Einführung in die Physikalische Chemie für Studierende der Natur- und Nanowissenschaften

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1 Einführung in die Physikalische Chemie für Studierende der Natur- und Nanowissenschaften Bücher mit Übungsaufgaben / Lösungen: Neuere Ausgaben von Atkins Physical Chemistry Clyde R. Metz, Schaum s Outline of Physical Chemistry, McGraw-Hill 1988 (auch auf Deutsch erhältlich)

2 Einführung in die Physikalische Chemie: Organisation Einführung in die Physikalische Chemie: Inhalt

3 Repetition Chapter 1 Repetition Chapter 1: Multipole Expansion of the Intermolecular Potential At large intermolecular distances there is no significant overlap between the molecular wavefunctions. The intermolecular interactions are purely electrostatic and determined by the intrinsic charge distribution of the molecules. In electrostatics every charge distribution can be expressed as a sum of multipole moments (m=1,2,4,8,.. = order of the multipole) Hence the intermolecular potential can be formulated as a sum of interaction terms between multipoles: + - m=1 m=2 m=4 Vn <0... interaction coefficients, R... intermol. distance m=8 The interaction index can be calculated according to: static multipole moments m1, m2: n=m1+m2-1 induced multipole moments m1 (i), m2 (i) : n=2(m1 (i) +m2 (i) -1) These non-covalent interactions are termed van der Waals interactions

4 Chapter 2: States of Matter Chapter 2: States of Matter Contents: Literature: 2.1 Introduction 2.2 Intermolecular Model Potentials 2.3 Real Gases 2.4 Liquids 2.5 Solids and Crystals 2.6 Structure of Condensed Phases 2.7 Molecular Dynamics Simulations P. Atkins, J. de Paula, Atkins Physical Chemistry, 8th Ed., Oxford University Press 2007 Chapters 1, 18, 20 I. Tinico et al., Physical Chemistry, Principles and applications in biological sciences, 4th Ed., Prentice-Hall 2002 Chapter 9 G. C. Maitland et al., Intermolecular Forces, Clarendon Press 1981 Supplementary Material: Tutorials Potential and Molecular-Dynamics Simulation on the web

5 2.1 Introduction 2.1 Introduction Topic: how do the interactions between molecules determine the structure and state of matter State of matter = aggregate state: solid, liquid, gaseous Pair additivity: the total potential energy Epot,tot can be formulated as a sum of interactions between pairs of particles Epot, ij: (2.1) i i j j The force F is defined as (2.2) Hence, pair additivity also applies to forces. Many-particle contributions (n>2) to Epot,tot frequently amount to only a few per cent (however, in systems like water they can be as high as 30%! Why?) Aim: we want to calculate the macroscopic properties of matter based on our knowledge of the microscopic interactions between the particles

6 2.2 Intermolecular Model Potentials 2.2 Intermolecular Model Potentials The multipole-expansion coefficients Vn depend on the specific properties of the molecules. Their accurate determination from spectroscopic measurements or theoretical calculations is often very elaborate -> use model potentials. multipole expansion Potential between two spherical atoms/molecules: Lennard-Jones Epot>0: repulsive interaction Epot<0: attractive interaction Potential terms are usually of the form Vn (R) = Vn/R n (see chapter 1) The larger the exponent n, the faster the decrease of the potential as a function of R (e.g., Coulomb potential V R long range!) neutrale Teilchen Coulomb-Potential 0 R

7 2.2 Intermolecular Model Potentials Simple Model Potentials The attractive potential between free neutral particles is usually of the form -R -6 or faster (i.e., n>6) (see multipole expansion, chapter 1) The repulsive potential is caused by the electronic and nuclear repulsion at short range. Examples for repulsive model potentials: hard spheres elastic spheres Frequently formulated empirically as: with k 9-12 Alternative ansatz: (2.3a) (2.3b) σ... diameter of the spheres = collision cross section σ

8 2.2 Intermolecular Model Potentials Summation of Attractive and Repulsive Contributions: Formation of Potential Wells Example: (2.4) using A=100 and B=10 potential well Weakly bound pairs of molecules (dimers) can be formed when the total interaction potential exhibits at least one potential well (=minimum). The equilibrium separation Re between the two molecules corresponds to the position of the minimum.

