Atomic structure

Atomic Structure

Synopsis: Fundamental Particles

• According to Dalton, an atom is the smallest indivisible particle, but discharge tube experiments have proven that atoms consist of even smaller particles.
• Electrons, protons, and neutrons are the fundamental particles of an atom.
• The electron was discovered in the cathode ray experiment.
• Ideal conditions to produce cathode rays in a discharge tube are very low pressure ($0.01 mmHg$) and high electric discharge ($10,000 V$) potential.
• Cathode rays can be deflected by electric and magnetic fields, indicating they are negatively charged particles.
• The particle nature of cathode rays was proven by:
  • Their ability to cause mechanical motion.
  • Photo-electric effect
  • Compton effect
• Cathode rays are negatively charged and consist of electrons.
• Anode rays are positively charged ions.
• Protons and neutrons are present in the nucleus and are called nucleons.
• Protium contains only an electron and a proton. Except for protium, all atoms contain electrons, protons, and neutrons.
• Electrons are negatively charged particles with a unit negative charge and negligible mass.
• Protons are positively charged particles with unit mass.
• The proton was discovered in the anode ray experiment.
• Anode rays, also called canal rays or positive rays, were discovered by E. Goldstein.
• Anode rays contain material particles obtained by the removal of one or more electrons from the gaseous atoms/molecules present in the tube.
• The positively charged particles present in the anode rays produced when Hydrogen gas is present in the discharge tube were called protons by Rutherford (proton = first particle).
• The specific charge on the anode rays was found to be maximum when the gas present in the discharge tube was hydrogen.
• Neutrons are neutral particles with unit mass.

Fundamental Particle Properties

Specific Charge and Atomic Number

• The ratio of charge to mass is called specific charge.
• The electron has the highest specific charge because of its negligible mass.
• The mass of an electron increases with an increase in velocity. Thus, the e/m of an electron decreases with an increase in velocity.
• If an electron moves with a velocity equal to that of light, then its mass becomes infinity and e/m becomes zero.
• The e/m of cathode rays is independent of the nature of the gas in the discharge tube because electrons are universal constituents.
• The e/m of anode rays depends on the nature of the gas in the discharge tube.
• The number of electrons or protons present in an atom of an element is called its atomic number.
• A neutral atom contains an equal number of electrons and protons.
• The atomic number is denoted by Z.
• The atomic number is equal to the nuclear charge of an element.
• Moseley proposed a relationship between the frequencies of the characteristic x-rays of an element and its atomic number: $\nu = a(Z – b)$
  • $\nu$ is the frequency of characteristic x-rays.
  • Z = atomic number
  • 'a' and 'b' are constants with definite values for that element.
• The sum of the number of protons and neutrons in the atom of an element is called its mass number and is denoted by 'A.'
• Number of neutrons = A – Z.
• The mass number is always a whole number.
• Atoms of elements having the same atomic number but different mass numbers are called isotopes.
• Isotopes of an element have the same number of protons and electrons but differ in the number of neutrons.
• Isotopes of an element have the same chemical properties but different physical properties.

Rutherford's Atomic Model

• This model is called the planetary model or nuclear model of the atom.
• Rutherford's atomic model is based on the findings of the alpha-ray scattering experiment.
• An atom is a hollow sphere, and the entire mass and positive charge are concentrated at the center of the atom in the smallest region called the nucleus.
• Electrons revolve around the nucleus in a circular path.
• This model failed to explain the stability of atoms and the line spectra of atoms.
• As per the laws of electrodynamics, an electron moving around the nucleus must radiate energy continuously and must spiral down into the nucleus.
• If an electron radiates energy continuously, the atomic spectrum should be a continuous spectrum.
• The atom should collapse if this happens.
• But all the atoms give line spectra.

