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 within the portion of the electromagnetic spectrum. Visible light is only a small portion of this spectrum. All lights constitute the transmission of energy in the form of waves and are characterized by their 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 successives waves
  • Frequency (ν) is the number of waves that pass through a point per unit time
  • Amplitude is the vertical distance from the midline of a wave to the top of the peak

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 electric and magnetic components. These 2 field components are both 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 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 =  hcλ

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

 

Photons

A quantum of light is referred to as a photon (particle of light)

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, 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, including all the wavelengths within a particular range, or a line spectrum, consisting only of certain discret wavelengths
 

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

 

Electronic transition:

A 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 initial excited state ni to 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 1n2​​​

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 1nf2 - 1ni2

 

 

The Rydberg-Balmer equation

The transition results in the emission of a photon of frequency ν and energy hν

|ΔE| = Ephoton = hν = hcλ

hcλ = ​​​​​2.1799 x 10-18 1nf2 - 1ni2

 

 

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

1λ = R 1nf2 - 1ni2

λ = wavelength (in m)
R = Rydberg constant = 2.1799 × 10-18hc = 1.097 x 10-7 m-1

Wave-Particle Duality

The de Broglie theory:

Through phenomena observed from light, de Broglie suggested that matter has properties similar to particles and waves and obeys to the equation:
 

λ = hp = hmv

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