Monday, January 30, 2023

Energy Levels: UNIZOR.COM - Physics4Teens - Atoms - Electronic Structure...

Notes to a video lecture on http://www.unizor.com

Energy Levels

In the previous lecture we have addressed how electrons are positioned around nucleus.
We mentioned that, according to Bohr's model, they are only located on stationary orbits within shells and subshells.

We had a nice formula for the maximum number of electrons that can be located on each subshell, depending on its order number, traditionally designated by a letter (subshell #1 is designated letter s, the next #2 is p, then d, f, g etc.) For subshell #m the maximum number of electrons on it is 4m−2.
The number of subshells within shell #N is N.
Consequently, the maximum number of electrons held in each shell is 2·N².

This mathematically perfect picture gets more complicated, when we consider the energy level associated with each subshell.
The main principle of distribution of energy among subshells, that is, the greater radius of a subshell - the greater energy electrons within this subshell possess, is absolutely true.
The problem is, the further we go from the nucleus - the closer to each other are the shells and, subshells of greater radius of a shell #N might overlap with subshells of smaller radius of shell #N+1.

Let's consider a few elements in a sequence of their atomic numbers, that is the number of protons in their nucleus or electrons within shells on different levels.

Hydrogen has only one proton in the nucleus and one electron within the first subshell (designated by a letter s) of the only shell #1.
So, the electron structure of the atom of hydrogen, where the number specifies the shell number and a letter specifies the subshell, where the superscript at the subshell letter indicates the number of electrons within the same subshell, is:
1s1

Helium has two protons in the nucleus and two electrons within the first subshell (designated by a letter s) of the only shell #1.
The subshell s (the first one) can hold maximum of 2 electrons, so everything is fine.
The electron structure of the atom of helium is:
1s2

Lithium has three protons in the nucleus and three electrons.
The first (and only) subshell s of the shell #1 can hold only 2 electrons. Therefore, a new shell #2 should exist and one electron should be in its first subshell s.
The subshell s of the second shell can hold maximum of 2 electrons, so everything is fine.
The electron structure of the atom of lithium is:
1s2 2s1

Let's skip a few elements and consider carbon with atomic number 6.
Its 6 electrons should fill the shell #1 with its only subshell s holding 2 electrons, then the shell #2 with its subshell s holding another 2 electrons, then, considering shell #2 has two subshells, s and p, the second subshell p will hold the remaining 2 electrons.
The subshell p, as the second subshell, can hold maximum of 6 electrons, so we still have room.
The electronic structure of an atom of carbon is, therefore,
1s2 2s2 2p2

Skip a few more elements to silicon with atomic number 14.
We already know that the first shell can hold 2 electrons in one and only subshell s.
The second shell can hold 2 electrons in subshell s and 6 electrons in subshell p.
That totals 10 electrons. To accommodate 4 other electrons the third shell should be used.
Its subshell s will hold 2 electron and subshell p will hold remaining 2 ones.
The electronic structure of silicon is
1s2 2s2 2p6 3s2 3p2

So far, increasing in atomic number is synchronous with sequential filling of shells and subshells within shells.
But let's consider element argon. Its atomic number is 18 and it completely fills one subshell s of the first shell, both subshells of the second shell and two (out of three) subshells of the third shell.
Its electronic structure is
1s2 2s2 2p6 3s2 3p6

The next element is potassium with atomic number 19.
It would be reasonable to expect that, since the second shell's last subshell p is already filled as well as two first subshells of shell #3, and the third shell allows three subshells, the extra electron #19 should be in the third subshell d (that is the third subshell's letter) of the shell #3.
However, the experiments showed that the last electron #19 goes to the fourth shell's first subshell s instead. Why?

Here we face the fact that shells overlap, and higher energy level of the third subshell d of shell #3 exceeds the lower energy level of the first subshell s of shell #4.
That's why the location of electron #19 of potassium is not 3d1 but 4s1.
That is the reason why the electronic structure of potassium is
1s2 2s2 2p6 3s2 3p6 4s1

The order of filling the electronic subshells, as the number of electrons in the atom grows, under normal conditions corresponds to an order of increasing their energy levels. This is known as the Aufbau principle or Madelung or Klechkovsky rule.