9 2.2 Intermolecular Model Potentials Inclusion of the Kinetic Energy Ekin: Total Energy Etot ε... potential well depth = interaction energy = bonding energy = dissociation energy σ... collision cross section Re... equilibrium distance Etot = Ekin + Epot (2.5) Etot < 0 : dimer is bound Etot > 0: dimer dissociates When a dimer dissociates at a given Etot, its kinetic energy Ekin is given by Ekin=Etot-Epot according to Eq. (2.5)

10 2.2 Intermolecular Model Potentials The Lennard-Jones (LJ) Potential Simple model potential of the form: (2.6) repulsive attractive LJ-m,n-potential The molecules are treated as spheres with R representing the distance between their centres - For neutral particles: n=6 (-> dispersion interaction, see chapter 1) - For the repulsive term m=12 is usually chosen -> LJ-12,6-Potential Note: for ions the Coulomb interaction has to be taken into account (2.7) Coulomb interaction term between positive and negative ions

11 2.2 Intermolecular Model Potentials The Lennard-Jones (LJ) Potential: Alternative Representation (2.8) Compare Eq. (2.8) with : (2.9) It is easy to verify by inserting that V(R)=0 R=σ Find a general expression for Re (-> Übungen) Example: determination of the LJ potential parameters for the HCl. Ar complex -> blackboard

12 2.2 Intermolecular Model Potentials Lennard-Jones Potentials for Different Dimers All examples in the diagram on the left are non-polar neutral dimers! Dimer von: LJ LJ works best for non-polar gases, liquids and solids in which dispersion is the dominant interaction between the particles. exact potential Compare fitted LJ potential for He. He with exact potential (not bad!). Which trends in the well-depth ε can you observe? LJ parameters can most accurately be determined by fitting them to reproduce experimental results like virial coefficients, see later.

13 2.2 Intermolecular Model Potentials Lennard-Jones Parameters for Different Dimers Dimer von: Compare: typical bond energies: 500 kj/mol ( K) H-bond energies: 20 kj/mol ( K) CO2 dimer dissociation energy: 1.6 kj/mol ( 190 K) Ne dimer dissociation energy: 0.3 kj/mol ( 36 K)

14 2.3 Real Gases 2.3 Real Gases Until now: ideal gases (2.10) molar Volume Vm = V/n molecules are treated as point-like non-interacting particles without volume Problems: 1. Vm 0 when T 0 or p (unphysical, because particles have an intrinsic volume of their own) 2. no interactions between the gas particles no condensation or phase changes (unphysical!)

15 2.3 Real Gases The van der Waals Equation The van der Waals equation represents a description for real gases which approximatively corrects these two deficiencies: 1. The volume Vm is reduced by a parameter b corresponding to the intrinsic volume of the particles 2. The pressure is reduced by the attraction between the particles which is taken into account by a negative term a/vm 2 : (2.11) The properties of the van der Waals Equation will be discussed in Übungen 2,3 Advantage: simple equation with an intuitive interpretation Disadvantage: only approximate and cannot be improved systematically more accurate approach: virial expansion

16 2.3 Real Gases The Virial Expansion Motivation: In the pv vs. p diagram, ideal gases show a straight line irrespective of the temperature, i.e., Real gases show a linear dependence on p at low pressures develop pvm as a power series in p or Vm -1 : (2.12) 2. virial coefficient 3. virial coefficient (2.13)

17 2.3 Real Gases (2.13) In contrast to the van der Waals equation, the accuracy of Eq. (2.13) can be arbitrarily increased by taking into account more terms in the expansion. However, the virial expansion does not converge for liquids (more complex structure than gases, see later) The virial coefficients B, C,... can be determined a) experimentally (by measuring the pv vs. p diagram and fitting to Eq. (2.12)) b) from the intermolecular potential, e.g., (2.14) (Note: this formula can be derived from statistical thermodynamics and is given without proof)