Nature of Light

• The two theories that explain the nature of light are:
  • Wave theory
  • Corpuscular theory
• The wave theory of light could satisfactorily explain diffraction and refraction.
• Corpuscular theory could explain the photoelectric effect and Compton effects.
• Wave theory is superior to corpuscular theory.
• Visible light is only a small portion of the electromagnetic spectrum.
• All radiant energy is in the form of electromagnetic waves.
• These radiations are associated with electric and magnetic fields.
• The vertical component of the wave (E) indicates the variation of electric field strength.
• The horizontal component of the wave (H) indicates the variation of magnetic field strength.
• The distance between two successive crests or troughs is called the wavelength ($\lambda$).
• Wavelength is measured in Angstrom units or nanometers.
• $1 Å = 10^{-8} cm = 10^{-10} m$; $1 nm = 10^{-9} m = 10 Å$
• The number of waves passing through a given point in one second is called the frequency of the wave. $\nu = c / \lambda$ Units of frequency is Hz.
• The velocity of light in air or in vacuum is $3 \times 10^{8} ms^{-1}$ or $3 \times 10^{10} cms^{-1}$.
• Frequency * wavelength = velocity; $\nu \lambda = c$
• The reciprocal of wavelength is called wave number. $\bar{\nu} = \frac{1}{\lambda}$
• The units of wave number is $cm^{-1}$ or $m^{-1}$.
• $\bar{\nu} = \frac{1}{\lambda} ; \lambda = \frac{1}{\bar{\nu}} ; \nu = c \cdot \bar{\nu}$
• 'A' is the amplitude of the wave or intensity of the light.
• The intensity of color depends on the amplitude, and the color of the light depends on the frequency. $(I \propto A^2)$

Sources of Different Radiations

• A high-pressure hydrogen or deuterium discharge tube is the source of ultraviolet rays.
• The wavelength range of ultraviolet rays is $1850 Å$ to $3750 Å$.
• To obtain high energetic U.V., a xenon arc lamp or mercury vapor lamp can be used.
• The glass-enclosed tungsten filament is the source of visible radiation. An incandescent lamp is the source of I.R. radiation.
• The best source of near-infrared radiations is a black body.
• To produce far-infrared radiations, a Nernst glower or globar source is used.
• The wavelength of these radiations is about $14,000 Å$.
• A mixture of Zirconium and Yttrium oxides shaped into a small hollow rod is used in the Nernst glower.
• The glower is heated to $1500 °C$ to $2000 °C$.
• The globar source is a rod of sintered silicon carbide, which is heated to $1300 °C$ to $1700 °C$.

Planck's Quantum Theory

• Planck's quantum theory explains black body radiation.
• A hollow sphere coated inside with platinum black and having a pinhole acts as a nearer black body.
• A black body is not only a perfect absorber but also a perfect emitter of radiant energy.
• A black body kept at a high temperature gives radiations in a wide range of different wavelengths.
• Curves are obtained at different temperatures when the intensity of radiations is plotted against wavelength.
• If the energy is emitted continuously, the curve should be as shown by the dotted lines.
• The study of the curves shows that the nature of the radiation depends on temperature.
• At a given temperature, the intensity of radiation increases with wavelength, reaches a maximum, and then decreases.
• As the temperature increases, the peak of the curve shifts to lower wavelengths (i.e., towards the left).
• Based on the above observations of black body radiation, Planck proposed the quantum theory of radiation. The salient features of the theory are:
  • The vibrating particle in the black body does not emit energy continuously.
  • It is emitted in the form of small discrete packets called quanta.
  • The emitted radiant energy is propagated in the form of waves.
  • If the vibrating particles oscillate with a frequency $\nu$, then the energy associated with a quantum is $E \propto \nu; E = h\nu$ ($h$ = Planck's constant. $h = 6.625 \times 10^{-27} ergs-sec$, $h = 6.625 \times 10^{-34} J-sec$)
  • Energy is emitted or absorbed in some simple integral multiples of a quantum, i.e., $E = 1h\nu$ (or) $2h\nu$ (or) $3h\nu$, but not a fractional multiple of $h\nu$.
  • This is called quantization of energy.

Einstein's Generalization of Planck's Quantum Theory

• Planck's quantum theory was extended to all types of electromagnetic radiations by Einstein.
• According to Einstein, energy is released in the form of photons, and they continue to exist as photons until they are absorbed by another body.
• According to Max Planck, energy is emitted in the form of packets and propagated in the form of waves.
• According to Einstein, both emission and propagation of energy take place in the form of photons.
• Einstein explained the photoelectric effect with the help of his generalized quantum theory.
• Emission of electrons from the metal surface when it is exposed to light is called the photoelectric effect.
• Such emitted electrons are called photoelectrons.
• According to Einstein, an electron is ejected from a metal when it is struck by a photon that has sufficient energy.
• If the photon has insufficient energy, it cannot eject the electron, and the photoelectric effect is not observed.
• A photon of violet light has higher energy than that of red light. Violet light can eject electrons from potassium, but red light has no effect.
• When the photon having energy $h\nu$ strikes the metal surface, some part of it is utilized to eject the electron, and the remaining part is utilized to increase the K.E. of the photoelectron.
• If the frequency of incident radiation increases, the K.E. of photoelectrons increases.
• If the intensity of incident radiation increases, the rate of photoelectric emission increases.
• $h\nu = W + K.E$ where:
  • $h\nu$ = energy of the striking photon
  • $W$ = energy required to eject the electron (work function)
  • $K.E.$ = kinetic energy of the emitted electron.