According to this principle, and taking into consideration that shells get closer and closer to each other as we move away from a nucleus, while the number of subshells is increasing linearly with the shell number, the order of filling the electronic structure of an atom is not the numerical order of shell/subshell numbers, but as follows: 1s→2s→2p→3s→3p→4s→
→3d→4p→5s→4d→5p→6s→
→4f→5d→6p→7s→5f→
→6d→7p→...

In the lecture "Orbiting Electron" of the chapter "Building Blocks of Matter" of this course we have derived a formula for a total energy of an orbiting electron:
E = −k·e²/(2·r)
where
k is Coulomb constant
e is the electric charge of an electron
r is the radius of an orbit.

From the above formula we see that absolute value of the energy is decreasing with increasing of a radius of an orbit, but, since it's negative, the energy itself is increasing.
As a radius of an orbit of an electron is increasing, its (negative) energy is increasing, getting asymptotically closer to zero. At the same time the number of subshells of each shell is increasing with the shell number.
That explains the overlapping energy characteristic of the shells.

Thursday, January 26, 2023

Electrons and Shells: UNIZOR.COM - Physics4Teens - Atoms - Electronic St...

Notes to a video lecture on http://www.unizor.com

Electrons and Shells

First of all, there are some very complicated theories of the electronic structure of an atom. They are experimentally confirmed and, therefore, are considered as true representation of how electrons are arranged around a nucleus.
These theories are above the level of this course, but certain facts based on them we will mention without any proof.

Recall the Bohr's model of an atom and electrons that are supposed to be on stationary orbits (or, rather, within stationary shells), where they do not emit any energy.

Shells of a larger radius can hold more electrons, and electrons in those higher orbit shells have higher level of energy.

Shells are numbered in order of increasing radius as shell #1, shell #2, ...shell #N.

Any shell can have one or more subshells. The number of subshells in each shell corresponds to this shell's number, that is:
shell #1 has 1 subshell,
shell #2 has 2 subshell,
shell #3 has 3 subshell,
shell #4 has 4 subshell,
shell #5 has 5 subshell,
shell #6 has 6 subshell,
shell #7 has 7 subshell,
etc.

For historical reasons subshells within each shell are not enumerated, but rather assigned a letter. The first 4 subshells are called s, p, d and f, then the letter is assigned in alphabetical order.
So, the first few subshell names are:
s for subshell #1,
p for subshell #2,
d for subshell #3,
f for subshell #4,
g for subshell #5,
h for subshell #6,
i for subshell #7,
etc.

To bring a bit of math into this structure, let's use letter N for a shell's number and letter m for a subshell number within a shell.
Using these symbols, we can state the following:
shell #N has N subshells:
#1, #2,...,#N
.

Each subshells has its own capacity to hold electrons. There are theoretical reasons for this based on Quantum Theory. We just state the result of this theory confirmed by experiments:
subshell #m has 4·m−2 electrons.

So,
subshell #1 (s) has 2 electrons,
subshell #2 (p) has 6 electrons,
subshell #3 (d) has 10 electrons,
subshell #4 (f) has 14 electrons,
subshell #5 (g) has 18 electrons,
subshell #6 (h) has 22 electrons,
subshell #7 (i) has 26 electrons,
etc.

Now let's calculate the maximum number of electrons in each shell.

Shell #1 has 1 subshell #1(s) and, therefore, can hold no more than 2 electrons.

Shell #2 has 2 subshells #1(s), #2(p) and, therefore, can hold no more than 2+6=8 electrons.

Shell #3 has 3 subshells #1(s), #2(p), #3(d) and, therefore, can hold no more than 2+6+10=18 electrons.

