observable) properties of fluids

(This analysis can then by followed up by statistical thermodynamics, or statistical mechanics, and then quantum mechanics, all of which increasingly allow the calculation of rates of both physical and chemical processes from “first principles”… but this is, of course, beyond the scope of our discussion.)

Collision frequency/flux (from kinetic theory of gases):

This equation leads to the collision theory of chemical kinetics, whose key equations are the following:

Related (and more basic) equations are those for the mean molecular speed, mean free path, and the diffusivity (of an ideal gas):

Exercise 1. A recent physical chemistry textbook (Silbey et al., 2005) reports the following values for the interaction between O₂ and H₂ molecules at 25 ^oC.

(a) Collision frequency of H₂ with O₂ molecule = 4.63x10⁹ s^-1;

(b) Collision frequency of O₂ with H₂ molecule = 9.258x10⁹ s^-1;

(c) Collision density between H₂ and O₂ molecules = 1.244x10⁸ mol/L/s.

Verify them! [The complete text of the problem statement is as follows (p. 638): “A gas mixture contains H₂ at 0.666 bar and O₂ at 0.333 bar at 25 ^oC. (a) What is the collision frequency z₁₂ of a hydrogen molecule with an oxygen molecule? (b) What is the collision frequency z₂₁ of an oxygen molecule with a hydrogen molecule? (c) What is the collision density Z₁₂ between hydrogen molecules and oxygen molecules in mol L^-1 s^-1? The collision diameters of H₂ and O₂ are 0.272 nm and 0.360 nm, respectively.”]

Exercise 1a. See here the Mathematica solution to Example 24.1 in Atkins. Here you can see, as in Exercise 2 below, that the fraction of effective collisions is small, of the order of 10^-6.

Exercise 1b (McQuarrie and Simon, Physical Chemistry, Ch. 27). Consider a mixture of methane and nitrogen in a 10 L container at 300 K with partial pressures of 65 and 30 mbar, respectively. Show whether the collision frequency of a methane molecule with N₂ molecules is indeed 2.8x10⁸ s^-1, and the frequency of CH₄/N₂ collisions is 4.4x10³² molecules/m³/s.

Here is a template for numerical exercise #21.15 in Atkins…

Exercise 2. Calculate the number of collisions of methane molecules with O₂, in moles/cm³/s, assume zero-order reaction and estimate the time required for complete consumption of methane, assuming that all collisions are ‘effective’. Based on the order of magnitude obtained for this time, discuss whether the latter assumption is reasonable. (Hint: Combustion of natural gas is fast, but not that fast! The fraction of effective collisions turns out to be ca. 1/10⁶.)

Exercise 3. A typical value for the collision frequency (which in some cases can be equated with the preexponential factor in the Arrhenius rate expression) is 10¹³ s^-1. The following experimental results confirm this for the decomposition reaction 2N₂O₅ -> 2N₂O₄ + O₂. Determine the activation energy and the preexponential factor. (What is the order of most decomposition reactions? Is this fact consistent with the law of mass action? See the handout on the Lindemann-Hinshelwood mechanism.)

T (K) 288 298 313 323 338

k x 10⁵ (s^-1) 1.04 3.38 24.7 75.9 487

Silbey et al. (Physical Chemistry, 4^th edition) report the following results for the reaction N₂O₅ --> 2NO₂ + 0.5O₂:

T (K) 273 298 308 318 328 338

k x 10⁵ (s^-1) 0.0787 3.46 13.5 49.8 150 487

Is this consistent with the previous results?

McQuarrie and Simon (Physical Chemistry, 1997, p. 1146) report the following results for the same reaction at 318 K. Do they agree with the previous results?

t/min 0 10 20 30 40 50 60 70 80 90 100

[N₂O₅]/10^-2 mol dm^-3 1.24 0.92 0.68 0.50 0.37 0.28 0.20 0.15 0.11 0.08 0.06

Exercise 4. Froment and Bischoff (1990 CRE textbook) report the following kinetic data for thermal cracking (pyrolysis, decomposition) of ethane:

T (^oC) 702 725 734 754 773 789 803 810 827 837

k (s^-1) 0.15 0.27 0.33 0.59 0.92 1.49 2.14 2.72 4.14 4.67

(a) Determine the activation energy and the preexponential term and comment whether these values are ‘reasonable’. (See also Exercise #5 below.)

(b) Use the Web of Science to find out whether there is agreement in the literature about these values.

Ch. 22 (The rates of chemical reactions):

An alternative theory to that of “effective collisions” is the transition state theory. Let’s now summarize its key equations.

Van’t Hoff has provided an elegant, thermodynamic interpretation of the temperature dependence of the reaction rate. (Of course, the name of Arrhenius is associated with this equation; he verified it experimentally.) Let’s start with the Law of Mass Action (Guldberg and Waage, 1864):

aA + bB + ... F pP + qQ + ...

At equilibrium, by definition,

Therefore, since

we have a neat link between kinetics and thermodynamics (at least in principle):

Remember one of our ‘mystery’ graphs!