18 2.3 Real Gases Example: determination of B for a square-well potential Square-well potential The parameters ε, r1, r2 can be determined from a fit to experimentally determined B values, e.g., for the Xe dimer : (-> blackboard) from fitted square-well potential Conclusion: it is not possible to reconstruct the potential unambiguously from the virial coefficients, but it is possible to calculate the virial coefficients if the potential is known Relationship between the virial expansion and the van der Waals equation -> blackboard

19 2.3 Real Gases Example: determination of B for a square-well potential Square-well potential The parameters ε, r1, r2 can be determined from a fit to experimentally determined B values, e.g., for the Xe dimer : from fitted square-well potential Conclusion: it is not possible to reconstruct the potential unambiguously from the virial coefficients, but it is possible to calculate the virial coefficients if the potential is known Relationship between the virial expansion and the van der Waals equation -> blackboard

20 2.3 Real Gases Accurate determination of the interaction potential for gaseous dimers/clusters: high-resolution spectroscopy (outlook to chapter 4) The intermolecular potential well holds bound levels with well-defined quantised energies (see chapter 3). The energy of these levels is very sensitive to the precise shape of the potential curve. Light is absorbed at well-defined frequencies corresponding to the difference between the energy levels -> molecular spectroscopy, see chapter 4. spectroscopic transitions induced by light The potential curve can be reconstructed from the measured spectroscopic transition frequencies by fitting them to those calculated with a model potential, Example: use a Lennard-Jones potential and adjust the ε, σ parameters until the energy levels calculated from the potential reproduce the experimental transition frequencies.

21 2.3 Real Gases Example: spectroscopic determination of the interaction potential in Xe. Xe + P. Rupper, O. Zehnder, F. Merkt, J. Chem. Phys. 121 (2004), 8279 Experimental photoelectron spectrum of the transition Xe2 -> Xe2 + showing vibrational energy levels in the lowest electronic state of Xe2 + : Energy Xe2 + + e - Xe2 Experimentally determined potential energy curves: circles: prediction from quantum-mechanical calculations curves for excited electronic states experimentally determined curve for electronic ground state

22 2.4 Liquids 2.4 Liquids Certain properties of liquids can be connected with properties of the intermolecular potential function: Boiling point for a particle to evaporate from a liquid it must overcome the intermolecular attractive forces hence, the boiling point is in a rough approximation equal to the potential-well depth ε (expressed in units of K)

23 2.4 Liquids Hence: The larger ε, the larger the intermolecular attraction, the higher the boiling point: increasing potential-well depth ε

24 2.4 Liquids Surface Tension The intermolecular forces pull molecules from the surface of a liquid into its centre: Hence, liquids always tend to minimise their surface area formation of spherical droplets. The energy ΔE needed to change the surface area A by an amount ΔA is given by (2.15) The proportionality constant γ is the surface tension (units: [γ]=[j m -2 ]=[N m -1 ]) Obviously, γ is related to the magnitude of the intermolecular interactions and hence to the potential-well depth ε: the larger ε, the larger γ.

25 2.4 Liquids Viscosity The viscosity η is the resistance of a liquid towards deformation ([η]=[poise]=[10-1 kg m -1 s -1 ]). The stronger the molecules in a liquid attract each other (i.e., the larger ε), the higher the viscosity. However, other properties like the molecular geometry strongly affect the viscosity as well Enthalpy of Vaporisation The enthalpy of vaporisation (the heat required to vaporise a liquid) like the boiling point is also approximately proportional to ε.

