Chapter 43

1. Chapter 43 Molecules and Solids

2. Molecular Bonds – Introduction • The bonding mechanisms in a molecule are fundamentally due to electric forces • The forces are related to a potential energy function • A stable molecule would be expected at a configuration for which the potential energy function has its minimum value

3. Features of Molecular Bonds • The force between atoms is repulsive at very small separation distances • This repulsion is partially electrostatic and partially due to the exclusion principle • Due to the exclusion principle, some electrons in overlapping shells are forced into higher energy states • The energy of the system increases as if a repulsive force existed between the atoms • The force between the atoms is attractive at larger distances

4. Potential Energy Function • The potential energy for a system of two atoms can be expressed in the form • r is the internuclear separation distance • m and n are small integers • A is associated with the attractive force • B is associated with the repulsive force

5. Potential Energy Function, Graph • At large separations, the slope of the curve is positive • Corresponds to a net attractive force • At the equilibrium separation distance, the attractive and repulsive forces just balance • At this point the potential energy is a minimum • The slope is zero

6. Molecular Bonds – Types • Simplified models of molecular bonding include • Ionic • Covalent • van der Waals • Hydrogen

7. Ionic Bonding • Ionic bonding occurs when two atoms combine in such a way that one or more outer electrons are transferred from one atom to the other • Ionic bonds are fundamentally caused by the Coulomb attraction between oppositely charged ions

8. Ionic Bonding, cont. • When an electron makes a transition from the E = 0 to a negative energy state, energy is released • The amount of this energy is called the electron affinity of the atom • The dissociation energy is the amount of energy needed to break the molecular bonds and produce neutral atoms

9. Ionic Bonding, NaCl Example • The graph shows the total energy of the molecule vs the internuclear distance • The minimum energy is at the equilibrium separation distance

10. Ionic Bonding,final • The energy of the molecule is lower than the energy of the system of two neutral atoms • It is said that it is energetically favorable for the molecule to form • The system of two atoms can reduce its energy by transferring energy out of the system and forming a molecule

11. Covalent Bonding • A covalent bond between two atoms is one in which electrons supplied by either one or both atoms are shared by the two atoms • Covalent bonds can be described in terms of atomic wave functions • The example will be two hydrogen atoms forming H2

12. Wave Function – Two Atoms Far Apart • Each atom has a wave function • There is little overlap between the wave functions of the two atoms when they are far away from each other

13. Wave Function – Molecule • The two atoms are brought close together • The wave functions overlap and form the compound wave shown • The probability amplitude is larger between the atoms than on either side

14. Active Figure 43.3 • Use the active figure to move the individual wave functions • Observe the composite wave function PLAY ACTIVE FIGURE

15. Covalent Bonding, Final • The probability is higher that the electrons associated with the atoms will be located between them • This can be modeled as if there were a fixed negative charge between the atoms, exerting attractive Coulomb forces on both nuclei • The result is an overall attractive force between the atoms, resulting in the covalent bond

16. Van der Waals Bonding • Two neutral molecules are attracted to each other by weak electrostatic forces called van der Waalsforces • Atoms that do not form ionic or covalent bonds are also attracted to each other by van der Waals forces • The van der Waals force is due to the fact that the molecule has a charge distribution with positive and negative centers at different positions in the molecule

17. Van der Waals Bonding, cont. • As a result of this charge distribution, the molecule may act as an electric dipole • Because of the dipole electric fields, two molecules can interact such that there is an attractive force between them • Remember, this occurs even though the molecules are electrically neutral

18. Types of Van der Waals Forces • Dipole-dipole force • An interaction between two molecules each having a permanent electric dipole moment • Dipole-induced dipole force • A polar molecule having a permanent dipole moment induces a dipole moment in a nonpolar molecule

19. Types of Van der Waals Forces, cont. • Dispersion force • An attractive force occurs between two nonpolar molecules • The interaction results from the fact that, although the average dipole moment of a nonpolar molecule is zero, the average of the square of the dipole moment is nonzero because of charge fluctuations • The two nonpolar molecules tend to have dipole moments that are correlated in time so as to produce van der Waals forces

20. Hydrogen Bonding • In addition to covalent bonds, a hydrogen atom in a molecule can also form a hydrogen bond • Using water (H2O) as an example • There are two covalent bonds in the molecule • The electrons from the hydrogen atoms are more likely to be found near the oxygen atom than the hydrogen atoms

21. Hydrogen Bonding – H2O Example, cont. • This leaves essentially bare protons at the positions of the hydrogen atoms • The negative end of another molecule can come very close to the proton • This bond is strong enough to form a solid crystalline structure

22. Hydrogen Bonding, Final • The hydrogen bond is relatively weak compared with other electrical bonds • Hydrogen bonding is a critical mechanism for the linking of biological molecules and polymers • DNA is an example

23. Energy States of Molecules • The energy of a molecule (assume one in a gaseous phase) can be divided into four categories • Electronic energy • Due to the interactions between the molecule’s electrons and nuclei • Translational energy • Due to the motion of the molecule’s center of mass through space

24. Energy States of Molecules, 2 • Categories, cont. • Rotational energy • Due to the rotation of the molecule about its center of mass • Vibrational energy • Due to the vibration of the molecule’s constituent atoms • The total energy of the molecule is the sum of the energies in these categories: • E = Eel + Etrans + Erot + Evib