Light Spectra

• Spectrum: It is the pattern of lines produced on the photographic plate by the dispersion of a beam when it is passed through the prism.
• Spectrometer: It is the device used to record the spectrum.
• Spectrograph: It consists of a source of light, prism, and photographic plate.
• Spectra are of two types:
  • Emission spectrum
  • Absorption spectrum

Emission Spectrum

• When the substances are in the excited state, they emit light. The spectrum obtained with this emitted light is called the emission spectrum.
• The emission spectrum is obtained by heating the substances on a flame or by passing an electric discharge through the gases.
• The emission spectrum consists of bright lines on a dark background.

Absorption Spectrum

• It is due to the absorption of light.
• When the substances are in the ground state, they absorb radiation and go to the excited state, the spectrum so obtained is called the absorption spectrum.
• The absorption spectrum consists of dark lines on a bright background.
• In the absorption spectrum, lines are formed at the same wavelengths as those of the emission spectrum.
• The emission spectrum or absorption spectrum is of two types:
  • Continuous spectrum or band spectrum
  • Discontinuous spectrum or line spectrum

Continuous Spectrum

• In this spectrum, the formation of lines is continuous.
• Each color fades into the next color as in a rainbow.
• A beam of white light when passed through a prism gives a continuous spectrum of seven colors, i.e., VIBGYOR.
• An incandescent lamp or hot solids at high temperatures will give a continuous spectrum.

Discontinuous Spectrum

• Line spectrum consists of sharp, distinct, and well-defined lines.
• Gases or vapors of elements when heated in a flame or by passing electric discharge through them, a line spectrum is obtained.
• The line spectrum is given by atoms and so it is called the atomic spectrum.
• Each element has its own characteristic line spectrum, by which the element can be identified.

Band Spectrum

• It consists of a series of bands where each band is a group of lines merged together.
• The band spectrum is given by molecules and so it is called the molecular spectrum.

Hydrogen Spectrum

• It consists of a number of lines.
• They can be classified into various series.
• Only one such series is visible to the naked eye and is termed the visible region of the hydrogen spectrum, i.e., the Balmer series.
• The wavelength or wave number of various lines in the visible region can be expressed by an equation (Reydberg – Ritz equation): $\frac{1}{\lambda} = R \left[ \frac{1}{n<em>1^2} - \frac{1}{n</em>2^2} \right]$
  • where $n<em>1 = 2$ which is constant for all the lines in the Balmer series; $n</em>2 = 3, 4, 5…..$

Series in Hydrogen Spectrum

• The other series in the hydrogen spectrum are invisible.
• The wavelength or wave number of all the lines in all the series can be calculated by using Rydberg's equation or Rydberg-Ritz equation:
  $\frac{1}{\lambda} = R \left[ \frac{1}{n<em>1^2} - \frac{1}{n</em>2^2} \right]$
• The value of $R = 1,09,677 cm^{-1}$ is valid only for the lines in the hydrogen spectrum.
• For a spectral line of one electron species like $He^+$, $Li^{2+}$, the value of $R = 1,09,677 \times Z^2 cm^{-1}$.
• The first line in the Balmer series is called the $H_\alpha$ line, and its wavelength is $6563 Å$.
• The second line is called the $H_\beta$ line, and its wavelength is $4861 Å$.
• The spectral lines get closer when the $n_2$ value is increased.

