Periodic Law of Atoms (English)

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3-3: The Periodic Law of Elements

In the preceding section,

we have learned

that the structure of

hydrogen atom can completely be

explained by quantum mechanics.

Then, how about

the other kinds of elements?

To see this,

the periodic law of elements

seems one of the most suitable subjects.

In the present section,

let us see that the periodic law

is beautifully derived

from the shell structure

of atoms.

[The Periodicity]

A Russian chemist,

D. I. Mendeleev

(Russia, 1834 - 1907)

discovered

the periodic law of elements

that the properties

of the elements

are periodic functions

of their atomic weights (1869),

i.e., by arranging the elements

in order of increasing

atomic weights,

elements having similar properties

occur at fixed intervals.

On the basis of this

periodicity of elements,

the periodic table

arranged the elements

in order of atomic numbers

to emphasize the chemical

relationships between the elements

with similar configurations.

Since the periodic table

must be very familiar to you

in textbooks of chemistry

in your highschool time,

its details will be omitted here

except for a few examples

of the periodicity of the elements

as shown below.

Some examples

of the periodicity

of the elements

are shown in

Fig. (A) and

Fig. (B).

We can find similar features

in other various experimental data.

What

Fig. (A).

shows

is the ionization energy

which is necessary to

ionize an atom

by removing an electron

out of an atom.

The elements

He, Ne, Ar, Kr, Xe and Rn,

are especially difficult

to be ionized

and they are called the

noble gases or

rare gases.

On the other hand,

the neighboring elements

whose atomic numbers

are larger by one than those

of the noble gases

are the alkali metals,

Li, Na, K, Rb, Cs and Fr,

which have high chemical

activity and

are quite easy

to combine with other elements.

Fig. (B)

shows the atomic radii

estimated from the interatomic distances

in crystals.

This data shows

that the atomic radii

of the alkali metals

are remarkably large.

We can see from these data

that the atomic numbers

= 2, 10, 18, 36, 54 and 86

may have a special meaning

which can be explained

by quantum mechanics

as discussed below.

[Shell Structure of Atoms]

As mentioned before,

the eigenstates

of a hydrogen atom

are specified by

a set of quantum numbers,

n ,

l ,

and

and the energy eigenvalues, ,

are determined only by

an integer n.

(See Eq. (5) on the preceding page.)

For example, the ground state

is of

n = 1 and

l = m = 0.

The first excited states

are of

n = 2,

but there are four

possible states of

l = m = 0, and

l = 1 and

m = -1, 0, 1.

These four states have

the same energy eigenvalue.

Namely, the first excited

states are "fourfold".

Generally, such "multifold" states

are called degenerate states,

and its multiplicity is called

degeneracy.

So that the first

excited states of a hydrogen atom

are quadruply degenerate.

In the hydrogen atom,

all the eigenstates

with the same quantum number

are degenerate,

and their degeneracy is given by .

The reason why

such degenerate excited states occur

in the hydrogen atom

is because the force

exerted on the electron

is purely the Coulomb force

from the nucleus (proton).

In the general kinds

of atoms other than hydrogen,

the situation

is somewhat different.

There the force exerted

on electrons

are not pure Coulomb

atractive force

between the electron

and the nucleus (proton),

but there are many interactions

being worked from other electrons.

Even in general atoms

other than hydrogen,

although the eigenstates

of electrons are still

specified by the quantum numbers

the energy levels

would be splitted.

Although the energy eigenvalues

in the hydrogen atom

are determined only by

n ,

those in general atoms

other than hydrogen

are determined by the quantum numbers

and

l ,

(but do not depend on

m ).

The energy levels

of the general atoms

are listed in

Table 1 below

in ascending order

from the lowest one.

And their schematic drawing is

shown in

Fig. (C).

Fig. (C): The energy levels of electrons in hydrogen atom and other atoms in general

The energy levels

in the hydrogen atom and

the general atom are compared

in this figure.

Here, the energy levels

on the right-hand side

are of the hydrogen atom

and those on the left-hand side

the general atoms

other than hydrogen.

The detailed explanation

is given below.

The energy levels

in the hydrogen atom and

the general atom are compared

Fig. (C),

in which the energy levels

on the right-hand side

are of the hydrogen atom

and those on the left-hand side

the general atoms

other than hydrogen.

Every level is specified

by the quantum numbers

and l .

The states with

l = 0, 1, 2, 3, ...

are denoted by

symbols,

s, p, d, f, ... ,

respectively.

In the hydrogen atom,

the energy eigenvalues

of the states with the same

but different l

are degenerate

(of a equal energy).

