Video Lesson

Lesson Objectives

At the end of this lesson, you will be able to :

• Explain electromagnetic radiation, atomic spectra and Bohr models of the atom.
• The quantum mechanical model of the atom and the related postulates and principle.
• Characterize electromagnetic radiation (EMR) in terms of wavelength, frequency, and speed.
• Calculate the wavelength and Frequency of EMR
• Explain the dual nature of light.
• Describe emission spectra of atoms as consisting a series of lines.
• Define a photon as a unit of light energy.
• Distinguish how the photon theory explains the photoelectric effect.
• Identify the relationship between a photon absorbed and an electron released.
• State Bohr’s assumption of energy of the electron in a hydrogen atom.
• Calculate the radius of electron orbit, the electron velocity and the energy of an electron using Bohr’s model.
• Explain that the line spectrum of hydrogen atom demonstrates the quantized nature of the energy of its electron.
• Explain that atoms emit or absorb energy when they undergo transitions from one state to another.
• Compose the limitations of Bohr’s theory.

Brainstorming Questions

1. Which type of rays are used to determine if a bone is broken?

2. “How do variations in electromagnetic radiation affect the atomic spectra of elements, and what can the resulting spectral lines reveal about the arrangement and behavior of electrons in an atom?”

key terms and concepts

1.5 Electro magnetic radiation & Atomic Spectra

1.5.1 Electromagnetic Radiation

• In 1873, James Clerk Maxwell proposed that light consists of electromagnetic waves.
• According to his theory, an electromagnetic wave has an electric field and magnetic field components.
• EMR is the emission and transmission of energy in the form of electromagnetic waves.
• Radiation refers to the emission or transmission of energy in the form of waves or particles through space or a material medium.
• Electromagnetic waves have three primary characteristics: wavelength, frequency and speed.
• Wavelength(λ,Greek lambda),is the distance the wave travels during one cycle
• Frequency(ν, Greek letter nu) is the number of cycles the wave undergoes per second and is expressed in units of l/second (1/s; also called hertz, Hz).
• The speed of a wave depends on its type and the nature of the medium it travels through, such as air, water, or a vacuum.
• The Speed (c) of a wave is the product of its wavelength and it’s frequency, C = ν×λ

• In vacuum, light travels at a speed of 𝟑.𝟎×𝟏𝟎^𝟖m/s
• Electromagnetic radiation are also characterized by their amplitude or intensity

• The Electromagnetic waves in the different spectral region travel at the same speed but differ in frequency and wavelength.
• Light of a particular shade of red, for instance, always has the same frequency and wavelength, but it can be dim (low amplitude) or bright (high amplitude).
• EMR comes in a broad range of frequencies called the electromagnetic spectrum.
• A rainbow is an example of a continuous spectrum.
• Different Wavelengths Invisible light have different Colors From Red(λ= 750 nm) to violet (λ= 380 nm)

Electromagnetic Spectrum:

• Fig.1.8 Electromagnetic spectrum
• Electromagnetic spectrum encompasses all the different types of electromagnetic waves.
• The enregies in Electromagnetic spectrum are arranged according to their frequencies or wavelengths. This spectrum includes radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays, and gamma rays.

• Radiation Provides an Important means of energy transfer for instance, the energy from the sun reaches the earth Mainly In The form of visible and ultraviolet radiation.

• Radiation provides an important means of energy transfer.
• For instance, The energy from the sun reaches the earth mainly in the form of visible and ultraviolet radiation.
• The glowing coals of a fireplace transmit heat energy by infrared radiation.
• n microwave ovens, microwave radiation is used to heat water in foods, causing them to warm up quickly.
• The glowing coals of a fireplace transmit heat energy by infrared radiation
  • EXAMPLES

1. Ethiopian National Radio , Addis Ababa station broadcasts its AM signal at a frequency of 2400kHz . what is the wavelength of the radio wave expressed in meters ?