Bohr's Atomic Theory

• Bohr recognized the relationship between the nature of the series of spectral lines and the arrangement of electrons in the atom.
• Bohr applied Planck's quantum theory to the electrons revolving around the nucleus. He retained the basic concept of Rutherford's model of the atom, that electrons revolve round the positively charged nucleus.
• Bohr proposed his theory to explain the structure of the atom. The important postulates of his theory are:
  • Electrons revolve around the nucleus with definite velocities in concentric circular orbits. These orbits are called stationary orbits as the energy of the electron remains constant. As long as the electron revolves in the same circular orbit, it neither radiates nor absorbs energy.
  • The angular momentum of the electron is quantized. The electronic motion is restricted to those orbits where the angular momentum of an electron is an integral multiple of $h/2\pi$ or $mvr = \frac{nh}{2\pi}$. This is called Bohr's quantum condition or quantization of angular momentum.
  • The energy of the electron changes only when it moves from one orbit to another orbit.
  • Energy is absorbed when an electron jumps from a lower orbit to a higher outer orbit.
  • If the electron is in the 1s orbit, it can only absorb but cannot emit energy.
  • Energy is released when an electron jumps from a higher orbit to a lower orbit.
  • The released or absorbed energy is equal to the difference between the energies of the two orbits.
  • If $E<em>2$ is the energy of the electron in the outer orbit ($n</em>2$) and $E<em>1$ is the energy of the electron in the inner orbit ($n</em>1$), then $E<em>2 – E</em>1 = \Delta E = h\nu$.
  • Where n is called the principal quantum number, and it represents the main energy level.
  • It takes all positive and integral values 1, 2, 3, 4… etc.
• With the help of these postulates, Bohr derived the expression for the radius of the circular orbit, the energy of the electron in a circular orbit, and the velocity of the electron in a circular orbit.
• Bohr's theory could satisfactorily explain the formation of different series of lines in the hydrogen spectrum.
• The wavelengths and the frequencies of the lines determined experimentally are in excellent agreement with those calculated by using Bohr's equation.

Radius of Orbit

• The hydrogen atom contains one proton in the nucleus and one electron revolving around the nucleus in a circular orbit of radius r.
• The electron maintains the same circular motion in a given orbit as centripetal and centrifugal forces are equal in magnitude and opposite in direction.
• Centripetal force = centrifugal force (Columbic forces of attraction provide necessary centripetal force) i.e. $\frac{-e^2}{r^2} = \frac{-mv^2}{r}$
• According to Bohr's quantum condition: $mvr = \frac{nh}{2\pi}; v = \frac{nh}{2\pi mr}; v^2 = \frac{n^2h^2}{4\pi^2m^2r^2}$
• $\frac{e^2}{r} = m \cdot \frac{n^2h^2}{4\pi^2m^2r^2}$
• $r = \frac{n^2h^2}{4\pi^2me^2}$
• The radius of the nth orbit is given by: $r_n = n^2 \left( \frac{h^2}{4\pi^2me^2} \right) = 0.529 \times 10^{-8} n^2 cm$
  • Where h = Planck's constant; m = mass of electron; e = charge of electron; $r_n$ = radius of nth orbit
• The radius of the first orbit of the hydrogen atom is called Bohr's radius, which is denoted by $r<em>0. r</em>0 = 0.529 \times 10^{-8} cm = 0.529 Å$

Energy of Electron

• The total energy of the electron in a stationary orbit is equal to the sum of its kinetic and potential energies.

• Total energy of electron E = K.E + P.E.

• K.E. is always positive and P.E is always negative.

• K.E. is half to that of P.E. in magnitude.

• $E = \frac{1}{2}mv^2 - \frac{e^2}{r} = -\frac{1}{2} \frac{e^2}{r}$ (∵ $\frac{1}{2}mv^2 = \frac{e^2}{r}$)

• Energy of electron for a single electron species is:
  $E<em>n = -\frac{2\pi^2e^4m}{n^2h^2}$$E</em>n = -\frac{k}{n^2} eV/atom$
  $E<em>n = -\frac{2.18 \times 10^{-11}}{n^2} ergs/atom$$E</em>n = -\frac{2.18 \times 10^{-18}}{n^2} J/atom$
  $E<em>n = -\frac{313.6}{n^2} kcal/mole$$E</em>n = -\frac{1312}{n^2} kJ/mole$

• The energy of electron is negative in the atom.

• As the value of n increases, energy increases.

• When n is infinity, the value of E is zero.

• When the n value decreases, the energy of electron also decreases.

