Energy Quantization – SS3 Physics Lesson Note (Second Term)

Energy quantization means that an electron in an atom can have only certain fixed amounts of energy, called energy levels. Energy is emitted in separate packets called quanta, and each quantum is given by E = hf, where h is Planck’s constant and f is the frequency. An electron moves between levels when it gains or loses energy.

Behavioural objectives

By the end of the lesson, students should be able to:

• state what energy quantization is and use the relation E = hf;
• distinguish between the ground state and an excited state of an atom;
• define the electron volt and change electron volts to joules;
• calculate the frequency and wavelength of a photon from the change in energy level of an electron;
• describe emission spectra, absorption spectra and the spectra of a discharge lamp.

What is energy quantization?

Bohr suggested that an electron in an atom can exist only in certain discrete energy states, which is known as quantization, and that the electron can move from one level to another. Energy in such bodies is emitted in separate or discrete energy packets called energy quanta (E).

E = hf

where h = Planck’s constant and f = frequency.

What are the energy levels in an atom?

Electrons in atoms are arranged around their nuclei in positions known as energy levels or electron shells. More energy is needed to remove an electron from the first energy level than from any of the higher levels. The energy of an electron is given by the relation:

E = -R/n²

where n = the electron quantum number and R = a constant.

The minus sign shows that work must be done on the electron to remove it from the atom.

Ground state and excited state

The ground state is the stable state of an atom, corresponding to its minimum energy. When an atom is heated or bombarded with an energetic particle, the atom is excited. An excited state is an allowed state of higher energy in which the atom is unstable.

The electron volt

One electron volt (1 eV) is the energy acquired by an electron in falling freely through a p.d. of 1 volt.

1 eV = 1.6 × 10^-19 J

Energy change between levels

During excitation from a lower energy level, potential energy is converted into kinetic energy, so the electron acquires a velocity given by:

E = ½mv² = eV

The energy gained by the electron = charge × p.d. = eV. The electron moves from one level to another according to the relation:

E_n – E₀ = hf = hc/λ

Worked examples

Example 1

The change in the energy level of an electron in an atom is 6.2 × 10^-21 J. Calculate (a) the frequency of the photon, (b) the wavelength. (c = 3.0 × 10⁸ m s^-1, h = 6.625 × 10^-34 Js)

ΔE = E_n – E₀ = 6.2 × 10^-21 J

ΔE = hf

f = ΔE/h = (6.2 × 10^-21) / (6.625 × 10^-34)

f = 9.358 × 10¹² Hz

But c = fλ

λ = c/f = (3.0 × 10⁸) / (9.358 × 10¹²)

λ = 3.21 × 10^-5 m

Example 2

An atom excited to an energy level E₂ = -12.42 × 10^-19 J falls to a ground level of energy E₀ = -30.3 × 10^-19 J. Calculate the frequency and the wavelength of the emitted photon.

ΔE = E₂ – E₀

= -12.42 × 10^-19 – (-30.3 × 10^-19) J

= 17.88 × 10^-19 J

f = ΔE/h = (1.788 × 10^-18) / (6.625 × 10^-34)

f = 2.699 × 10¹⁵ Hz

λ = c/f = (3.0 × 10⁸) / (2.699 × 10¹⁵)

λ = 1.11 × 10^-7 m

Example 3

The ground state of hydrogen is -26.3 eV and the second state is -10.3 eV. Calculate the wavelength of the radiation if the electron returns to the ground state.

ΔE = E₂ – E₀ = -10.3 eV – (-26.3 eV) = 16 eV

1 eV = 1.6 × 10^-19 J

So 16 eV = 16 × 1.6 × 10^-19 J = 2.56 × 10^-18 J

ΔE = hf = hc/λ

λ = hc/ΔE = (6.625 × 10^-34 × 3.0 × 10⁸) / (16 × 1.6 × 10^-19)

= (1.9875 × 10^-25) / (2.56 × 10^-18)

λ = 7.76 × 10^-8 m

Example 4

The p.d. through which an electron moves is 1.5 kV. Calculate (a) the velocity with which the electron moves, if the ratio of its charge to mass is 1.8 × 10¹¹ C kg^-1, (b) the kinetic energy.

KE = ½mv² = eV

2eV = mv²

v² = 2eV/m

But e/m = 1.8 × 10¹¹

v = √(2 × 1.5 × 10³ × 1.8 × 10¹¹)

v = 2.3 × 10⁷ m/s

KE = eV = 1.6 × 10^-19 × 1.5 × 10³

KE = 2.4 × 10^-16 J

What are line spectra from hot bodies?

If the light of an excited gas is examined in a spectrometer, an emission spectrum is seen. For example, neon produces a line spectrum. The spectrum consists of a number of well-defined lines, each having a particular wavelength and frequency.

What are absorption spectra?

Absorption spectra are dark lines in the emission spectrum of a heated substance, caused by the absorption of radiation. Electrons absorb radiation when they jump from a lower energy level to a higher energy level.

What are the spectra of a discharge lamp?

A gas is put into a discharge tube and its pressure is progressively reduced using a vacuum pump. When a high p.d. is applied at low pressure, an electric discharge takes place in the tube and the gas glows.

The gas glows because its atoms are given energy by the flow of electricity through the tube. The colour of the glow depends on the nature of the gas used.

Evaluation

1. Define the ground state and an excited state of an atom.
2. State the value of one electron volt in joules.
3. What does the minus sign in E = -R/n² show?
4. Distinguish between an emission spectrum and an absorption spectrum.

Assignment

Reading assignment

• New School Physics, pages: (to be added).

Weekend assignment

1. An atom in the ground state is said to be
   • (A) excited
   • (B) stable
   • (C) ionised
2. Which of the following has the greatest penetrating power?
   • (A) Beta rays
   • (B) Gamma rays
   • (C) X-rays
3. Which of the following gives rise to the line spectra obtained from atoms?
   • (A) change of electrons from a higher to a lower energy level
   • (B) the potential energy of the electron inside an atom
   • (C) excitement of an electron in the atom
4. Which of the following statements is correct about cathode rays? They are fast-moving
   • (A) atoms
   • (B) ions
   • (C) electrons
5. The nucleon number and the proton number of a neutral atom are 238 and 92 respectively. What is the number of neutrons in the atom?
   • (A) 146
   • (B) 330
   • (C) 73

Theory

1. Explain the term excitation.
2. An electron jumps from one energy level to another in an atom, radiating 9.0 × 10^-19 J. If h = 6.6 × 10^-34 Js and c = 3.0 × 10⁸ m/s, what is the wavelength of the radiation?