Which Transition Causes The Absorption Line At The Shortest Wavelength

Of all the possible ways an atom can absorb a photon, the transition that produces an absorption line at the shortest wavelength is the one involving the largest possible energy jump. In the context of atomic spectroscopy, this is most famously and fundamentally demonstrated by the Lyman-alpha transition in a hydrogen atom, where an electron is excited from the ground state (n=1) to the first excited state (n=2).

This article will dig into the physics behind this phenomenon, explaining why this specific transition holds the key to the shortest wavelength absorption line and exploring its significance in astrophysics and quantum mechanics.

The Foundation: Atomic Energy Levels and Transitions

To understand absorption lines, we must first grasp the quantum nature of atoms. Electrons within an atom are not free to occupy any energy level; they are confined to specific, discrete "orbitals" or energy states, often labeled by a principal quantum number, n (where n = 1, 2, 3, ...And ). All other states (n=2, 3, etc.The state with the lowest possible energy (n=1) is called the ground state. ) are excited states Worth keeping that in mind..

An atom absorbs a photon of light when that photon's energy exactly matches the energy difference between two allowed states. This is expressed by the fundamental equation:

E_photon = E_high - E_low

Where:

• E_photon is the energy of the absorbed photon.
• E_high is the energy of the higher (excited) state.
• E_low is the energy of the lower state.

The energy of a photon is inversely proportional to its wavelength, a relationship defined by the Planck-Einstein relation:

E = hc / λ

Where:

• h is Planck's constant.
• c is the speed of light.
• λ is the wavelength of the light.

From this equation, it's clear: the larger the energy difference (ΔE), the shorter the wavelength (λ) of the absorbed photon. So, our quest for the shortest wavelength absorption line is a quest for the transition with the greatest possible energy jump.

The Hydrogen Atom: A Quantum Model

The hydrogen atom, with its single electron, is the simplest system and serves as the perfect model. Its energy levels are described by the Bohr model and, more accurately, by the Schrödinger equation. The energy of an electron in a state n is given by the formula:

E_n = -R_H / n²

Where R_H is the Rydberg constant for hydrogen (approximately 13.6 electron volts, or eV). The negative sign indicates that the electron is bound to the nucleus; energy must be added to free it (reach an energy of 0 eV) The details matter here..

Let's calculate the energy difference for a transition from a lower level n_low to a higher level n_high:

ΔE = E_high - E_low = (-R_H / n_high²) - (-R_H / n_low²) = R_H (1/n_low² - 1/n_high²)

This formula is the key to unlocking the answer Not complicated — just consistent..

Identifying the Largest Energy Jump

To maximize ΔE, we need to make the term (1/n_low² - 1/n_high²) as large as possible. Let's analyze the variables:

1. Minimize n_low: The value of 1/n_low² is largest when n_low is as small as possible. The smallest possible value for n_low is 1 (the ground state). So, we set n_low = 1. This gives us 1/1² = 1 Worth keeping that in mind..

2. Maximize n_high: The value of 1/n_high² becomes smaller as n_high increases. To maximize the difference, we want to subtract the smallest possible value. The smallest value for 1/n_high² approaches zero as n_high approaches infinity. This represents the transition from the ground state to a free, unbound electron (ionization).

While the transition to infinity (n=∞) represents the absolute maximum energy jump (the ionization energy, 13.But 6 eV), it does not produce a sharp "line" in the absorption spectrum. In real terms, instead, it creates a continuous absorption edge. For a distinct, quantized absorption line, we need a transition to a specific, bound excited state.

Because of this, the transition that creates the shortest wavelength line is the one from the ground state (n=1) to the first available excited state, which is n=2. This is the Lyman-alpha (Ly-α) transition.

The Lyman Series and the Shortest Wavelength Line

Transitions that start from the ground state (n=1) form a series of spectral lines named after Theodore Lyman, who discovered them. This is the Lyman series.

• Lyman-alpha (Ly-α): n=1 → n=2
• Lyman-beta (Ly-β): n=1 → n=3
• Lyman-gamma (Ly-γ): n=1 → n=4
• ...and so on.

Using our energy formula, we can calculate the wavelength for the Lyman-alpha transition:

ΔE = R_H (1/1² - 1/2²) = R_H (1 - 1/4) = (3/4) R_H

ΔE = (3/4) * 13.6 eV = 10.2 eV

Now, convert this energy to wavelength (λ = hc / ΔE). Using hc ≈ 1240 eV·nm:

λ = 1240 eV·nm / 10.2 eV ≈ 121.6 nanometers (nm)

This wavelength falls squarely in the ultraviolet (UV) part of the electromagnetic spectrum, far shorter than the visible light we see (400-700 nm). It is the shortest wavelength absorption line for hydrogen No workaround needed..

For comparison, the next line in the series, Lyman-beta (n=1 → n=3), has an energy difference of R_H (1 - 1/9) = (8/9) R_H ≈ 12.In real terms, 1 eV, giving a wavelength of about 102. 5 nm, which is even shorter. That said, the Lyman-alpha line is the most prominent and significant because it involves the most probable transition from the ground state.

Short version: it depends. Long version — keep reading And that's really what it comes down to..

Comparison with Other Series

Hydrogen has other named series of spectral lines, each corresponding to transitions ending on a different lower energy level:

• Balmer Series: Transitions ending on n=2. These lines are in the visible spectrum (e.g., the red H-alpha line at 656 nm).
• Paschen Series: Transitions ending on n=3. These are in the infrared (IR).
• Brackett and Pfund Series: Ending on n=4 and n=5, respectively, also in the IR.

Because all these series start from a higher n_low value (2, 3, etc.), the term 1/n_low² is smaller (1/4, 1/9, etc.), making the overall energy difference ΔE smaller and the resulting wavelength longer than any line in the Lyman series. Because of this, **the Lyman series consistently produces the shortest wavelength absorption lines for hydrogen Simple as that..

Why This Matters: The Cosmic Significance

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