# Early Quantum Theory | General Chemistry 1

Early quantum theory is studied in this chapter: properties of waves, wave model of light, quantization of energy, photons and photoelectric effect, atomic line spectra, electronic transition, the line spectrum of hydrogen, Rydberg-Balmer equation, wave-particle duality

## Wave Model of Light

Light:

An electromagnetic radiation in the portion of the electromagnetic spectrum. Visible light is only a small part of this spectrum. All light is a transmission of energy in the form of waves and is characterized by its wavelength, frequency, and amplitude

The different forms of electromagnetic radiation are:

• gamma rays (< 10-2 nm)
• X rays (10-2 nm – 101 nm)
• ultraviolet light (101 nm – 400 nm)
• visible light (blue 400 nm - 750 nm red)
• infrared light (750 nm – 5 x 105 nm)
• microwaves (5 x 105 nm – 108 nm)
• radio waves (> 108 nm)

Properties of waves:

• Wavelength (λ) is the distance between identical points on successive waves
• Frequency (ν) is the number of waves that pass through a point per unit time
• Amplitude is the vertical distance between the centerline of a wave and the peak
• Period (T) is the measure of the time it takes for the wave to complete its cycle

Relationship between speed of light, wavelength and frequency:

c = λν

c (in m.s-1) = speed of light = 2.9979 x 108 m.s-1
λ (in m) = wavelength
ν (in s-1) = frequency

Electromagnetic wave:

A wave that has both electrical and magnetic components. These 2 field components are mutually perpendicular and in phase

## Photons and Photoelectric Effect

Quantization of energy

In 1900, Max Planck proposed that the energy of light can only have certain values. A quantum is the smallest amount of energy that can be emitted or absorbed as electromagnetic radiation

Energy E of a single quantum (in J):

E = hν =  $\frac{\mathrm{hc}}{\mathrm{\lambda }}$

h = Planck’s constant = 6.626 x 10-34 J.s
ν (in s-1) = frequency
c (in m.s-1) = speed of light = 2.9979 x 108 m.s-1
λ (in m) = wavelength

Photons

A quantum of light is called a photon (light particle). Photons are therefore the smallest possible packets of electromagnetic energy

Energy E of a group of photons (in J):

E = nhν

n = number of photons
h = Planck’s constant = 6.626 x 10-34 J.s
ν (in s-1) = frequency

Photoelectric effect:

A phenomenon in which electrons are ejected from the surface of a metal exposed to light of a certain minimum frequency called the threshold frequency ν0

Kinetic energy Eof the ejected electrons (in J):

Ek = 0 (when ν < ν0)

Ek = hν - hν(when ν > ν0)

h = Planck’s constant = 6.626 x 10-34 J.s
ν0 (in s-1) = threshold frequency
ν (in s-1) = frequency of photons
E = hν0 = minimum energy required to eject an electron

## Atomic Line Spectra

Emission spectrum:

The light emitted by a substance in an excited electronic state. It can be either a continuum, comprising all wavelengths in a particular range, or a line spectrum, consisting of only certain discrete wavelengths

White light has no gaps ⇒ continuous spectrum
Atoms absorb or emit energy at only specific wavelengths ⇒ line spectrum

Electronic transition:

The change of an electron from one energy level to another within an atom. Atoms emit or absorb electromagnetic radiation when they undergo electronic transitions. The energy of the transition from the initial excited state ni to the final state nf (where Ei > Ef ⇒ emission) is given by Ei = Ef + Ephoton

The ground state is the lowest possible energy state for an atom. An excited state is any energy level higher than the ground state

## The Line Spectrum of Hydrogen

Energy of an electron

Electrons are allowed to occupy only certain orbits of specific energies ⇒ their energy is quantized. The energy of the orbitals of the hydrogen atom is given by:

En = -2.1799 x 10-18 $\left(\frac{1}{{\mathrm{n}}^{2}}\right)$​​​

En (in J) = energy of the orbital n
n = 1, 2, 3 ... = orbital number

En values are the energy states of electrons in a hydrogen atom. The higher the absolute value of En, the more stable the electron in the orbital n. The orbital n = 1 has the more stable electrons, this is the ground state. An electron in an orbital n > 1 is said to be in an excited state

Emission - Energy of a photon:

When the electron moves from a higher energy state to a lower energy state, the atom emits photons. The difference between the energies of the initial (ni) and final (nf) states is:

ΔE = Ef – Ei

ΔE = - 2.1799 x 10-8

The Rydberg-Balmer equation

The Rydberg-Balmer equation is a mathematical formula used to predict the wavelength of light resulting from the movement of an electron between energy levels in an atom. The transition results in the emission of a photon of frequency ν and energy hν

|ΔE| = Ephoton = hν = $\frac{\mathrm{hc}}{\mathrm{\lambda }}$

$\frac{\mathrm{hc}}{\mathrm{\lambda }}$ = ​​​​​2.1799 x 10-18

The Rydberg-Balmer equation predicts line spectrum of hydrogen atom:

λ = wavelength (in m)
${\mathrm{R}}_{\infty }$ = Rydberg constant =  = 1.097 x 10-7 m-1

## Wave-Particle Duality

The de Broglie theory:

The de Broglie equation is one of the equations commonly used to define the wave properties of matter. Through phenomena observed from light, de Broglie suggested that matter has properties similar to particles and waves and obeys to the equation:

λ = $\frac{\mathrm{h}}{\mathrm{p}}$ = $\frac{\mathrm{h}}{\mathrm{mv}}$

λ (in m) = de Broglie wavelength
h = Planck’s constant = 6.626 x 10-34 J.s
p (in kg.m.s-1) = momentum = mv (mass in kg x velocity in m.s-1)

Shortly after de Broglie's proposal, experiments showed that electrons also exhibit wavelike properties as diffraction

Light in science is an electromagnetic radiation in the portion of the electromagnetic spectrum. Visible light is only a small part of this spectrum. All light is a transmission of energy in the form of waves and is characterized by its wavelength, frequency and amplitude

The different forms of electromagnetic radiation are:

• gamma rays (< 10-2 nm)
• X rays (10-2 nm – 101 nm)
• ultraviolet light (101 nm – 400 nm)
• visible light (blue 400 nm - 750 nm red)
• infrared light (750 nm – 5 x 105 nm)
• microwaves (5 x 105 nm – 10nm)
• radio waves (> 108 nm)

The most common characteristics of waves are wavelength, frequency, amplitude, and period.

• Wavelength (λ) is the distance between identical points on successive waves
• Frequency (ν) is the number of waves that pass through a point per unit time
• Amplitude is the vertical distance from the centerline of a wave to the peak
• Period (T) is the measure of the time it takes for the wave to complete its cycle

Frequency and wavelength are inversely proportional to each other:

λ = $\frac{\mathrm{c}}{\mathrm{\nu }}$

c (in m.s-1) = speed of light = 2.9979 x 108 m.s-1
λ (in m) = wavelength
ν (in s-1) = frequency

A quantum is the smallest amount of energy that can be emitted or absorbed as electromagnetic radiation

Energy E of a single quantum (in J):

E = hν =  $\frac{\mathrm{hc}}{\mathrm{\lambda }}$

h = Planck’s constant = 6.626 x 10-34 J.s
ν (in s-1) = frequency
c (in m.s-1) = speed of light = 2.9979 x 108 m.s-1
λ (in m) = wavelength

A quantum of light is called a photon (light particle). Photons are therefore the smallest possible packets of electromagnetic energy

Energy E of a group of photons (in J):

E = nhν

n = number of photons
h = Planck’s constant = 6.626 x 10-34 J.s
ν (in s-1) = frequency

The photoelectric effect is a phenomenon in which electrons are ejected from the surface of a metal exposed to light of a certain minimum frequency called the threshold frequency ν0

Kinetic energy Eof the ejected electrons (in J):

Ek = 0 (when ν < ν0)

Ek = hν - hν(when ν > ν0)

h = Planck’s constant = 6.626 x 10-34 J.s
ν0 (in s-1) = threshold frequency
ν (in s-1) = frequency of photons
E = hν0 = minimum energy required to eject an electron

The emission spectrum is the light emitted by a substance in an excited electronic state. It can be either a continuum, comprising all wavelengths in a particular range, or a line spectrum, consisting of only certain discrete wavelengths

The change of an electron from one energy level to another within an atom. Atoms emit or absorb electromagnetic radiation when they undergo electronic transitions. The energy of the transition from the initial excited state ni to the final state nf (where Ei > Ef ⇒ emission) is given by Ei = Ef + Ephoton

The ground state is the lowest possible energy state for an atom. An excited state is any energy level higher than the ground state

The energy of the orbitals of the hydrogen atom is given by:

En = -2.1799 x 10-18 $\left(\frac{1}{{\mathrm{n}}^{2}}\right)$​​​

En (in J) = energy of the orbital n
n = 1, 2, 3 ... = orbital number

The Rydberg-Balmer equation is a mathematical formula used to predict the wavelength of light resulting from the movement of an electron between energy levels in an atom. The transition results in the emission of a photon of frequency ν and energy hν:

$\frac{\mathrm{hc}}{\mathrm{\lambda }}$ = ​​​​​2.1799 x 10-18

The Rydberg equation only works for hydrogen and species with one electron

The de Broglie equation is one of the equations commonly used to define the wave properties of matter. De Broglie suggested that matter has properties similar to particles and waves and obeys the equation:

λ = $\frac{\mathrm{h}}{\mathrm{p}}$ = $\frac{\mathrm{h}}{\mathrm{mv}}$

λ (in m) = de Broglie wavelength
h = Planck’s constant = 6.626 x 10-34 J.s
p (in kg.m.s-1) = momentum = mv (mass in kg x velocity in m.s-1)