Learning Outcomes
By the end of this lesson, students will be able to:
i. Derive an expression for the radius of an electron's orbit in the Bohr model, recognizing the relationship between the radius and the nuclear charge and the electron's energy level.
ii. Calculate the energy of an electron in a particular energy level using the Bohr model equation, understanding the connection between energy and the electron's orbit radius.
iii. Determine the frequency of light emitted when an electron transitions between energy levels, applying the Bohr model and the relationship between energy and frequency.
iv. Calculate the wavelength of emitted light using the Bohr model and the relationship between frequency and wavelength.
v. Define the concept of wave number and its connection to wavelength, recognizing that wave number is inversely proportional to wavelength.
vi. Derive an expression for the wave number of emitted light using the Bohr model, understanding the relationship between wave number, energy levels, and the Rydberg constant.
Introduction
The Bohr model of the atom, with its introduction of quantized energy levels, has provided a framework for understanding atomic structure and the emission of light from excited atoms. In this lesson, students will delve into the mathematical underpinnings of the Bohr model, exploring the relationships between various atomic properties, such as electron orbit radius, electron energy, frequency of emitted light, wavelength of emitted light, and wave number.
i. Orbit Radius: A Measure of Electron's Circular Path
The radius of an electron's orbit in the Bohr model is determined by the balance between the electrostatic attraction between the positively charged nucleus and the negatively charged electron and the centrifugal force exerted by the electron's circular motion. By equating these forces, an expression for the orbit radius can be derived:
r_n = (n^2 * h^2)/(4 * π^2 * m_e * Ze^2)
where:
r_n is the radius of the orbit for the nth energy level
n is the principal quantum number, representing the energy level
h is Planck's constant
m_e is the mass of an electron
Z is the atomic number, representing the number of protons in the nucleus
e is the elementary charge
This equation reveals that the orbit radius increases with increasing energy level (n).
ii. Electron Energy: A Quantum Leap in Understanding
The energy of an electron in the Bohr model is quantized, meaning it can only exist in specific discrete energy levels. The energy of an electron in the nth energy level is given by:
E_n = -13.6 eV/n^2
where:
E_n is the energy of an electron in the nth energy level
n is the principal quantum number
eV represents electronvolts, a unit of energy
This equation indicates that the energy of an electron decreases as the energy level (n) increases.
iii. Frequency of Emitted Light: A Measure of Energy Transition
When an electron transitions from a higher energy level (n_i) to a lower energy level (n_f), it emits light with a frequency (ν) given by:
ν = (E_i - E_f)/h
where:
ν is the frequency of emitted light
E_i is the energy of the electron in the initial energy level (n_i)
E_f is the energy of the electron in the final energy level (n_f)
h is Planck's constant
This equation demonstrates that the frequency of emitted light increases as the energy difference between the initial and final energy levels increases.
iv. Wavelength of Emitted Light: A Dance of Energy and Distance
The wavelength (λ) of emitted light is inversely proportional to its frequency (ν):
λ = c/ν
where:
λ is the wavelength of emitted light
c is the speed of light in a vacuum
ν is the frequency of emitted light
This equation implies that shorter wavelengths correspond to higher frequencies, and vice versa.
v. Wave Number: A Measure of Spectral Lines
Wave number (ν̃), a measure of the frequency of light, is defined as the inverse of the wavelength (λ):
ν̃ = 1/λ
where:
ν̃ is the wave number
λ is the wavelength
Wave number is often used to represent the positions of spectral lines in emission spectra.
Bohr Model and Wave Number: A Unifying Connection
Combining the equations for electron energy and the energy of emitted light, an expression for the wave number (ν̃) of emitted light can be derived:
ν̃ = R/(Z^2 * (1/n_f^