26 2.4 Liquids Some General Rules Most of the macroscopic properties discussed so far are proportional to the potential-well depth ε and hence the magnitude of the intermolecular interactions. Rules of thumb: 1. ε scales with the polarity of a molecule. 2. For non-polar species ε scales with the size of the molecule. (electrostatic interactions!) (dispersion!) Mind that exceptions to these rules can occur because of other factors, e.g., geometric and steric effects. Example: apply these rules to estimate the relative order of boiling points for the following substances: CH4 < C2H6 < C3H8 <... benzene < toluene diethyl ether < ethanol ethanol < water size size polarity H bonds

27 2.5 Solids and Crystals 2.5 Solids and Crystals In this chapter, we restrict ourselves to ionic crystals (i.e., 3D lattices consisting of cations and anions) Determination of the Lattice Energy from a Model Potential Motivation: we would like to calculate the total energy of a crystal lattice based on our knowledge of the intermolecular interactions Approximate model for the pair-interaction energy in an ionic crystal: (2.16) Z... charge of ion repulsive term (empirical, C>0, m>1) attractive term (Coulomb) The potential-well depth ε can be calculated to be: (-> blackboard) (2.17)

28 2.5 Solids and Crystals The equilibrium distance Re can be estimated from tabulated ionic radii: (2.18) The total energy of the crystal lattice (per mole) EL is given by: (2.19) The Madelung constant M is a proportionality factor which takes into account the deviation of the total energies from the sum of pair energies Epair. (Breakdown of the principle of pair additivity. Why?) M depends on the symmetry of the crystal lattice and is always greater than 1 (M>1). Example: calculate the lattice energy of NaCl (-> blackboard)

29 2.5 Solids and Crystals Determination of the Lattice Energy from Thermochemical Data: the Born-Haber Cycle Principle: because energies are additive, the lattice energy can be calculated from tabulated thermochemical quantities using an energy cycle : Ionisation energy Electron affinity Dissociation energy Enthalpy of sublimation Lattice energy EL Enthalpy of formation

30 2.5 Solids and Crystals The different energy contributions can be found in thermochemical tables listed in, e.g., Atkins: The lattice energy can thus be calculated: EL = ( * ) kj mol -1 = kj mol -1 Compare with result derived from model potential (EL=782 kj mol -1 ). The Born-Haber cycle can be used as a general approach to calculate unknown thermochemical quantities from known ones!

31 Excursion: unusual solid phases: purely positively charged ionic crystals at extremely low temperatures ( Coulomb crystals ) 2.5 Solids and Crystals Fluorescence images of laser-cooled Ca + ions at T 5 mk in an ion trap: Laser cooling: Experiment: velocity vector of atom Applications: ion trap quantum computing chemical reactions at ultralow temperatures manipulations of single atoms and molecules

32 2.6 Structure of Condensed Phases 2.6 Structure of Condensed Phases Problem: we would like to have a mathematical criterion to characterise the spatial ordering of molecules in condensed phases: the radial distribution function g(r). g(r)r 2 dr is the probability of finding a molecule in an infinitesimal slice dr at a distance R from another molecule Derivation of g(r): 1. Count the number of molecules at a given distance R 2. Divide by 4πR 2 ρ to obtain a normalised g(r) (ρ... number density) Number of molecules Radial distance R

33 2.6 Structure of Condensed Phases g(r) for different aggregate states Gas: random motion of the gas particles no order Liquid: formation of solvation shells short-range order Crystal: lattice structure with a well-defined symmetry long-range order What is g(r) for an ideal gas?

34 2.7 Molecular Dynamics Simulations 2.7 Numerical Simulations of Bulk Matter: Molecular Dynamics (MD) Methods Aim: calculate the macroscopic properties of matter starting from a model for the microscopic interactions between molecules in a sample Ideally, one would like to have a rigourous numerical solution of the quantummechanical dynamics of the molecules (see chapter 3). However, this is still not possible for the large number of molecules in a macroscopic sample because of insufficient computer power. Instead, calculate classical trajectories of the molecules numerically by solving Newton s equation of motion: mi... mass of molecule i (2.20) xi... coordinate vector of molecule i Vij... intermolecular potential between molecules i,j i... Nabla operator: =(d/dx, d/dy, d/dz) mass. acceleration = Force The intermolecular potential used is usually a model potential as the ones discussed in this chapter,e.g., Lennard-Jones. Further information: - Web-Tutorial Molecular Dynamics Simulation -> see also demonstration - Vertiefungsvorlesung Computational Chemistry (Prof. M. Meuwly)

