Question

Given that the hydrogen atom has an infinite number of energy levels, why can't a hydrogen atom in the ground state absorb all possible wavelengths of light?

Short answer

Answer: A hydrogen atom in the ground state cannot absorb all possible wavelengths of light because the energy differences between the energy levels of the atom restrict the absorption to specific photon wavelengths. The atom can only absorb photons that correspond to the energy differences between the ground state (n=1) and higher energy levels, resulting in a finite number of possible energy levels and absorbed photon wavelengths.

Step-by-step solution

Understanding the hydrogen atom energy levels

In a hydrogen atom, the energy levels are described by the formula: E_n = -13.6 eV/n^2, where n is the principal quantum number and E_n is the energy of the nth level. The ground state corresponds to n=1, and as n increases, the energy levels become closer together. In the ground state, the atom has the lowest possible energy.

Absorption of a photon by a hydrogen atom

When a hydrogen atom absorbs a photon, it gains energy and undergoes a transition to a higher energy level. The energy of the absorbed photon is equal to the energy difference between the initial and final energy levels. Mathematically, it can be represented as: \(E_{photon} = E_{final} - E_{initial}\) This energy difference corresponds to a specific wavelength of light given by the equation: \(\lambda = \frac{c}{\nu} = \frac{hc}{E_{photon}}\) Where \(\lambda\) is the wavelength of the absorbed photon, \(c\) is the speed of light, \(h\) is the Planck's constant, and \(\nu\) is the frequency of the absorbed photon.

Wavelengths absorbed by hydrogen atom in ground state

For a hydrogen atom in the ground state (n=1), it can only absorb photons that correspond to transitions from n=1 to higher energy levels (n=2, 3, 4, ...). However, it cannot absorb a photon that induces a transition from n=1 to n=1, as there is no energy gain in such a transition. Therefore, only specific wavelengths of light corresponding to these specific energy transitions are absorbed.

Why can't the hydrogen atom absorb all possible wavelengths of light?

Since the energy levels in a hydrogen atom correspond to specific energy differences, only certain wavelengths of light can be absorbed. As the atom transitions to higher energy levels, the energy differences become smaller, and the absorbed photon wavelengths become closer together. However, there will always be a finite number of possible energy levels and corresponding absorbed photon wavelengths. Therefore, a hydrogen atom in the ground state cannot absorb all possible wavelengths of light because the energy differences between levels restrict the absorption to specific photon wavelengths.

Key concepts

Quantum Mechanics

Quantum mechanics forms the foundation of our understanding of how subatomic particles behave, including electrons within atoms. Unlike everyday objects that seem to have a continuous range of motion and position, quantum mechanics tells us that at a minuscule scale, like that of a hydrogen atom, these particles exist in discrete states. This means that electrons can't just possess any random energy value.

Instead, their energies are quantized, similar to a ladder's rungs: an electron can only 'jump' from one rung (energy level) to another without resting in between. Therefore, when we consider the behavior of a hydrogen atom's electron in quantum terms, we understand why it cannot just absorb any wavelength of light. Each wavelength would need to match exactly the energy gap between its quantized energy levels.

Atomic Spectral Lines

The atomic spectral lines are a visual representation of the quantization inherent in an atom's energy states. When an electron jumps between energy levels, it either absorbs or emits a photon with very specific characteristics. The energy of this photon is equal to the energy difference between the two levels. The spectral lines are essentially the 'fingerprints' of an element, with each line corresponding to a transition between energy levels.

These lines form when an electron transitions and releases energy in the form of a photon, resulting in an emission spectrum, or when it absorbs energy and moves to a higher energy state, creating an absorption spectrum. The discrete spectral lines associated with the hydrogen atom underscore the fact that only photons with certain energy, and thus specific wavelengths, can be absorbed, corresponding to the set energy level transitions.

Photon Absorption

Photon absorption is a process where an electron in an atom takes up energy from a photon and moves to a higher energy level. This is a bit like catching a ball that fits perfectly in your hand: only photons with the exact energy to match the energy 'gap' between allowed energy levels of an electron will be captured.

These 'gaps' are the energy differences which the electron can 'jump' over. If a photon's energy is too high or too low, it won't be absorbed because it doesn't match the required jump. Just as you can't comfortably catch a ball that's too big or too small for your hand, an electron can't absorb a photon unless its energy is precisely the right amount to move into an unfilled higher energy state. This selectivity in energy is why a hydrogen atom doesn’t absorb every possible wavelength.

Energy Level Transitions

In a hydrogen atom, the energy level transitions are the movements of an electron between different orbits around the nucleus. These transitions aren't arbitrary; each one corresponds to a specific change in energy. When an electron absorbs energy, it jumps to a higher orbit; when it loses energy, it falls back to a lower one.

The energies of these orbits are defined by the principal quantum number 'n' and can be calculated with the formula \(E_n = -13.6 \text{ eV}/n^2\). Because the jump must be from one exact energy level to another, and since the energy levels become denser as they go higher, the absorbed photons create a series of lines, not a continuous spectrum. This is analogous to how you can jump from one step to another on a staircase but not to a space in between steps. This stepwise transition precludes the absorption of wavelengths that don't perfectly match the energy required to move an electron from one energy level to another specific one.