The Nature of Light and Atomic Spectroscopy: Study Notes

The Nature of Light

Photoelectric Effect and Work Function

The photoelectric effect describes the emission of electrons from a metal surface when light of sufficient energy shines upon it. The minimum energy required to remove an electron from the surface is called the work function (φ).

• Work Function (φ): The threshold energy needed to eject an electron from a metal.

• Kinetic Energy of Ejected Electron: The energy of the electron after ejection is the difference between the energy of the incoming photon and the work function.

Equation:

$KE = h\nu - \phi$

Where h is Planck's constant, ν is the frequency of the light, and φ is the work function.

Example: Increasing the frequency (or decreasing the wavelength) of incident light increases the kinetic energy of ejected electrons, provided the energy exceeds the work function.

Photon Energy Calculations

Light consists of photons, each carrying a discrete amount of energy. The energy of a photon depends on its wavelength or frequency.

• Energy of a Photon: $E = h\nu = \frac{hc}{\lambda}$

• Energy of a Mole of Photons: Multiply the energy of a single photon by Avogadro's number ($N_A$).

Example:

• A red laser pointer emits light at 650 nm. Calculate the energy of a single photon and of one mole of photons.

Photoelectric Effect Example

Given: Light of wavelength 30.4 nm shines on platinum, ejecting electrons with kinetic energy $5.522 \times 10^{-18}$ J. Find the work function for platinum.

• Use $KE = h\nu - \phi$ to solve for φ.

Atomic Spectroscopy and the Bohr Model

Hydrogen Line Spectrum and Rydberg Equation

Hydrogen emits light at specific wavelengths, forming a line spectrum. The Rydberg equation predicts these lines:

$\frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$

• $R_H$: Rydberg constant ($1.096776 \times 10^7$ m$^{-1}$)

• $n_1$ and $n_2$: Positive integers, $n_2 > n_1$

Purpose: Predicts the wavelengths of hydrogen's spectral lines.

Bohr Model of the Atom

Niels Bohr proposed that electrons occupy fixed orbits at set distances from the nucleus, each with a specific energy.

• Allowed Orbits: Electrons can only exist in certain orbits (energy levels).

• Quantized Energies: Each orbit has a specific energy value.

Energy of an Electron in Orbit:

$E_n = -hcR_H \left( \frac{Z^2}{n^2} \right)$

or

$E_n = (-2.178 \times 10^{-18} \text{ J}) \left( \frac{Z^2}{n^2} \right)$

Where Z is the nuclear charge and n is the principal quantum number.

Absorption and Emission of Energy

Electrons absorb or emit energy only when transitioning between allowed energy levels.

• Absorption: Electron moves to a higher energy level (unstable).

• Emission: Electron falls to a lower energy level, emitting light.

Energy Change for Transition:

$\Delta E = E_{final} - E_{initial}$

$\Delta E = (-2.178 \times 10^{-18} \text{ J}) \left( \frac{Z^2}{n_f^2} - \frac{Z^2}{n_i^2} \right)$

Sign Conventions and Photon Energy

• Emission: $\Delta E$ is negative (energy released).

• Absorption: $\Delta E$ is positive (energy absorbed).

• Photon Energy: Always positive, $E_{photon} = |\Delta E|$

Example: Electron Transition in Hydrogen

• Calculate the energy change (kJ/mole) for an electron moving from $n=5$ to $n=2$.

• Find the frequency (Hz) and wavelength (nm) of the emitted light.

Hydrogen Emission Series

The hydrogen emission spectrum is divided into series based on the final energy level:

Name	nfinal
Lyman	1
Balmer	2
Paschen	3
Brackett	4

• Highest energy photon: Lyman series (UV region)

• Lowest energy photon: Brackett series (IR region)

Electromagnetic Spectrum Regions

Name	nfinal	EM region
Lyman	1	UV
Balmer	2	Visible
Paschen	3	IR
Brackett	4	IR

Example: The Brackett series contains an emission at $\lambda = 2620$ nm. Determine the initial and final values of n for this spectral line.

Additional info: The notes cover foundational concepts in quantum theory, including the quantization of energy, the photoelectric effect, and the Bohr model, which are essential for understanding atomic structure in General Chemistry.