These calculations can be generalized in a formula for the maximum number of electrons in shell #N:
Σm∈[1,N](4m−2) = 2·N²

This formula can be easily proven by induction.
Indeed, it's correct for N=1 because
4·1−2 = 2 = 2·1²
Assuming the formula is correct for some number N, let's check it for N+1.
Σm∈[1,N+1](4m−2) =
=
Σm∈[1,N](4m−2)+4(N+1)−2 =
= 2·N² + 4·(N+1)−2 =
= 2·N² + 4·N + 2 =
= 2·(N² + 2·N + 1) =
= 2·(N+1)²

which is the same formula, but for N+1.

Monday, January 23, 2023

Nucleus of Atom: UNIZOR.COM - Physics4Teens - Atoms - Building Blocks of...

Notes to a video lecture on http://www.unizor.com

Nucleus of Atom

In 1897 J.J.Thomson, experimenting with cathode ray tubes, came to a conclusion that atoms contain tiny negatively charged particles.
He had demonstrated that cathode rays consist of negatively charged particles - electrons.
He then suggested a plum pudding model of an atom with these tiny negatively charged particles embedded into positively charged "soup".

In 1911 Rutherford experimented with a beam of positively charged particles (called alpha particles) directed toward a thin foil. He found that some particles go through a foil, while some are reflected back.
As a result, he came up with a planetary model of an atom with positively charged nucleus and negatively charged electrons rotating around a nucleus on a substantial (relatively to a size of a nucleus) distance, so an atom is substantially empty.

In 1917 Rutherford proved experimentally that nucleus of hydrogen atom is present in atoms of all substances he dealt with.
Later on the nucleus of hydrogen atom was called proton - another building block of an atom together with electron.

A few years later Rutherford suggested that another type of particle - an electrically neutral combination of tightly coupled together proton and electron, called by him neutron - must be present inside an atom's nucleus. While this hypothesis was not confirmed experimentally, the term "neutron" remained and used later on.

In 1932 James Chadwick discovered electrically neutral particles emitted from beryllium bombarded by alpha particles. These electrically neutral particles, in turn, were used to bombard paraffin wax and liberated hydrogen nuclei - protons.
That was a proof of existence of electrically neutral neutrons.

Right after that, in 1932, Dmitry Ivanenko and Werner Heisenberg proposed a proton-neutron structure of an atom's nucleus. This model of a nucleus together with electrons outside of a nucleus moving within stationary shells, each with a specific energy level, remains as the main atom's model.

The number of protons in a nucleus of any element under normal conditions is equal to the number of electrons around a nucleus to maintain electric neutrality of an atom. The properties of any element very much depend on this number and, actually, characterize the element's properties. This number is called an atomic number of an element.

The atomic number of a hydrogen is 1, its nucleus contains 1 proton and there is 1 electron outside a nucleus.
The atomic number of a helium is 2, its nucleus contains 2 proton and there are 2 electrons outside a nucleus.
The atomic number of a carbon is 6, its nucleus contains 6 proton and there are 6 electrons outside a nucleus.
The atomic number of a gold is 79, its nucleus contains 79 proton and there are 79 electrons outside a nucleus.
The atomic number of a uranium is 92, its nucleus contains 92 proton and there are 92 electrons outside a nucleus.

The number of neutrons inside a nucleus is also very important, but can vary for the same element.
Thus, a nucleus of an atom of hydrogen, besides one proton, can have no neutrons, one neutron and two neutrons. Their atomic numbers are the same. To differentiate them, another characteristic is used - a sum of the number of protons and the number of neutrons called atomic mass.

Different kinds of the same element with different numbers of neutrons (that is, different atomic mass) are called isotopes. So, there are three isotopes of hydrogen - with atomic masses of 1 (1 proton, no neutrons), 2 (1 proton, 1 neutron) and 3 (1 proton, 2 neutrons).

Isotopes of the same element have close but not identical properties. To fully identify an element, including its isotope, the following notation is used.
The element is identified by its abbreviated Latin name.
For example,
He for helium,
Au for gold ("aurum" in Latin),
Fe for iron ("ferrum" in Latin) etc.

