# 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

## Nature of Light

Light:

Light can be defined as a form of electromagnetic radiation that encompasses a wide range of wavelengths within the electromagnetic spectrum. While visible light is a small portion of this spectrum, including wavelengths between approximately 400 nm (violet) to 750 nm (red), light also extends beyond the visible range, encompassing other forms of electromagnetic radiation.

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 (400 nm - 750 nm)
• infrared light (750 nm – 5 x 105 nm)
• microwaves (5 x 105 nm – 108 nm)
• radio waves (> 108 nm)

Properties of waves:

Light exhibits wave-like behavior, characterized by its wavelength (λ), frequency (ν), and amplitude:

• Wavelength (λ) is the distance between successive wave peaks.
• Frequency (ν) is the number of wave peaks that pass a given point per unit time.
• Amplitude is the height of the wave maximum from the center.

Relationship between speed of light, wavelength and frequency:

The speed of light (c) in a vacuum is a constant value, approximately 2.9979 x 108 m.s-1. This speed is determined by:

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:

An electromagnetic wave has an electric field and a magnetic field components. These 2 field components are mutually perpendicular and in phase (same wavelength and frequency).

## Photons and Photoelectric Effect

Quantization of energy

In 1900, Max Planck proposed the idea that energy is quantized, meaning it can only exist in discrete, specific amounts called quanta. A quantum is the smallest quantity of energy that can be emitted or absorbed in the form of 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 of the radiation
c (in m.s-1) = speed of light = 2.9979 x 108 m.s-1
λ (in m) = wavelength of the radiation

Photons:

A photon is the quantum of light, representing the smallest possible packet 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 of the radiation

Photoelectric effect:

The photoelectric effect is a phenomenon where electrons are ejected from the surface of a metal when exposed to light of a certain minimum frequency, known as 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 = hν = energy of the incident photon
E = hν0 = minimum energy required to eject an electron

## Atomic Line Spectra

Emission spectrum:

An emission spectrum refers to the light emitted by a substance when its electrons transition from higher to lower energy levels. It can manifest as either a continuum or a line spectrum:

• A continuum spectrum covers a broad range of wavelengths without any gaps, as seen in white light.
• In contrast, a line spectrum consists of discrete, specific wavelengths, indicating that atoms absorb or emit energy only at certain distinct frequencies.

Electronic transition:

Electronic transitions occur when electrons within an atom move from one energy level to another. During these transitions, atoms either emit or absorb electromagnetic radiation. The energy of the transition, Ephoton​, is determined by the difference in energy between the initial excited state (Ei) and the final state (Ef), following the equation: Ei = Ef + Ephoton

The ground state of an atom represents its lowest energy level, while any energy level higher than the ground state is termed an excited state.

## The Line Spectrum of Hydrogen

Energy of an electron in a hydrogen atom:

Electrons in atoms can only occupy certain orbitals with specific energies, a concept known as quantization. Bohr's model showed that the energy of the orbitals in a hydrogen atom is determined by the formula:

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

En (in J) = energy of the nth orbital of a hydrogen atom
n = 1, 2, 3 ... = orbital number (principal quantum number)

• The lowest energy state of an electron in an atom is called the ground state, which corresponds to n = 1. When an electron occupies an orbital n > 1, it is considered to be in an excited state.
• The radius of each circular orbit in Bohr's model depends on n2. As n increases, the orbit radius increases very rapidly. The higher the excited state, the farther the electron is from the nucleus.

Emission - Energy of a photon:

When an electron transitions from a higher energy level (ni​) to a lower one (nf​), the hydrogen atom emits photons. The difference between the energy of the initial state and the final state is:

ΔE = Ef – Ei

ΔE = - 2.1799 x 10-8

Rydberg-Balmer equation:

The Rydberg-Balmer equation predicts the wavelength (λ) of light emitted during electron transitions in a hydrogen atom, and thus the line spectrum of hydrogen atoms. The wavelength is expressed as:

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

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

λ (in m) = wavelength of light emitted or absorbed ​​​​​​
${\mathrm{R}}_{\infty }$ = Rydberg constant =  = 1.097 x 10-7 m-1
ni = integer representing the initial energy level
nf = integer representing the final energy level

## Wavelike Properties of Matter

De Broglie hypothesis:

Proposed by Louis de Broglie, this hypothesis suggests that particles, such as electrons, exhibit both particle-like and wave-like behavior. According to de Broglie, the motion of an electron in an atom can be described as a standing wave, with the circumference of its orbit being a multiple of its wavelength.

Relationship between the circumference of an allowed orbit and the wavelength:

2πr = nλ

r (in m) = radius of an allowed orbit
n = positive integer
λ (in m) = wavelength of the electron wave

De Broglie wavelength formula:

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)

Diffraction of electrons:

Experiments conducted shortly after de Broglie's proposal demonstrated that electrons exhibit wavelike properties similar to light. When a beam of electrons is passed through a narrow slit or around an obstacle, it diffracts, spreading out and interfering with itself. This diffraction pattern consists of alternating bright and dark spots, confirming the wave-like nature of electrons.

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)