Solution; λ = C/v =3.0×10⁸ ms^-1/2400 × 10³ sec = 126m

2. Find the frequency in Hz of the Gamma radiation from a radioactive cobalt- 60 source. If its wave length is 1.0 × 10⁻⁹𝑚.

Solution; wavelength 𝜆 = 1.0 × 10⁻⁹ 𝑚 — Given

c = 3.0 × 10⁸ 𝑚s^-1 ……………….speed of light— Given
ν =C/λ

ν = 3.0×10⁸ms^-1 / 1.0×10⁻⁹m

ν = 3.0 × 10¹⁷Hz

1.5.2 The Quantum Theory And Photon

• In 1900, Max Planck, the German physicist, came to an entirely view of matter and energy.
• He made a groundbreaking proposal that, like matter, energy is also discontinuous.
• According to Planck, atoms and molecules could emit or absorb energy only in discrete quantities, like small packages or bundles. Called Quantum
• The energy of a quantum is proportional to the frequency of the radiation, E=hν
• Where h is called Planck’s constant and ν is the frequency of radiation. The value of Planck’s constant is 6.63 × 10^-34J.s
• When we say that energy is quantized, it means that energy can only exist in specific, discrete amounts or “quanta.”
  • Since ν=c/λ, Energy can also be expressed as, E=hc/𝝀
• According to quantum theory,energy is always emitted or absorbed in integral multiples of hν; for example, hν, 2hν, 3h etc

EXAMPLES

1. Calculate the energy, in joules, of a photon of red light that has afrequency of 3.73 × 10¹⁴s−1

Solution: E = hv

E = 6.63 × 10^-34 J. s ×3.73 × 10¹⁴s−1

E= 2.47×10^-21 Joule.

2. The blue color in fireworks is often achieved by heating copper (I) chloride (CuCl) to about 1200 °C. Then the compound emits blue light having a wavelength of 600 nm. What is the increment of energy that is emitted at 600 nm by CuCl?

Solution:

The quantum of energy can be calculated from the Equation , E = hv

The frequency (ν) for this case can be calculated as follows: v=c/λ

So, E = hv = hc/λ

E= 6.63 × 10^-34 J. s × 3.0 × 10⁸ ms^-1 /600×10^-9 m

E= 3.315 × 10^-19 J.

The Photoelectric Effect

• In 1905, Albert Einstein used the quantum theory to explain the photoelectric effect.

• Light shining on a clean metallic surface can cause the surface to emit electrons. This phenomenon is known as the photoelectric effect.
• The Minimum amount of Energy required to remove an electron from the metal surface is called work function(𝜱), 𝜱=hv_o
• v_o is the Minimum Frequency required to remove an electron from the metal surface which is also called threshold frequency.
• Photons are particles of light or energy packet. A photon with energy less than E_o(ν < ν_o) cannot remove an electron, or a light with a frequency less than the ν_o produces no electrons.
• If we supply more energy than 𝜙 by using radiation with a higher frequency than 𝜈0, then the excess energy goes into the kinetic energy of the ejected electrons.

Where KE_e – is the kinetic energy of an electron,

m- is mass of an electron,

v – is the velocity of an electron,

hν – is the energy of an incident photon, and

hν_ois the energy required to remove an electron from the metal surface.

• Intensity of light is a measure of the number of photons present in a given part of the beam:
• Now consider two beams of light having the same frequency (which is greater than the threshold frequency) but different intensities.
• The more intense a beam of light, the larger the number of photons it contains , consequently, it ejects more electrons from the metal surface than a weaker beam of light.
• The higher the frequency of the light, the greater will be the kinetic energy of the emitted electrons

EXAMPLE

1. The threshold frequency for metallic potassium is5.46 × 10¹⁴𝑠−1 . Calculate the maximum kinetic energy that the emitted electron has when the wavelength of light shining on the potassium surface is 350nm.