To fully identify an element, in-front of this abbreviated name two indices are used: top for atomic mass (sum of the numbers protons and neutrons) and bottom for atomic number (the number protons).
Examples:
24He for helium
(2 protons and 2 neutrons),
92238U for uranium
(92 protons and 146 neutrons),
79197Au for gold
(79 protons and 118 neutrons),
2656Fe for iron
(26 protons and 30 neutrons).

Positively charged nucleus with certain number of protons keeps the same number of negatively charged electrons inside an atom because opposite charges attract.

Why a nucleus is held together, when similarly charged protons are lumped together and repel each other?

Apparently, there other attractive forces acting specifically on very small distances between particles inside a nucleus, which are stronger than repelling electrical forces. These forces are called nuclear or strong forces.
They are short range forces of attraction between any particles inside a nucleus, protons and neutrons (collectively called nucleons), and they are millions of times stronger than electric forces. They keep nucleons together inside a nucleus.

Bohr's Atom Model: UNIZOR.COM - Physics4Teens - Building Blocks of Matter

Notes to a video lecture on http://www.unizor.com

Bohr's Atom Model

In the previous lecture we briefly mentioned the planetary model of an atom, authored by Ernest Rutherford in 1911 and supported by many physicists at that time. We also mentioned two fundamental problems with this model.

Firstly, a purely theoretical problem with planetary model of an atom was related to the fact that an electron rotating around a nucleus, that is going with centripetal acceleration, should produce oscillations of an electromagnetic field and, therefore, is supposed to lose energy. This would cause its falling onto a nucleus, which destroys the fundamental structure of matter.

Secondly, the spectrum of radiation produced by an electron falling onto a nucleus had to be continuous, which contradicted experimental results that showed discrete spectrum.
Experiments showed that gases emit light when exposed to intensive electric field. The electric field supplies energy to electrons, and they increase the radius of their orbits. Then they spontaneously release this energy as visible light (electromagnetic field oscillations in visible spectrum of frequencies) and lower their orbits. This light, going through a prism, produces distinct spectral lines of monochromatic light specific for each gas and independent of an intensity of the electric field applied to it, gas temperature or density.
This consistency of spectral lines had to be explained and planetary model failed to do it.

In 1900 Max Plank, based on his experiments with radiation caused by heat, has suggested that radiation is carried in chunks, and each chunk has an amount of energy E that depends only on the frequency f of this radiation:
E = h·f, where
h=6.62607015·10−34 m²·kg/s is Planck's constant

At the same time, analyzing the process of photoelectric emission, Albert Einstein used the idea of quantum character of the electromagnetic oscillations to introduce a photon as an indivisible unit of absorbed or emitted electromagnetic energy.

Combining the theoretical knowledge and results of experiments, Niels Bohr suggested a new atom model that seemed to be capable of explaining all the experimental results on a new theoretical foundation.

The Bohr's atom model was an enhancement of the Rutherford's planetary model and repeats its geometrical configuration of a central positively charged nucleus and orbiting around it negatively charged electrons.
The main modification to that model was using a quantum character of energy carried by electromagnetic field oscillations.

The development of quantum concepts of electromagnetic energy was a collective effort of famous physicists Planck, Lorentz, Einstein, Haas, Nicholson and others. Bohr used these concepts, applying them to formulate his model of an atom.

Bohr's model of an atom is based on these main principles:

I. For each type of an atom there are certain stable electron orbits (or shells), called stationary, where electron, as long as it stays on such an orbit, emits no energy.
This proposition contradicts the classical theory of electromagnetism, which states that accelerated electron must emit energy.

II. Each stationary orbit is associated with certain level of energy. The larger the radius of an orbit of an electron - the higher energy it possesses.
Energy is absorbed by an electron, when it jumps from a lower energy shell to a higher energy one.
Energy is emitted by an electron, when it jumps from a higher energy shell to a lower energy one.
The amount of energy absorbed or emitted by an electron when it jumps from one shell to another equals exactly the difference in energy levels of these shells.

