# Early Quantum Theory | General Chemistry 1

## 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 – 10^{1}nm) - ultraviolet light (10
^{1}nm – 400 nm) - visible light (blue 400 nm - 750 nm red)
- infrared light (750 nm – 5 x 10
^{5}nm) - microwaves (5 x 10
^{5}nm – 10^{8 }nm) - radio waves (> 10
^{8}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 10^{8} 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 10^{8} 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 ν_{0}

Kinetic energy E_{k }of the ejected electrons (in J):

E_{k} = 0 (when ν < ν_{0})

E_{k} = hν - hν_{0 }(when ν > ν_{0})

h = Planck’s constant = 6.626 x 10^{-34} J.s

ν_{0} (in s^{-1}) = threshold frequency

ν (in s^{-1}) = frequency of photons

E = hν_{0} = 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 n_{i} to the final state n_{f} (where E_{i} > E_{f} ⇒ emission) is given by E_{i} = E_{f} + E_{photon}

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:

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

E_{n} (in J) = energy of the orbital n

n = 1, 2, 3 ... = orbital number

E_{n} values are the energy states of electrons in a hydrogen atom. The higher the absolute value of E_{n}, 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 (n_{i}) and final (n_{f}) states is:

ΔE = E_{f} – E_{i}

ΔE = - 2.1799 x 10^{-8} $\left(\frac{1}{{{\mathrm{n}}_{\mathrm{f}}}^{2}}-\frac{1}{{{\mathrm{n}}_{\mathrm{i}}}^{2}}\right)$

**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| = E_{photon} = hν = $\frac{\mathrm{hc}}{\mathrm{\lambda}}$

$\frac{\mathrm{hc}}{\mathrm{\lambda}}$ = 2.1799 x 10^{-18} $\left(\frac{1}{{{\mathrm{n}}_{\mathrm{f}}}^{2}}-\frac{1}{{{\mathrm{n}}_{\mathrm{i}}}^{2}}\right)$

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

$\frac{1}{\mathrm{\lambda}}={\mathrm{R}}_{\infty}\left(\frac{1}{{{\mathrm{n}}_{\mathrm{f}}}^{2}}-\frac{1}{{{\mathrm{n}}_{\mathrm{i}}}^{2}}\right)$

λ = wavelength (in m)

${\mathrm{R}}_{\infty}$ = Rydberg constant = $\frac{2.1799\times {10}^{-18}}{\mathrm{hc}}$ = 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

### Check your knowledge about this Chapter

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 – 10^{1}nm) - ultraviolet light (10
^{1}nm – 400 nm) - visible light (blue 400 nm - 750 nm red)
- infrared light (750 nm – 5 x 10
^{5}nm) - microwaves (5 x 10
^{5}nm – 10^{8 }nm) - radio waves (> 10
^{8}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 10^{8} 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 10^{8} 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 ν_{0}

Kinetic energy E_{k }of the ejected electrons (in J):

E_{k} = 0 (when ν < ν_{0})

E_{k} = hν - hν_{0 }(when ν > ν_{0})

h = Planck’s constant = 6.626 x 10^{-34} J.s

ν_{0} (in s^{-1}) = threshold frequency

ν (in s^{-1}) = frequency of photons

E = hν_{0} = 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 n_{i} to the final state n_{f} (where E_{i} > E_{f} ⇒ emission) is given by E_{i} = E_{f} + E_{photon}

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:

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

E_{n} (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} $\left(\frac{1}{{{\mathrm{n}}_{\mathrm{f}}}^{2}}-\frac{1}{{{\mathrm{n}}_{\mathrm{i}}}^{2}}\right)$

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})