Energy, Frequency, Wavelength, and the Hydrogen Spectrum (HSC SSCE Physics): Revision Notes

Energy, Frequency, Wavelength, and the Hydrogen Spectrum

Introduction to hydrogen spectral series

When electrons in hydrogen atoms transition between different energy levels, they emit or absorb light at specific wavelengths. These transitions create distinct patterns called spectral series. Three main series are observed in the hydrogen spectrum, each corresponding to transitions ending at different energy levels.

The table below shows the three major spectral series for hydrogen:

Series Name	Part of Spectrum	Shortest Wavelength (nm)	Longest Wavelength (nm)
Lyman	Ultraviolet	91.1	121.6
Balmer	Visible	364.5	656.3
Paschen	Infrared	820.1	1870

The Balmer series produces visible light and occurs when electrons fall to (or jump from) the second energy shell ($n = 2$). The Lyman series involves transitions to the first energy shell ($n = 1$) and produces ultraviolet radiation. The Paschen series involves transitions to the third energy shell ($n = 3$) and produces infrared radiation.

Energy levels in the hydrogen atom

Understanding energy level representations

Energy levels in hydrogen atoms can be represented in two different ways, both equally valid but with different reference points.

Method 1 - Ionisation energy as zero: In this representation, all energy states have negative values. The ground state (lowest energy level, $n = 1$) has an energy of $-13.6$ eV. Higher energy levels are less negative, approaching zero as the electron moves further from the nucleus. When the electron is completely removed (ionised), its energy is zero.

Method 2 - Ground state as zero: Here, the ground state is set to zero energy, and all other energy levels have positive values. The ionisation energy (energy needed to remove the electron) is $+13.6$ eV.

Key terms

Ground level (ground state): The lowest possible energy level in an atom, corresponding to the electron being in the orbit with the smallest radius. For hydrogen, this is the $n = 1$ level.

Ionisation: The process of removing an electron from an atom completely, so it is infinitely far away (or far enough that electrostatic attraction becomes negligible).

Energy transitions and spectral lines

The diagram above shows how different spectral series arise from electron transitions. The Lyman series (with lines labelled $\alpha$, $\beta$, $\gamma$, etc.) corresponds to electrons falling to the ground state ($n_f = 1$). The Balmer series (with lines labelled $H_\alpha$, $H_\beta$, $H_\gamma$, $H_\delta$, etc.) corresponds to electrons falling to the $n = 2$ level.

The energy of a photon

When an electron moves between energy levels, it either absorbs or emits a photon. The energy of this photon is directly related to its frequency through Planck's equation:

$E = hf$

where:

• $E$ = energy in joules (J) or electron-volts (eV)
• $h$ = Planck's constant = $6.626 \times 10^{-34}$ J·s or $4.14 \times 10^{-15}$ eV·s
• $f$ = frequency of the photon in hertz (Hz)

When an electron transitions from a higher energy level (initial, $E_i$) to a lower level (final, $E_f$), the energy difference appears as a photon:

$E = E_i - E_f = hf$

The relationship between energy and wavelength

For electromagnetic waves, the speed of light ($c$), frequency ($f$), and wavelength ($\lambda$) are related by:

$v = f\lambda$

For light, $v = c = 3.00 \times 10^8$ m/s, so:

$c = f\lambda$

Combining this with $E = hf$ gives us another useful relationship:

$E = \frac{hc}{\lambda}$

Rydberg's equation

Historical context

In the 1880s, Johannes Rydberg developed a mathematical formula that accurately predicted the wavelengths of spectral lines in hydrogen. He derived this equation empirically - meaning he fitted it to observed data rather than deriving it from a theoretical model. At that time, no atomic theory could explain why spectral lines existed at all. Rydberg's work was crucial in leading to the development of the Bohr model of the atom.

The Rydberg formula

The Rydberg equation predicts the wavelength of emission and absorption lines for hydrogen:

$\frac{1}{\lambda} = R\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$

where:

• $\lambda$ = wavelength of the spectral line
• $R$ = Rydberg constant $= 1.097 \times 10^7$ m$^{-1}$
• $n_f$ = final energy level (an integer: 1, 2, 3, ...)
• $n_i$ = initial energy level (an integer greater than $n_f$)

For different series:

• Lyman series: $n_f = 1$ (transitions to ground state)
• Balmer series: $n_f = 2$ (transitions to first excited state)
• Paschen series: $n_f = 3$ (transitions to second excited state)

Conservation of energy in atomic transitions

The law of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. This fundamental principle applies to electron transitions in atoms.

When an electron falls from a higher energy level to a lower one, it loses energy. This lost energy doesn't disappear - it is transformed into a photon of electromagnetic radiation. The energy of the photon exactly equals the energy difference between the two levels:

$E_{\text{photon}} = E_i - E_f$

Conversely, when an atom absorbs a photon, the electron gains energy and jumps to a higher level. The photon's energy must exactly match the energy difference between levels for absorption to occur.

Exam tips

Formula selection:

• Use $E = hf$ when you know or need frequency
• Use $E = \frac{hc}{\lambda}$ when you know or need wavelength
• Use Rydberg's equation when dealing with specific energy level transitions in hydrogen

Unit conversions:

• Always convert wavelengths to metres before calculations
• Remember: 1 nm = $10^{-9}$ m
• Watch for energy in eV vs J - use the appropriate value of Planck's constant

Significant figures:

• Energy level numbers ($n$) are exact, so don't limit your significant figures based on them
• Match significant figures to the given constants (usually 3-4 significant figures)