35 2.7 Molecular Dynamics Simulations Particle trajectories in different aggregate states Gas Liquid Solid

36 For revision and better understanding: Web-Tutorials Potential and Molecular Dynamics Simulation Sheets Zusammenhänge and Lernziele 2 to be downloaded from the website! E i n f ü h r u n g i n d i e P h y s i k a l i s c h e C h e m i e E i n f ü h r u n g i n d i e P h y s i k a l i s c h e C h e m i e Zusammenhänge Zusammenhänge Zusammenhänge Zusammenhänge Zusammenhänge Zusam Diese Blätter zeigen Grössen und Formeln verschiedener Kapitel in einem Zusammenhang und weisen auf die Verknüpfung Mikrokosmos-Makrokosmos hin. D. E. Woon berechnete 1993 quantenchemisch ein Ar-Ar-Potential (Chem.Phys.Letters 204, 29 (1993)). Mit Moleküldynamik-Simulationen im Computer wäre es möglich, daraus die verschiedensten Eigenschaften von flüssigem und festem Argon zu berechnen. Hier sollen durch Vereinfachungen die Zusammenhänge ohne grosse Computerrechnungen direkt gezeigt werden. V Wir benutzen von Woon's Potential die Tiefe! = me h = 1179! r J/mol = K, den Gleichgewichtsabstand r e = a o = pm und die Distanz, wo das Potential null ist, " = a o = pm. Zur Vereinfachung nähern wir das Potential durch dasjenige in nebenstehender Figur mit! und " von Woon an. "' nehmen wir als!2", was z.b. als zweite Schale im kubisch flächenzentrierten Argon-Kristall interpretiert werden kann. " "' Die obige Topftiefe! und der Radius " können mit den entsprechenden empirischen Parametern des Lennard-Jones Potentials (s. ausgeteilte Tabelle) verglichen werden:! = 124 K und " = 342 pm. Die Grössenordnungen stimmen, aber die Tiefe weicht beträchtlich ab. Potential und Dispersionswechselwirkung Wie in der Vorlesung gezeigt wurde, ist der Parameter B in der Darstellung V = A/r 12 -B/r 6 gegeben durch B = 4!" 6. Aus den Woon'schen Werten erhalten wir B"= Jm 6 und aus den Lennard-Jones Parametern B = Jm 6. Da hier reine Dispersionwechselwirkung vorliegt, kann B auch mit der entsprechenden Formel aus dem Polarisationsvolumen #' = 1.63 Å 3 und dem Ionisationspotential I = J näherungsweise berechnet werden (London-Näherung): B = 3 / 4 (#') 2 I = Jm 6. Die Dichte von festem und flüssigen Argon aus den Potentialdaten Um die Dichte von kristallinem Argon zu berechnen, benötigen wir von oben nur r e und die Tatsache, dass Argon im kubisch flächenzentrierten Gitter kristallisiert. Die Seitenlänge der Einheitszelle mit 4 Argonatomen beträgt!2r e. Damit wird das Volumen pro Atom r e3 /!2 und mit einem Molgewicht von Argon von m = g/mol erhalten wir eine Dichte von d = 39.95/(N A. re3 /!2) = 1.74 g/cm 3. Diese stimmt gut mit den experimentellen Werten von 1.79 g/cm 3 bei 1"K und 1.62 g/cm 3 am Schmelzpunkt (84 K) überein. Wenn wir annehmen, dass wegen der geringeren Ordnung in der Flüssigkeit nur 10 statt der 12 Nachbaratome im Kristall vorhanden sind, reduziert sich die Dichte auf 10/ 12, d.h g/cm 3. Dies kann mit dem experimentellen Wert von 1.41 g/cm 3 für die Flüssigkeit am Siedepunkt (86 K, 1 bar) verglichen werden. Verdampfungswärme und Siedepunkt von Argon aus den Potentialdaten Die Energie, um ein Atom von allen andern zu