However, in general atoms

other than hydrogen,

this degeneracy is broken

and these energy levels

are splitted.

The energy splitting

are seen in

Fig. (C),

in which the relative

position or order

of the levels

is roughly drawn

and you should ignore

the absolute values

of the energy eigenvalues.

The circled numbers in

Fig. (C),

show the total electron numbers

below which all levels

are occupied or

filled up by electrons.

These figures exactly

reproduce the atomic numbers

of the noble gases.

The same number

of electrons as the atomic

number Z

are filled in the levels

in ascending

(or increasing) order

of energies

of the levels

and thus an atom

might be made up.

Roughly speaking,

the "spatial" positions

of these levels

would be queued up

(or layered)

from inner to outer layers.

This might look something like

the layer structure in an onion.

Each layer is called

a shell

and this type of structure

the shell structure.

[The Pauli Principle, Spin]

How many electrons can be filled in each shell ?

In quantum mechanics,

different electrons are not

distinguished.

Not only electrons

but all kinds of

identical particles

are indistinguishable

in quantum mechanics

when being interchanged.

Considering this

indistinguishability

of identical particles,

all kinds of particles

are classified

into two groups;

one is the group of fermions

and the other the group

of bosons,

and

all particles in quantum mechanics

are either fermions

or bosons.

Which group,

the fermion group

or the boson group,

a particle belongs to

is uniquely dependent

on the kind of the particle.

Such a property of particles

is called the statistics.

For example, electrons

and protons are fermions,

and photons are bosons.

Now, let us go back

to electrons

that are fermions.

In the case of fermions,

no two identical fermions can be in the same quantum state.

In other words,

only 0 or 1 fermion can enter into one quantum state.

This rule is called the

Pauli principle,

which was first proposed

in 1925 by

W. Pauli

(Austria, Switzerland, 1900 - 58).

It is also called the

Pauli exclusion principle.

Readers who want to learn

more details of such statistics

are recommended to read

some other textbooks

on quantum mechanics.

The lowest-energy level

is 1s as seen in

Fig. (C).

This 1s level is not degenerate.

There is only one state

with the same energy eigenvalue.

According to the Pauli principle,

only one electron

can therefore stay

in this 1s level.

However, two electrons

can occupy this level

in practice.

The reason is because

there is another degree of freedom

called spin

which was proposed

by G. Uhlenbeck

(The Netherlands)

and

S. A. Goudsmit

(The Netherlands)

in 1925.

More details concerning

the spin would be referred to

other textbooks

of quantum mechanics.

Thus, if both the Pauli principle

and the spin

are taken into account,

then one state characterized

by the quantum numbers

n ,

l and

could contain a maximum

of two electrons.

The maximum numbers

of electrons which

can be contained

in a level characterized

by the quantum numbers

and

are listed in the fourth column,

"Max of electron number",

in the above Table 1.

[Closed Shells, the Noble Gases, the Periodic Law]

Filling Z

electrons into the electron levels

in Table 1 or

in Fig. (C)

in ascending order

from the lowest-energy level,

we can make up the

electron configuration

of an atom with

the atomic number Z.

For example,

filling one electron

in the lowest-energy

1s shell (level),

we have a hydrogen atom, H.

If we fill two

electrons in the same 1s shell,

then this shell would be

filled up to be

a closed shell

according to the Pauli principle.

This is a helium

atom, He.

Because the energy

of the next level

is rather high,

the helium atom, He,

with Z = 2

is quite stable.

This is the most

typical noble gas.

If we put an electron

in the next 2s shell

(level),

we would have

the electron configuration of

the lithium atom,

Li.

Lithium is one of

the alkali metals

which is

of very high chemical activity.

If the 2p shell

is filled up to be

the closed shell

after putting electrons

successively in the levels,

the total electron number

would be 10

and we have the next

noble gas of Z

= 10,

which is neon, Ne.

The next is a sodium

atom, Na,

whose configuration

is that one electron

is contained

in the next 3s shell.

Sodium is also one of

the alkali metals

and has

quite high chemical activity.

Thus, as the atomic number

increases,

we would successively

have the closed shells

which correspond

to the noble gases,

and one electron

being added, we would have

very active elements,

i.e. the alkali metals,

and another electron

being increased in addition,

we have the alkaline earth metals

that are of rather

high activity.

The elements just before

the closed shells

are also chemically

very active and called

the halogens

which have one less electrons

than the noble gases.

In this way,

as the electron number increases,

the property of the element varies

periodically.

This is the periodic law of elements.

Thus we conclude that

the periodic law

is completely explained

by quantum mechanics.

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