III. When electron jumps from an orbit of higher energy level Ehi to an orbit of lower energy level Elo, it emits electromagnetic radiation of frequency f, determined by an equation
Ehi − Elo = h·f
where h is Planck's constant.
Obviously, to jump from an orbit of lower energy level Elo to an orbit of higher energy level Ehi, electron absorbs this amount of energy from outside.

IV. The next principle is more complicated and was formulated by Bohr as follows.
The angular momentum of an electron rotating on a stationary orbit equals to an integer multiple of reduced Planck constant ħ=h/(2π) (Latin letter h with a horizontal stroke).
This principle is known as quantization of angular momentum.
As we know, a momentum of a body of mass m moving along a straight line with speed v equal to m·v.
An angular momentum L of a body of mass m uniformly rotating with linear speed v along a circular trajectory of radius r equals to m·v·r.
So, this principle of Bohr's model can be expressed in an equation
L = m·v·r = n·ħ
where n is a positive integer number and ħ is a reduced Planck's constant.

This was Bohr's hypothesis given based on some experimental facts and certain theoretical derivations from them.
At the same time it corresponded to experimentally obtained formula for radii of electron orbits of an atom of hydrogen suggested by Rydberg (see previous lecture on Rydberg Formula).

The theoretical explanation of this last Bohr's principle that quantizes the angular momentum was suggested later on by de Broglie in 1924.

Contemporary explanation, based on duality of a particle and a wave, can be shortened to the following.

The full energy E of a particle of mass m, using the Theory of Relativity, can be expressed as
E = m·c²
where c is the speed of light.

From the Quantum Theory the energy E of a quantum of light (photon) of frequency f and wave length λ is
E = h·f = h·c/λ
where h is Planck's constant.
Therefore,
m·c² = h·c/λ
m·c = h/λ

Expression p=m·c is a momentum of a particle of mass m moving with speed c.
Therefore,
p = h/λ

When an object of mass m rotates with linear speed v along an orbit of radius r. its angular momentum is
L = p·r = h·r/λ

Electron, rotating around a nucleus, from the wave theory, is analogous to a string fixed at both ends, like on a guitar. The wave length of a sound this string produces must fill the length of a string integer number of times, otherwise it will interfere with itself.
Using this principle, the wave length of an electron λ and a radius of its orbit must be in a relation
2π·r = n·λ
where n any positive integer number.

Using the above, we obtain
L = h·r/λ =
= h·r·n/(2π·r) =
= h·n/(2π) = n·ħ

where
n is any positive integer number and
ħ is a reduced Planck constant.

Orbiting Electrons: UNIZOR.COM - Physics4Teens - Atoms - Building Blocks...

Notes to a video lecture on http://www.unizor.com

Orbiting Electron

Let's analyze the dynamics of an electron rotating around a nucleus of a hydrogen atom on a circular orbit.

Considering the strength of electric forces significantly exceeds the strength of gravitational forces, we will ignore the gravity between a nucleus and an electron.

Assume, the mass of an electron is m (it's a known constant), its negative electric charge is −e (also a known constant) and a radius of its orbit is r (variable).

An atom of hydrogen has only one negatively charged electron. Therefore, to maintain electric neutrality, its nucleus has to have positive electric charge equal in magnitude to a charge of an electron, that is, e.

The Coulomb's Law states that the magnitude of the force of attraction between a nucleus of an atom of hydrogen and its electron equals to
F = k·e·e/r² = k·e²/
where k is a Coulomb's constant.

On the other hand, according to Rotational Kinematics, that same force gives an electron a centripetal acceleration
a = v²/r
where v is a linear speed of an electron circulating around a nucleus.