25. Spectra of Molecules • The translational energy is unrelated to internal structure and therefore unimportant to the interpretation of the molecule’s spectrum • By analyzing its rotational and vibrational energy states, significant information about molecular spectra can be found

26. Rotational Motion of Molecules • A diatomic model will be used, but the same ideas can be extended to polyatomic molecules • A diatomic molecule aligned along an x axis has only two rotational degrees of freedom • Corresponding to rotations about the y and x axes

27. Rotational Motion of Molecules, Energy • The rotational energy is given by • I is the moment of inertia of the molecule • µ is called the reduced mass of the molecule

28. Rotational Motion of Molecules, Angular Momentum • Classically, the value of the molecule’s angular momentum can have any value L = Iω • Quantum mechanics restricts the values of the angular momentum to • J is an integer called the rotational quantum number

29. Rotational Kinetic Energy of Molecules, Allowed Levels • The allowed values are • The rotational kinetic energy is quantized and depends on its moment of inertia • As J increases, the states become farther apart

30. Allowed Levels, cont. • For most molecules, transitions result in radiation that is in the microwave region • Allowed transitions are given by the condition • J is the number of the higher state

31. Active Figure 43.5 • Use the active figure to adjust the distance between the atoms • Choose the initial rotational energy state of the molecule • Observe the transition of the molecule to lower energy states PLAY ACTIVE FIGURE

32. Vibrational Motion of Molecules • A molecule can be considered to be a flexible structure where the atoms are bonded by “effective springs” • Therefore, the molecule can be modeled as a simple harmonic oscillator

33. Vibrational Motion of Molecules, Potential Energy • A plot of the potential energy function • ro is the equilibrium atomic separation • For separations close to ro, the shape closely resembles a parabola

34. Vibrational Energy • Classical mechanics describes the frequency of vibration of a simple harmonic oscillator • Quantum mechanics predicts that a molecule will vibrate in quantized states • The vibrational and quantized vibrational energy can be altered if the molecule acquires energy of the proper value to cause a transition between quantized states

35. Vibrational Energy, cont. • The allowed vibrational energies are • v is an integer called the vibrational quantum number • When v = 0, the molecule’s ground state energy is ½hƒ • The accompanying vibration is always present, even if the molecule is not excited

36. Vibrational Energy, Final • The allowed vibrational energies can be expressed as • Selection rule for allowed transitions is Δv = ±1 • The energy of an absorbed photon is Ephoton = ΔEvib = hƒ

37. Molecular Spectra • In general, a molecule vibrates and rotates simultaneously • To a first approximation, these motions are independent of each other • The total energy is the sum of the energies for these two motions:

38. Molecular Energy-Level Diagram • For each allowed state of v, there is a complete set of levels corresponding to the allowed values of J • The energy separation between successive rotational levels is much smaller than between successive vibrational levels • Most molecules at ordinary temperatures vibrate at v = 0 level

39. Molecular Absorption Spectrum • The spectrum consists of two groups of lines • One group to the right of center satisfying the selection rules ΔJ = +1 and Δv = +1 • The other group to the left of center satisfying the selection rules ΔJ = -1 and Δv = +1 • Adjacent lines are separated by h/2πI

40. Active Figure 43.8 • Use the active figure to adjust the spring constant and the moment of inertia of the molecule • Observe the effect on the energy levels and the spectral lines PLAY ACTIVE FIGURE

41. Absorption Spectrum of HCl • It fits the predicted pattern very well • A peculiarity shows, each line is split into a doublet • Two chlorine isotopes were present in the same sample • Because of their different masses, different I’s are present in the sample

42. Intensity of Spectral Lines • The intensity is determined by the product of two functions of J • The first function is the number of available states for a given value of J • There are 2J + 1 states available • The second function is the Boltzmann factor

43. Intensity of Spectral Lines, cont • Taking into account both factors by multiplying them, • The 2J + 1 term increases with J • The exponential term decreases • This is in good agreement with the observed envelope of the spectral lines

44. Bonding in Solids • Bonds in solids can be of the following types • Ionic • Covalent • Metallic

45. Ionic Bonds in Solids • The dominant interaction between ions is through the Coulomb force • Many crystals are formed by ionic bonding • Multiple interactions occur among nearest-neighbor atoms

46. Ionic Bonds in Solids, 2 • The net effect of all the interactions is a negative electric potential energy • α is a dimensionless number known as the Madelung constant • The value of α depends only on the crystalline structure of the solid

47. Ionic Bonds, NaCl Example • The crystalline structure is shown (a) • Each positive sodium ion is surrounded by six negative chlorine ions (b) • Each chlorine ion is surrounded by six sodium ions (c) • α = 1.747 6 for the NaCl structure

48. Total Energy in a Crystalline Solid • As the constituent ions of a crystal are brought close together, a repulsive force exists • The potential energy term B/rm accounts for this repulsive force • This repulsive force is a result of electrostatic forces and the exclusion principle

49. Total Energy in a Crystalline Solid, cont • The total potential energy of the crystal is • The minimum value, Uo, is called the ionic cohesive energy of the solid • It represents the energy needed to separate the solid into a collection of isolated positive and negative ions

50. Properties of Ionic Crystals • They form relatively stable, hard crystals • They are poor electrical conductors • They contain no free electrons • Each electron is bound tightly to one of the ions • They have high melting points