Solution::- Threshold frequency, 𝜈₀ = 5.46 × 10¹⁴𝑠⁻¹ … 𝑔𝑖𝑣𝑒𝑛

Minimum wavelength, 𝜆𝑚𝑖𝑛 = 350𝑛𝑚 … 𝐺𝑖𝑣𝑒𝑛

K. E = λν − λν₀

ν = C/λ = 3.0×10⁸ / 350𝑛𝑚 = 8.6×10⁵ s^-1

K. E = h (ν − ν₀) ,

K. E = 6.63 × 10^-34 J. s (8.6×10¹⁴ s^-1 – 5.46 × 10¹⁴𝑠⁻¹ )

K.E= 2.08x 10^-19 J.

1.5.3 Atomic or Line spectra

An atomic spectrum refers to the spectrum of electromagnetic radiation emitted or absorbed by an atom. This spectrum is unique to each element and is primarily a result of the transitions of electrons between energy levels within the atom.

Types of Spectra:

Emission Spectrum: Produced when an atom emits light as electrons drop to lower energy levels. It appears as a series of bright lines on a dark background.

Absorption Spectrum: Occurs when light passes through a gas and certain wavelengths are absorbed by electrons jumping to higher energy levels. This appears as dark lines on a continuous spectrum.

Energy Levels: Each electron in an atom occupies specific energy levels. When an electron moves from a higher to a lower energy level, energy is released in the form of light. Conversely, energy is absorbed when an electron moves to a higher level.

Line Spectrum: The emitted or absorbed light is not continuous but consists of discrete lines, corresponding to specific wavelengths. These lines can be analyzed to determine the composition of the substance.

Applications of Atomic Spectra

Identification of Elements: Atomic spectra are used in spectroscopy to identify the elemental composition of stars and other celestial bodies.

• Quantum Mechanics: The study of atomic spectra provides evidence for quantum theory and the quantization of energy levels.Examples: The hydrogen emission spectrum is one of the most studied, consisting of several series of lines (e.g., Balmer series) in the visible range.

The 1.5.4 Bohr Model of Hydrogen Model

In 1913,Niels Bohr (a Danish physicist) explained why the orbiting electron does not radiate energy as it moves around the nucleus.

He introduced the fundamental idea that the absorption and emission of light by hydrogen atoms was due to energy changes of the electrons with in the atoms.

The fact that only certain frequencies are absorbed or emitted by an atom tells us that only certain energy changes are possible. Thus, energy changes in an atom are quantized.

The Bohr model was based on the following assumptions:

1. The electron in a hydrogen atom travels around the nucleus in a circular orbit.

2. The energy of the electron in an orbit is proportional to its distance from the nucleus. i.e. The further the electron is from the nucleus, the more energy it possesses.

3. Only a limited number of orbital’s with certain energies are allowed. In other words, the orbits are quantized.

4. The only orbits that are allowed are those for which the angular momentum of electron is an integral multiple of plank’s constant, h divided by 2π.

5. As long as an electron stays in a given orbit, it neither gains nor loses energy.
 When an electron jumps to a higher energy orbit, a definite amount of energy is Absorbed.
 When an electron falls into a lower energy orbit a definite amount of energy is Emitted.

The Energy Level of an Atom

• A very significant result from Bohr’s work is an equation for calculating the energy level of an atom:-

Where 𝑅_𝐻 – is the Rydberg constant, 𝑅_𝐻 = 2.18 × 10⁻¹⁸

n- is an integer and is called the principal quantum number. 𝑛 = (1,2,3,−− −)
z- is the charge of the nucleus for hydrogen atom Z=1

E_n= -2.18×10^-18J.1/n²

The energy difference between the two levels:

ΔE = E_final – E _initial

ΔE = -2.18×10^-18​(1/n²_final −1/n²_initial​)

The negative sign appears because as the electron gets farther away from the nucleus, the energy approaches to zero,

𝐸 = 0, when 𝑛 = ∞, 𝑠𝑜 𝐸_𝑛 < 0

EXAMPLES

1. Consider the n=2 state of the H-atom. Using the Bohr model, calculate the energy of the electron.

Solution; E₂ =2.18×10−18J/2²

E₂ = −0.545 × 10⁻¹⁸J

E₂ = −545 × 10⁻¹⁷J

2. How much energy in kilojoules per mole, is released when an electron makes a transition from n=5 to n=2 in H- atom?

Solution;

The Energy States

 Allowed energy states are called stationary states in quantum theory. This is due to the fact that each electron can take only certain discrete value (quantization).
 The stationary states of the lowest energy is called the ground state, at n=1. And the allowed states of the higher energies are called excited states.

