# Early Quantum Theory | General Chemistry 1

## 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 – 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 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 10^{8} 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 = $\frac{\mathrm{hc}}{\mathrm{\lambda}}$

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 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, 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 n_{i} to 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 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:**

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