separieren, besteht in unserem Modellpotential gerade aus dem Produkt der Anzahl Nachbaratome mal die Wechselwirkungsenergie! (die weiter entfernten Atome tragen nichts bei, da dort das Modellpotential null ist). Um die Energie zu erhalten, die benötigt wird, alle Atome zu separieren, müssen wir obigen Wert mit der Avogadrozahl multiplizieren und durch zwei teilen (da sonst die Energie für jedes Paar zweimal gezählt würde). Diese Energie ist die Verdamp-! K2-17 Bau der Materie Lernziele Kapitel 2 -! Was ist Paar-Additivität? -! Welche Beziehung gilt zwischen Kraft und potentieller Energie? -! Was wird in einer Potentialkurve auf der y- bzw. x-achse dargestellt? -! Wie sieht ein Potential zwischen harten Kugeln ohne Anziehung aus? -! Warum stossen sich Moleküle auf kurze Distanz ab? -! Wie sieht ein realistisches Potential zwischen zwei Teilchen, die sich anziehen, qualitativ aus? -! Wie kommt die kinetische Energie bei der Diskussion einer Potentialkurve ins Spiel? -! Wie sieht der anziehende Term des LJ-12-6 Potentials aus? -! Geben Sie eine Begründung, warum und unter welchen Bedingungen dieser Term vernünftig ist? -! Wie sieht der abstossende Term des LJ-12-6 Potentials aus? Warum wird er so gewählt? -! Was ist e bzw. s im LJ Potential? -! Was ist der Gleichgewichtsabstand Re? Wie lässt er sich für das LJ Potential aus s berechnen? -! Welchen Term müssen Sie mindestens noch zum LJ Potential hinzufügen, wenn Sie die Wechselwirkung zwischen Ionen beschreiben wollen? -! Wie können Sie B im LJ Potential für Ar2 berechnen? Welche Daten des Ar-Atoms brauchen Sie? -! Mit Hilfe welcher Daten können Sie die Grösse von s für ein Dimer abschätzen? -! Wie können Sie e berechnen, wenn Sie B und s kennen? -! Wieviele kj/mol beträgt eine zwischenmolekulare Wechselwirkung etwa? Vergleich mit Bindung? -! Wie sieht die Van der Waals Gleichung für ein reales Gas aus? -! Wie sieht die Virialentwicklung in Vm für ein reales Gas aus? -! Wie verhält sich (qualitativ) der zweite Virialkoeffizient B als Funktion der Temperatur? Hinweis: siehe Van der Waals Gas. -! Nennen Sie 3 Beispiele, wo Potentiale verwendet werden! -! Wie gross ist in gröbster Näherung der Siedepunkt einer Flüssigkeit mit einem Potential dessen Tiefe e = 400 K ist? -! Erklären Sie, wieso Oberflächenspannung und Verdampfungsenthalpie für verschiedene Flüssigkeiten sich proportional verhalten! -! Erklären Sie, wie Siedepunkt und Potentialtiefe für verschiedene Flüssigkeiten zusammenhängen! -! Schätzen Sie, ob Ethanol oder Methanol den höheren Siedepunkt hat! Warum? -! Schätzen Sie, ob Ethanol oder Ether die höhere Viskosität hat! Warum? -! Schätzen Sie, ob Essigsäure oder n-butansäure die höhere Oberflächenspannung hat! Warum? -! Wie berechnet sich die Gitternergie mit Hilfe der Madelungkonstante aus der Paarenergie? -! Was ist ein Born-Haber-Zyklus? -! Was besagt die radiale Paarverteilungsfunktion? -! Wie sieht sie qualitativ aus für ein Gas, eine Flüssigkeit, einen Festkörper?! -! Was versteht man ungefähr unter einer Molekül-Dynamik-Simulation? -! Wodurch unterscheiden sich die Trajektorien eines Moleküls in Gas, Flüssigkeit und Festkörper?