Rydberg Constant (R)

• When an electron jumps from an outer energy level ($n<em>2$) to an inner energy level ($n</em>1$), energy is released.
• i.e. $E<em>2 – E</em>1 = \Delta E = h\nu$
• $E<em>2$ = energy of electron in higher orbit ($n</em>2$)
• $E<em>1$ = energy of electron in lower orbit ($n</em>1$)
  $E<em>2 – E</em>1 = \frac{-2\pi^2e^4m}{n<em>2^2h^2} + \frac{2\pi^2e^4m}{n</em>1^2h^2}$
  $\Delta E = \frac{2\pi^2e^4m}{h^2} \left[ \frac{1}{n<em>1^2} - \frac{1}{n</em>2^2} \right]$
  $h\nu = \frac{2\pi^2e^4m}{h^2} \left[ \frac{1}{n<em>1^2} - \frac{1}{n</em>2^2} \right]$
  $\nu = \frac{2\pi^2e^4m}{h^3} \left[ \frac{1}{n<em>1^2} - \frac{1}{n</em>2^2} \right]$
  $\bar{\nu} = \frac{1}{\lambda} = \frac{\nu}{c}$
  $\bar{\nu} = \frac{2\pi^2e^4m}{ch^3} \left[ \frac{1}{n<em>1^2} - \frac{1}{n</em>2^2} \right]$
• $R = \frac{2\pi^2e^4m}{ch^3} = 1,09,681 cm^{-1}$.
• This value Rydberg constant (R) calculated by Bohr as above is in good agreement with the experimental value.

Velocity of Element in the nth Orbit

• As per Bohr’s quantum condition, $mvr = \frac{nh}{2\pi}$
  $v<em>n = \frac{nh}{2\pi mr}$$v</em>n = \frac{ \frac{2\pi e^2}{h}}{n}$ (for ‘H’ atoms or any other single electron species, $v_n = \frac{ \frac{2\pi e^2 z}{h}}{n}$
• Substituting the values of constants,
  $v_n = \frac{2.188 \times 10^{-8} z}{n} cm/sec$
  velocity = $\frac{circumference}{v}$
• Number of revolutions per second, made by the electron in a circular orbit is = $\frac{v}{2\pi r}$

Hydrogen Spectrum – Bohr’s Explanation

• When hydrogen gas is heated or exposed to light energy or subjected to electric discharge, different atoms absorb different amounts of energy, and electrons are excited to different higher energy levels.

• The bright light emitted when passed through a prism and received on a photographic plate is recorded as the atomic spectrum of hydrogen.

• The hydrogen spectrum is the simplest of all the atomic spectra. It is a line spectrum and emission spectrum, and it contains a number of series of lines.

• The electrons in the excited atoms may be completely knocked out of the atom if the absorbed energy is greater than or equal to 13.58 eV, which is the ionization potential of a hydrogen atom.

• If the energy available is less than 13.58 eV, the electron absorbs only a certain quantum of energy, and the electron jumps to a higher orbit.

• The electron in the higher quantum state tends to emit energy and come back to the lower energy level.

• This may happen in a single step or in multiple steps.

• If an electron jumps from any higher orbit to the 1st orbit, a Lyman series is formed in the u.v. region. Electron transitions: $∞ → 1 7 → 1 6 → 1 5 → 1$ and so on.

• If an electron jumps back from any higher orbit to the 2nd orbit, a Balmer series is formed in the visible region. Electron transitions: $∞ → 2; 7 → 2, 6 → 2$……

• If an electron jumps back from any higher orbit to the 3rd orbit, a Paschen series is formed in the near I.R. region. Electron transitions: $∞ → 3; 7 → 3, 6 → 3$; and so on.

• If an electron jumps back from any higher orbit to the 4th orbit, a Bracket series is formed in the I.R. region. Electron transitions: $∞→4; 7→4, 6 → 4$ and so on.

• If an electron jumps back from any higher orbit to the 5th orbit, a Pfund series is formed in the far I.R region. Electron transitions: $∞ → 5; 7 → 5, 6 → 5$

• If an electron jumps back from the infinite state to the corresponding lower orbit, the spectral line is called a limiting line or limiting series
  $v = R \left( \frac{1}{n_1^2} - \frac{1}{∞^2} \right)$
  $v = \frac{R}{n^2}$

• In a given series, the line of longest wavelength is the 1st line (2→1), and the line of shortest wavelength is the limiting line.

• In all the five series of the H-spectrum, the line of longest wavelength is the 1st line of the Pfund series (6→5), and the line of shortest wavelength is the limiting line of the Lyman series ($\infty$ to 1).