Applying the Newton's Second Law
F = m·a,
we obtain an equation that connects radius of an orbit, linear speed of an electron, its charge and mass:
F = m·a = m·v²/r = k·e²/
or
m·v²·r = k·e²

The direct consequence of this equation is an expression for a kinetic energy of an electron, as a function of a radius of its orbit:
Ekin = m·v²/2 = k·e²/(2·r)

Potential energy of a negative charge e in the centrally symmetrical electric field of a nucleus carrying a positive charge e of the same magnitude (that is, work needed to bring a charge −e from infinity to a distance r from a central charge +e) is
Epot = −k·e²/r

Total energy of an electron is, therefore,
E = Ekin + Epot = −k·e²/(2·r)

Satellite Speed: UNIZOR.COM - Physics4Teens - Mechanics - Gravity, Weight

Notes to a video lecture on http://www.unizor.com

Satellite Speed

Our task is to find out a linear speed V of a satellite that freely rotates around a planet of mass M on a circular orbit of radius R.

The gravity force, acting on a satellite of mass m and keeping it on a circular orbit with constant angular and linear speed, according to the Law of Universal Gravitation, equals to
F = G·M·m /
where G=6.67·10−11(N·m²/kg²) is a Universal Gravitational Constant.

On the other hand, according to Rotational Kinematics, that same force gives a satellite a centripetal acceleration
a = V²/R

Applying the Newton's Second Law
F = m·a,
we obtain an equation that connects radius of an orbit, linear speed of a satellite and mass of a planet:
F = m·a = G·M·m /

Using the expression of centripetal acceleration above, this results in the following:
m·V² /R = G·M·m /

Notice that mass of a satellite m cancels out and the resulting expression for a satellite linear speed on an orbit is
V² = G·M /R
V = √G·M/R

The above formula allows to calculate the period T of rotation of a satellite - the time required to make a complete circle around a planet:
T = 2πR/V = 2π√R³/(G·M)

Wednesday, January 18, 2023

Rydberg Formula: UNIZOR.COM - Physics4Teens - Atoms - Building Blocks of...

Notes to a video lecture on http://www.unizor.com

Rydberg Formula

Rydberg formula is a perfect example of how science is done.
It all started in 1880's with numerous experimental results of spectral lines of hydrogen, emitted after its atoms are excited by some external energy, like electric field or heat.
These spectral lines had certain wave lengths observed through experiments.

Johann Jacob Balmer attempted to connect the wave lengths of observed spectral lines of hydrogen with some kind of empirical formula and found the one:
λ = B·n²/(n²−2²)
where
λ is a wave length of an observable spectral line,
B=3.6450682·10−7 m is a constant that Balmer has suggested,
n ≥ 3 is a sequence number of a spectral line.

Here are a few first wave lengths λ and colors of Balmer series for different sequence number n
nλ (nm)Color
3656.1Red
4486.0Cyan
5433.9Violet
6410.1Violet
7396.9Violet
8388.8Violet
9383.4Violet

Jumping forward, with introduction of Bohr's model with specific stationary electron orbits with fixed energy levels for each orbit of an atom, it was apparent that Balmer has described electron emitting radiation when jumping from some higher orbit to orbit #2.

When electron jumps from any higher orbit to orbit #1 (the closest to nucleus), the emitted radiation is in ultraviolet part of a spectrum and was not observed by Balmer.

A few years later Johannes Rydberg generalized the Balmer formula and described any jump of an electron in a hydrogen atom from orbit #n to orbit #m:
1/λ = R·(1/m² − 1/n²)
where
n ≥ 2 is an orbit number an electron jumps from,
1 ≤ m ≤ n−1 is an orbit an electron jumps to,
R is Rydberg constant.
For m=2 the Rydberg formula is
1/λ = R·(n² − 2²)/2²·n²)
or
λ = (4/R)·n²/(n² − 2²)
which corresponds to Balmer formula if B=4/R.

All the above formulas are empirical, obtained in the process of analyzing the results of experiments. The theoretical foundation of them would be known only after Bohr introduced his atom model in the beginning of the 20th century and his model would undergo certain improvements based on quantum physics.