Molecular BondsMolecular Spectra

CHAPTER 10Molecules and SolidsMolecules and Solids

Johannes Diderik van der Waals (1837 – 1923)

“You little molecule!”

- Fred Flintstone (insulting his friend Barney Rubble

Molecular Bonds

The Coulomb force is the only one to bind atoms.

The combination of attractive and repulsive forces creates a stable molecular structure.

Force is related to potential energy F = −dV / dr, where r is the distance separation.

It is useful to look at molecular binding using potential energy V.

Negative slope (dV / dr < 0) with repulsive force.Positive slope (dV / dr > 0) with attractive force.

An approximation of the force felt by one atom in the vicinity of another atom is

where A and B are positive constants.

Because of the complicated shielding effects of the various electron shells, n and m are not equal to 1.

Molecular Bonds

A stable equilibrium exists with total energy E < 0. The shape of the curve depends on the parameters A, B, n, and m. Also n > m.

Molecular Bonds

A pair of atoms is joined.

One would have to supply energy to raise the total energy of the system to zero in order to separate the molecule into two neutral atoms.

The corresponding value of r of a minimum value is an equilibrium separation. The amount of energy to separate the two atoms completely is the binding energy which is roughly equal to the depth of the potential well.

Vibrations are excited thermally, so the exact level of E depends on temperature.

Molecular Bonds

Ionic bonds:

The simplest bonding mechanisms.

Periodic Table: Alkali metals (1st Column) bond with halogens (7th column)

Ex: Sodium (1s22s22p63s1) readily gives up its 3s electron to become Na+, while chlorine (1s22s22p63s23p5) readily gains an electron to become Cl−. That forms the NaCl molecule.

Ex: K gives up a 4s electron to 3p shell in Cl.

K->K+ ion 4.34 eV of energy input

electron is transferred

Cl -> Cl- ion 3.62 eV energy given back. 0.72 eV is the difference

K+ and Cl- combine to give neutral KCl. The dissociation energy is 4.42 eV.

Example

10.1 Find b if the PE between the two ions is

10.2 Calculate the PE for KCL at its equilibrium distance of

2

9

ke bPE

r r

2.79or A

Molecular Bonds

Covalent bonds:

The atoms are not as easily ionized.

Ex: Diatomic molecules formed by the combination of two identical atoms tend to be covalent.

Larger molecules are formed with covalent bonds.

H2 molecule: two electrons moving the electromagnetic field of two protons.

From Q.M the two electrons can either have spin zero or one. S=0 state has lower energy

The most probable location of the two electrons are in the middle of the two protons.

Molecular Bonds

Van der Waals bond:

Weak bond found mostly in liquids and solids at low temperature.

Ex: in graphite, the van der Waals bond holds together adjacent sheets of carbon atoms. As a result, one layer of atoms slides over the next layer with little friction. The graphite in a pencil slides easily over paper.

Two atoms that have zero dipole moment can induce dipole moment into each other. That is positive and negative charges can be separated due to proximity of another atom. The weak attractive force between the induced dipoles is called van der Waals bond.

Molecular Bonds

Hydrogen bond:

Holds many organic molecules together.

Metallic bond:

Free valence electrons may be shared by a number of atoms.

Example

10.3 Calculate the dipole moment of KCl for the equilibrium distance of

2.79or A

Rotational States

Molecular spectroscopy:

We can learn about molecules by studying how molecules absorb, emit, and scatter electromagnetic radiation.

From the equipartition theorem, the N2 molecule may be thought of as two N atoms held together with a massless, rigid rod (rigid rotator model).

In a purely rotational system, the kinetic energy is expressed in terms of the angular momentum L and rotational inertia I.

221 1 ( )

2 2rot rot

IE I

I

Rotational States

L is quantized.

The energy levels are

Erot varies only as a function of the quantum number ℓ.

Example

10.4 Calculate the order of magnitude of the rotational energy levels for H2. Let the equilibrium spacing be 0.74 angstorm.

Vibrational States

A vibrational energy mode can also be excited.

Thermal excitation of a vibrational mode can occur.

It is also possible to stimulate vibrations in molecules using electromagnetic radiation.

Assume that the two atoms are point masses connected by a massless spring with simple harmonic motion.

The energy levels are those of a quantum-mechanical oscillator.

The frequency of a two-particle oscillator is:

where the reduced mass is μ = m1m2 / (m1 + m2) and the spring constant is κ.

If it is a purely ionic bond, we can compute κ by assuming that the force holding the masses together is Coulomb.

Vibrational States

and

Vibration and Rotation Combined

It’s possible to excite rotational and vibrational modes simultaneously.

Total energy of simple vibration-rotation system:

Vibrational energies are spaced at regular intervals.

Transition from ℓ + 1 to ℓ:

Photons will have an energies at regular intervals:

An emission-spectrum spacing that varies with ℓ.

The higher the starting energy level, the greater the photon energy.

Vibrational energies are greater than rotational energies. This energy difference results in the band spectrum.

Vibration and Rotation Combined

The positions and intensities of the observed bands are ruled by quantum mechanics. Note two features in particular:

1) The relative intensities of the bands are due to different transition probabilities.

The probabilities of transitions from an initial state to final state are not necessarily the same.

2) Some transitions are forbidden by the selection rule that requires Δℓ = ±1.

Absorption spectra:

Within Δℓ = ±1 rotational state changes, molecules can absorb photons and make transitions to a higher vibrational state when electromagnetic radiation is incident upon a collection of a particular kind of molecule.

Vibration and Rotation Combined

ΔE increases linearly with ℓ.

Vibration and Rotation Combined

Vibration and Rotation Combined

In the absorption spectrum of HCl, the spacing between the peaks can be used to compute the rotational inertia I. The missing peak in the center corresponds to the forbidden Δℓ = 0 transition.

The central frequency

Example

10.5 Estimate the vibration energy levels of H2 (r=0.74 angstrom)