Bohr model of the Hydrogen atom quantitatively explained the line spectra of Hydrogen.
 The absorption or emissions in the line spectra correspond to transition from one orbit to another.

ΔE = hv = -2.18 x 10^-18​ (1/n_f² −1/n_i²)

1) When 𝑛_𝑓 > 𝑛_𝑖

a) ∆𝐸 is positive
b) The system has absorbed energy
c) Electron moves from lower to higher Energy

2. When 𝒏_𝒇 < 𝒏_𝒊

a) ∆𝑬 is negative

b) The system has emitted energy.

c) Electron moves from higher to lower energy.

Fig .1.4 Series of lines in the hydrogen spectrum

• N.B:- When an electron falls from any higher orbit
  a) To n=3 (2^nd excited state) IR series of spectral line is observed. i.e. Pastchen series.
  b) To n=2 (1st excited state) visible series of spectral line is observed. i.e. Balmer series.
  c) To n=1 (ground state) UV series of spectral line is observed. i.e. Lyman series

1.Find the energy of emission when an electron transits from the 3^rd excited state in:-
a) Lyman series
b) Balmer series
c) Pastchen series

Solution; 3rd excited state is at n=4
a) Lyman series produces a UV line spectrum
when an electron drops to the ground state n=1

b . Balmer series produces a visible line spectrum . when an electron jumps to the 1^st excited state or n=2.

C. Paschen series produces an IR line spectrum when an electron jumps to the 2^nd excited state or n=3

Limitations of the Bohr Theory
1) The model was adequate to explain atoms and ions with one electron such as H, He+1 and Li+2 , but it failed to explain the atomic spectra of more complicated atoms or ions.
2) Bohr could not explain the further splitting of spectral lines in the Hydrogen spectra on application of magnetic field and electric fields.
◌ Louis de Broglie postulated that a particle of matter of mass, m and speed, v has an associated wave length, by analogy with light, is obtained as:

E = mc² −− − −− −Einstein′s equation.

E =hC/λ−− − −− −Plank′s Equation

• Equating the two equations:-

The electromagnetic equation shows:-
1) Wave properties
2) Particle properties
◌ Electrons also show particles properties with associated wave length.
◌ Energy is a form of matter, and all matter show the same types of properties. i.e. All matter exhibits both particle and wave properties

EXAMPLES

1. Calculate the wave length, (in nm), associated with a proton travelling at a speed of 2.55×10⁶𝑚/𝑠.
   𝑚ass of 𝑝𝑟𝑜𝑡𝑜𝑛 = 1.67 × 10⁻²⁷𝑘𝑔
   Solution::- Using de-Broglie’s equation
   λ =h/mν
   λ =6.62 × 10⁻³⁴J. s/ 1.67 × 10⁻²⁷ × 2.55 × 10⁶ ms
   λ = 1.56 × 10⁻⁴nm

2. Electrons that have a wave length of 12pm are used for electron diffraction. What is the velocity of these electrons?
Solution:- Using de-Broglie’s equation
λ =h/mv but λ = 12pm = 12 × 10⁻¹²m , 𝑚ass of electron = 9.11 × 10⁻³¹𝑘𝑔 and ℎ = 6.62 × 10⁻³⁴𝐽.

𝑉 =ℎ/𝑚λ

V= 6.62 × 10⁻³⁴ /9.11 × 10⁻³¹𝑘𝑔 × 12 × 10⁻¹²𝑚
𝑉 = 6.1 × 10⁷ 𝑚/s