• No. of possible spectral lines $\frac{n(n +1)}{2}$; $n = n<em>2 – n</em>1$

• As the value of 'n' increases:

  • i) the total energy of the electron increases
  • ii) the energy difference between the successive orbits decreases
  • iii) P.E increases and K.E. decreases
  • iv) the radius of orbits increases
  • v) the velocity of the electron decreases

Merits of Bohr's Theory

• He could explain the spectra of H-atom and other single electron species like $He^+$, $Li^{2+}$ etc.

• He could determine the frequency, wavelength, and wave number of lines in the H-spectrum.

• He could calculate the value of Rydberg constant (R).

• He could determine the energy and velocity of the electron and the radius of orbits.

• He could explain the stability of atoms, that is why electrons are not falling into the nucleus and atoms are not collapsed.

Demerits of Bohr's Theory

• Bohr failed to explain the spectra of multi-electron species.

• He failed to explain the fine structure of the H-spectrum.

• He failed to consider the wave number of the electron.

• Bohr's theory contradicts Hisenberg's uncertainty principle.

Quantum Numbers

• To fully explain the motion of an electron and to locate its correct address, the following four quantum numbers are required.

Principal Quantum Number (n)

• It is proposed by Bohr and denoted by 'n'.

• It represents the main energy level.

• It determines the size of the orbit and the energy of the electron.

• It takes all positive and integral values from 1 to n.

• The maximum number of electrons in a main energy level is $2n^2$, and the number of orbitals is $n^2$.

• If ‘n’ is the principal quantum number, the energy of the electron in the principal quantum level is
  $E<em>n = -\frac{2\pi^2e^4m}{h^2} \frac{1}{n^2}$$E</em>n = -\frac{13.6}{n^2} eV/atom$

• As the value of 'n' increases, the energy of the electron increases.

• The energy of the electron in the ground state of hydrogen atom is – 13.6 eV/atom (or) –2.176 x10–11 erg per atom (or) –2.176 \
  10–18 joule per atom (or) –1312 kJ per mole (or) –313.6 kcal per mole.
  The energy of the electron in the second orbit of hydrogen atom is $\frac{13.6}{2^2} = -3.4 eV/atom$

Azimuthal Quantum Number (ℓ)

• It is also known as the angular momentum quantum number or orbital quantum number (or) subsidiary quantum number.

• To express the quantized values of the orbital angular momentum, the azimuthal quantum number was proposed.

• It is denoted by ℓ and takes values from 0 to n – 1.

• The number of values of ‘ℓ’ is equal to the value of n.

• It explains the fine structure in the H-spectrum.

• It determines the shape of orbitals.

• More fine lines in each main spectral line are seen.

• If n = 1, ℓ = 0 (s - sub-shell)

• If n = 2, ℓ = 0, 1 (s, p sub-shells)

• If n = 3, ℓ = 0, 1, 2 (s, p, d sub-shells)

• If n = 4, ℓ = 0, 1, 2, 3 (s, p, d, f - sub-shells)

• The orbital angular momentum of electron = $\frac{h}{2\pi}$

• The azimuthal quantum number determines the shape of the orbital.

• The number of orbitals in a subshell is (2 ℓ + 1).

• The maximum number of electrons in a subshell is 2(2 ℓ + 1).

Magnetic Quantum Number

• To explain Zeeman and Stark effects, Lande proposed the magnetic quantum number.

• It is denoted by m.

• It represents the sub-sub energy level or atomic orbital.

• It determines the orientation of the orbital in space.

• When the atom is placed in an external magnetic field, the orbit changes its orientation.

• The number of orientations is given by the values of the magnetic quantum number m.

• m takes the values from – ℓ to + ℓ through 0. Total values of m for a given value of ℓ = (2 ℓ + 1) values.

• A subshell having azimuthal quantum number ℓ, can have (2 ℓ + 1) space orientations.

• The number of orbitals in a subsheII = (2 ℓ + 1).

• If the changes in the axis in one direction are indicated by + m values, the changes in the axis in the opposite direction are indicated by – m.

Spin Quantum Number (s)

• In the fine spectrum of alkali metals, pairs of widely separated lines are observed which are different from duplet, triplet, and quadruplets observed in the hydrogen spectrum.

• To recognize and identify these pairs of lines, Goudsmit and Uhlenbeck proposed that an electron rotates or spins about its own axis.

• This results in the electron having spin angular momentum, which is also quantized