It is now intuitively obvious (and can be easily derived; see Atkins) that the rate constant from transition state theory (or activated complex theory) can be obtained using the Eyring equation (see handout on Reactivity):

You can now show that the preexponential factor, or the frequency factor here, is indeed typically of the order of 10¹³ s^-1.

Using quantum chemistry methods similar to, but more sophisticated than, the HMO method, the geometries and energies of transitions states can be obtained by solving the Schrödinger equation, and thus the activation energies and the rate constants can be estimated for an increasing number of practically interesting reactions.

Here is a bottom-line summary of the kinetics of liquid-phase rxns.

Here is a brief empirical (macroscopic) summary on diffusion…

For a fundamental microscopic view, a rewarding exercise is to reproduce the calculation of mean free path that Einstein made in his “miraculous-year” Brownian motion paper? For those of you who like to read only the “original stuff”, here it is, courtesy of google.com (and of Augsburg University). See also Sections 21.9-21.12 in Atkins_8.

And here is a brief summary on chain reactions. Very profitable reading are Hinshelwood’s and Semenov’s Nobel lectures given in 1956.

Ch. 23 (The kinetics of complex reactions):

Perhaps the most important concept in the whole of science and engineering is that of the RDS. Let’s first draw the instructive analogy between electric circuits (remember Ohm’s law?) and consecutive chemical reactions. And then let’s show how RDS makes easier our analysis of the most important chemical reaction in the world (and in EGEE): nC + [(n+1)/2]O₂ = CO₂ + (n-1)CO.

Let’s analyze a practical example: the difference between pulverized coal combustion (small particles, high T) and fluidized-bed combustion (larger particles, lower T). Study this Mathematica file carefully, so that you can get a feel for the importance of the key parameters.

Another key concept in this field is that of a chain reaction, with its initiation, propagation and termination steps. A related concept, which allows us to digest more easily such complex reactions, is SSA. Let’s analyze a representative reaction, ethane decomposition, using the Rice-Herzfeld mechanism (see also relevant class handout):

(1) C₂H₆ à 2CH₃ k₁

(2) CH₃ + C₂H₆ à CH₄ + C₂H₅ k₂

(3) C₂H₅ à C₂H₄ + H k₃

(4) H + C₂H₆ à H₂ + C₂H₅ k₄

(5a) H + C₂H₅ à C₂H₆ k₅

Of course, the math becomes quite involved when many such reactions must be considered (hundred or more elementary reactions are not uncommon), and software such as ChemKin becomes indispensable even for routine analyses (e.g., of concentration vs. time profiles). For simpler problems, such as the one shown here, a home-made program will suffice.

Exercise 5. Show that the reaction rate is indeed first-order! Instead of reaction (5) shown above, assume the following termination reaction:

(5b) 2C₂H₅ à C₄H₁₀ k₅ = 4.0x10⁸ Exp[-0/RT].

(a) Use the provided Mathematica template to recalculate the relevant concentration profiles. Which are the dominant products?

(b) Can you compare the results in Exercise 4 with the ones obtained here?

For example, verify that the SSA results in the following expression for the rate of decomposition of ethane: (2k₁ + k₃(k₁/k₅)^0.5) [C₂H₆].

Note: The kinetic expression for eq. 3 is k₃ [C₂H₅] [C₂H₆]^0.5 instead of the straightforward k₃ [C₂H₅]. Can you find out why (say, using the Web of Science)?

Now, reducing this expression, as well as the one using eq. 5b instead of eq. 5a, to a first-order equation, are the rate constants of the same order of magnitude?

(c) Does methane formation become more important at higher temperatures?

Here is an Excel file that compares the results of a mechanistic analysis of ethane decomposition (Rice-Herzfeld) with those of a phenomenological analysis (Exercise 4). Be sure to verify whether it is correct!

A nice example of a real-world situation in which the virtues of mixed flow and plug flow must be combined is the combustion chamber (e.g., in a power plant or an automobile engine). This is best understood by ‘playing’ with the parameters of the kinetic equation that describes an autocatalytic reaction (e.g., A + P à P + P), and reproducing the characteristic maximum in the rate vs. conversion behavior. Here is a template that you can use as a starting point in this exercise.

With this discussion we can now appreciate the fact that chemical kinetics applied to EGEE problems spans the (very wide but digestible and elegant) range from the Schrödinger equation -- e.g., the reactivity of sites in aromatic molecules or the geometry of the transition state (remember also the term k_BT/h in transition-state theory…) -- to the reactor performance (design) equation.

HETEROGENEOUS AND CATALYTIC REACTION KINETICS

-Here is a summary of the principal heterogeneous reaction/catalysis issues

-see also Ch. 25 in Atkins_8

In EGEE applications, many chemical reactions are heterogeneous, i.e., with reactants in different phases (e.g., gas/solid), and many are also catalyzed and are thus heterogeneous. In catalytic reactions, the rate most often increases as the catalyst particle size decreases (because catalyst dispersion thus increases).

