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Subject Chemistry

Paper No and Title 8 and Physical Spectroscopy

Module No and Title 8: Alkali spectra

Module Tag CHE_P8_M8

CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 8 ( SPECTRA)

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TABLE OF CONTENTS

1. Learning Outcomes 2. Introduction 3. Multi-

4. Electron Orbitals

5. Alkali Metal Spectra

6. Summary

CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 8 (ALKALI METAL SPECTRA)

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1. Learning Outcomes

In this module, you will study about the failures of the Bohr theory, which led to the formulation of the quantum theory. You will understand how the new theory could explain the fine structure in the spectra of hydrogen and hydrogen-like ions, and how this theory can be extended to atoms which have a single electron in their outermost shell, i.e. the alkali metal atoms. You should also be able to write the term symbols for simple one-electron systems.

2. Introduction

Bohr’s model could only explain the line spectra of hydrogen and hydrogen-like ions. However, it did not even attempt to explain the spectra of multi-electron atoms. Even in the case of hydrogen, higher resolution shows that each line is split into a doublet, which the Bohr theory was clearly unable to explain. Clearly, the Bohr theory was inadequate and a better, universally applicable, theory was required.

3. Multi-electron Atoms

The problem with multi-electron atoms is that, not only are there several getting attracted to the same nucleus, they are repelling each other as well. This repulsion is much harder to deal with. But the worst failure was that electrons just weren’t little charged particles flying around in trajectories as Bohr assumed. They were instead smeared out into waves!

This was first recognized by Louis de Broglie, who first formulated the wavelength equation for matter waves: λ = h/p.

4. Electron Orbitals

Electron waves, like light waves, have nodes, but while light is in constant motion (c), electrons trapped in atoms are stationary waves with fixed nodal patterns.

3.1 Schrödinger Equation

Schrödinger proposed an equation that contains both wave and particle terms. Solving the equation leads to wave functions and their energies. The wave function gives the shape of the electronic orbital. The square of the wave function gives the probability of finding the electron, i.e. it gives the electron density for the .

CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 8 (ALKALI METAL SPECTRA)

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Schrödinger’s equation requires three quantum numbers:

The Principal , n

That integer n from the Rydberg equation turns out to govern the bulk of all electronic energy by being (one more than) the number of nodal surfaces in the wave. Why should that influence energy? Because more nodes mean shorter wavelengths, and shorter wavelengths mean bigger momenta (de Broglie), and bigger momenta mean bigger kinetic energies!

And since the number of nodes is n - 1, the smallest n (= 1) implies the fewest nodes possible is zero.

But where are these nodes? To begin with, they are radial surfaces within the electron matter wave (centred on the nucleus) where the wave has zero amplitude. If n = 3, there are n – 1 = 2 such spherical nodes, but for n = 1, there's none at all, and that matter wave just dies out as one moves away from the nucleus.

As n becomes larger, the atom becomes larger and the electron is further from the nucleus.

CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 8 (ALKALI METAL SPECTRA)

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The Angular Quantum Number, l

But couldn't nodes be on planes as well as spheres? If so, they'd be angular nodes because you wouldn't see them moving away from the nucleus but rather around it! Since nodes have to be either spherical or angular, we get a second quantum number, l, which is the number of these n - 1 nodes that are angular. Since we can't have more angular nodes than there were nodes in the first place, l can never exceed n - 1. But it can take on any integer value from 0 to n - 1 meaning that the number of angular nodes can be from none to all of them.

The , ml

Finally, the detailed geometry of the angles at which the nodes lie is determined by yet a third quantum number, ml. Its name as the magnetic quantum number comes from its being the direction that the electron's points. Revolving charges (an electron with angular momentum about the nucleus) must generate magnetic fields (Maxwell) and their direction up (plus values), down (minus values), or perpendicular to (zero value) some external magnetic field would be energetically different...just as orienting two bar magnetics near one another, you can feel their attractions and repulsions!

CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 8 (ALKALI METAL SPECTRA)

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ml is actually the (quantized) shadow (projection) of the electron's angular momentum, l, along that imaginary external magnetic field. In a vector sense, ml is l's component along that direction. Since the component can never be larger than its vector, ml never exceeds l and never projects more negative than - l. Thus, ml takes on the 2l + 1 integer values from - l to + l.

We thus have the following relationships among values of n, l, and ml through n = 4. The orbitals with l = 0, 1, 2, 3, etc. are called s, p, d and f orbitals. We write the before the orbital, so that 3d, for example refers to an orbital with n = 3, l = 2. There can be 2l + 1 values of ml.

For a one-electron system, we have the following diagram.

CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 8 (ALKALI METAL SPECTRA)

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Orbitals of the same energy are said to be degenerate.

For multi-electron atoms, the s- and p-orbitals are no longer degenerate because the electrons interact with each other. For example, the figure below shows that the s- electrons penetrate more to the nucleus (the small blue peak) and hence feel the attraction of the nucleus more than the p- electrons, which in turn experience the attraction of the nucleus more than the d- electrons.

Therefore, the Aufbau diagram looks slightly different for many-electron systems.

CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 8 (ALKALI METAL SPECTRA)

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Electron Spin

Line spectra of many-electron atoms show each line as a closely spaced pair of lines. Samuel Goudsmit and George Uhlenbeck in Holland proposed that the electron must have an intrinsic angular momentum and therefore a magnetic moment (1925). In order to explain experimental data, they proposed that the electron must have an intrinsic s = ½. [Number of possible values: 2s + 1 = 2: ms= -½ or ms = ½]

Stern and Gerlach designed an experiment to verify this. A beam of atoms was passed through a slit and into a magnetic field and the atoms were then detected. Almost half the atoms were deflected in one direction, and the other half in an opposite direction, almost as if half the electrons were spinning in one direction and others in the opposite direction.

5. Alkali Metal Spectra

Alkali metal atoms have formally the same as a hydrogen atom (ns)1 and should display similar spectra. Only the n quantum number is different for the different alkali . The inner electrons are all paired and do not contribute to the angular momentum. CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 8 (ALKALI METAL SPECTRA)

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The selection rules for the transitions are Δn = anything, Δl = ±1, Δml =0, ±1. The Δml selection rule manifests itself only in the presence of an external field, when the lines are split into their various ml components (Zeeman effect in the presence of a magnetic field). In the absence of an external field, the different ml values are degenerate. The Δl = ±1 selection rule allows transitions from s → p, p → s, d → p and f → d in the emission spectrum. Historically, these lines were labelled sharp, principal, diffuse and fundamental, respectively, and the s, p, d and f notations for the atomic orbitals also originated from these labels. The possible transitions for an atom are usually depicted in a Grotrian diagram, which is shown in Figure 1 for sodium.

Figure 1 Grotrian diagram for sodium

We may now understand the splitting of spectral lines in the hydrogen spectrum. Since both the orbital angular momentum (represented by l) and the spin angular momentum (represented by s) are vector quantities, they may couple, giving a total angular momentum represented by a quantum number j. Vectorially

CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 8 (ALKALI METAL SPECTRA)

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! ! ! j = l + s

For example, for the 1s electron, l = 0 and s = ½, so that the total angular momentum is entirely due to the spin (j = ½). According to the selection rules (Δn = unrestricted, Δl = ±1), the first allowed transition from 1s is to the 2p orbital. Now, l = 1, s = ½ and the angular momenta may couple in two ways. The possible values of the j quantum number are given by

j = l + s , l + s −1,...l − s ,

so that there are two values 3/2 and ½ for j. The two states have slightly different values of their energies and hence the splitting of lines in the spectrum is readily explained. Spin-orbit coupling thus gives rise to a splitting of lines, called fine structure.

The splitting of lines is not so obvious for hydrogen, and only spectrometers with the highest resolving power can detect them. The splitting increases with increasing principal quantum -1 number, and for sodium, we have the famous D1 and D2 spectral lines, separated by 17.2 cm . In contrast, the splitting for H is only 0.243 cm-1 and that for Cs is 554.1 cm-1.

Just as for the l and s quantum numbers, each value of j is associated with 2j + 1 values of mj, which are degenerate in the absence of an external field. The selection rule for j is Δj = 0, ±1, with the caveat that j = 0 ↔ 0 transition is forbidden on account of conservation of angular momentum.

Electronic Configurations and Atomic Term Symbols

The complete electron configuration of sodium is 1s2 2s2 2p6 3s1, for it is 1s2 2s2 2p2. As the atomic number increases, it becomes more and more cumbersome to write down the complete 2S+1 electron configuration of an atom. The term symbol LJ is a more succinct way of writing down the angular momentum coupling in an atom. It contains three pieces of information: 2S + 1 signifies the spin , or simply the multiplicity, of the atom arising from the electron configuration. The symbol L refers to the orbital angular momentum and the symbols used are S, P, D, F, etc. for L = 0, 1, 2, 3, …The total angular momentum is given by the symbol J. Notice the use of capital letters for the term symbol in place of lower case letters used for single electrons.

For example, for H with the electron configuration 1s1, the total spin is ½ (single electron), so S = ½ and consequently 2S + 1 is 2, and the state is termed a doublet. All atoms with a single

CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 8 (ALKALI METAL SPECTRA)

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unpaired electron, therefore, have doublet states. Similarly, the states are termed singlet, triplet, quartet, quintet, etc. according to their S values (0, 1, 3/2, 2, etc., respectively). Usually L < S and hence the multiplicity is given by 2S + 1, though the multiplicity is 2L + 1 if S < L. Since the single electron is in the s orbital, so L = 0 and the symbol is S. The only possible J value is ½, 2 1 and so the term symbol is S1/2. Since alkali metals also have a similar ns configuration (all inner electrons are paired and do not contribute to the angular momentum), their term symbol is also 2 S1/2. In order to distinguish from hydrogen, sometimes the principal quantum number is written before the term symbol. With this notation, the term symbols for hydrogen, lithium and sodium 2 2 2 are 1 S1/2, 2 S1/2 and 3 S1/2, respectively.

6. Summary

• Although the could explain the coarse features of the hydrogen atomic spectrum, it failed to explain the final structure. • The quantum theory led to the exact solution of the hydrogen atom, with three quantum numbers. • However, a fourth, spin, quantum number had to be introduced to explain the finer details. • Coupling of the orbital and angular momenta could explain the appearance of doublets in the spectra of hydrogen and hydrogen-like ions. • The splitting of lines in the alkali metal spectra could also be readily explained.

CHEMISTRY PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 8 (ALKALI METAL SPECTRA)