Exercise 6. For a cubic solid with 5 faces exposed to an adsorbate (or a reactant), verify that catalyst dispersion (D, or fraction exposed) can be calculated using the following expression: five times the molecular (or atomic) weight divided by the product of particle diameter (d), density, molecular (or atomic) cross-section and Avogadro number. Show that, in many cases, this approximately reduces to D = 1/d, where d is in nm. (Note: D is often determined from chemisorption experiments, and d is most often determined from TEM experiments.)

Exercise 7. A TEM analysis of a carbon-supported CaO catalyst (with 10% CaO by weight) shows particles whose average diameter is 30 nm. (a) Determine the surface area and the dispersion (fraction exposed) of this catalyst. (b) What would be the chemisorption uptake of CO₂ (in cc(STP)/g catalyst) if the adsorption stoichiometry is 1/1?

For a summary of vessels (“reactors”) in which chemical reactions take place, see here. The objective here, of course and as appropriate in an engineering class, is to show the direct impact of chemical kinetics on reactor size and therefore on process economics.

Exam #2 (due in Angel dropbox by midnight 11/16): Let’s ‘dissect’ the classical paper(s) by Daniels and coworkers on the decomposition of N₂O₅: JACS 43, 53 (1921); 52, 1472 (1930). Many PChem textbooks also contain an analysis of this “first homogeneous first-order reaction to be investigated” (Laidler, in “Chemical Kinetics,” 1950, p. 219).

-Figs. 2 and 3 show the ‘raw’ data.

-Figs. 4 and 5 and Tables I-V show data analysis. Select at least one point in Figure 4 and explain how it was obtained from the raw data and the equation(s) presented in the text. Do the same in Table I by showing the sequence of calculations for a representative row; for example, at 25 ^oC and P_O2=5 torr, 1/(8KP_O2)=2.43, then a=0.761, 3P_O2 +2aP_O2=22.6, etc. As a reminder of your thermo expertise, explain why K decreases as T increases (p. 58; can you verify the validity of the equation used?). In Table II we see, for example, from Figure 2 that, indeed, at 25 ^oC and 1200 min, k=2.303/(1200-20) log[(288.7-23.9)/(288.7-268.7)]=0.0022 min^-1. Right? Another example?

-Table VI shows a typical kinetic correlation (calculation of activation energy and pre-exponential factor). Note, of course, that one of the k values for 25 ^oC should be 0.00201, and not 0.00801. (Are there any other typos or errors?) Do the k values in this table agree with those shown in Exercise 3 above?

-In addition to answering and commenting on the questions listed above, be sure to address the same six general questions that we had on Exam #1. (Of course, where it says ‘thermodynamics’ there, ‘kinetics’ applies here.)

-Select at least one citing reference and discuss the context (Appropriate? Too general? Specific enough?) in which the Daniels and Johnston paper is cited by the citing author(s).

End-of-semester activity (no class meetings on 11/19, 12/8 and 12/10):

(1) Let’s get a suitable (and affordable!) software (Gamess?) for doing quantum chemistry calculations (see Part 2 in Atkins8, especially Chs. 8-13).

(2) Let’s reproduce a suitable kinetics tutorial. (Remember the Ochterski example?)

(3) Let’s use this program to determine the rate constant of a suitable and relevant reaction (Exam #3?).

-For a hands-on introduction to Gamess, see, for example, http://www.msi.umn.edu/tutorial/chemistryphysics/Intro_to_GAMESS.pdf. We are going to be interested primarily in determining vibrational frequencies and -- as a follow-up (see the “Ochterski_2000.pdf” summary mentioned above) -- in thermochemical properties (of the reactants and of the activated complex, whose geometry has been optimized previously). And for the most authoritative introduction to Gamess, see http://www.msg.chem.iastate.edu/tutorials/gamessintro.pdf, especially p16.

-ChemViz is a convenient (and affordable!) tool for visualizing and analyzing the geometry of molecules.

-For generation of Z-matrix (necessary for Gamess input file) and its conversion to cartesian coordinates, see http://www.shodor.org/chemviz.

Here is a typical input file for the calculation of energy of a molecule, as well as the corresponding output file. Can you modify/adapt them to include the calculation of thermodynamic properties? (Have you been able to find similar or more appropriate ‘templates’? For example, at the San Diego Supercomputer Center, http://education.sdsc.edu/download/chemistry/gamess_logfile_tutorial_ex3.htm? And perhaps the tutorial “Studying Chemical Reactions with PC GAMESS” is useful as well…)

A judicious search within the Web of Knowledge can halp you identify a reaction of (your) interest for which much information is already available regarding the geometry of the transition state… Here are some examples:

-“transition state” (title) and “ab initio” (title)

-Gamess (topic)

-Gamess (topic) and “transition state” (title)

-Gamess (topic) and “vibrational frequenc*” (topic)

-Others? (e.g., Gamess (topic) and “[your favorite molecule]”)

Exam #3: due 12/19 (in Angel dropbox), at the latest.

(a) Submit your results of “End-of-semester activity” (see above), OR

(b) Use quantum chemistry software (Gamess?) to solve P11.18a and confirm the results of E11.5a and E11.5b in Atkins8. (Note: If Gamess is inconvenient to use, are the links provided by Atkins8 more user-friendly?)

LRR3@psu.edu (updated 12/03/2008)