Question

Without doing detailed calculations, indicate which of the following electron transitions requires the greatest amount of energy to be absorbed by a hydrogen atom: from (a) \(n=1\) to \(n=2 ;\) (b) \(n=2\) to \(n=4 ;\) (c) \(n=3\) to \(n=9 ;\) (d) \(n=10\) to \(n=1\)

Short answer

The electron transition that requires the greatest amount of energy to be absorbed by a hydrogen atom is from \(n=10\) to \(n=1\).

Step-by-step solution

Understand electron transitions

An electron transition in an atom is a change from one energy level or state to another. It typically involves the absorption or emission of energy in the form of a photon. In a hydrogen atom, the energy levels are given by \(n\) where \(n=1\) is the ground state and higher values of \(n\) represent higher energy states.

Analyze the given transitions

Given transitions are (a) \(n=1\) to \(n=2\) (difference=1), (b) \(n=2\) to \(n=4\) (difference=2), (c) \(n=3\) to \(n=9\) (difference=6), and (d) \(n=10\) to \(n=1\) (difference=9). It can be seen directly from these differences that transition (d) \(n=10\) to \(n=1\) would require the highest energy to be absorbed.

Final conclusion

Hence, without performing detailed calculations we can deduce that the transition that will require the most energy to be absorbed by a hydrogen atom will be the transition from \(n=10\) to \(n=1\). This is because this transition results in the greatest change in energy levels amongst the provided choices.

Key concepts

Hydrogen Atom

The hydrogen atom is the simplest atom, comprised of only one proton and one electron. This simplicity makes it a fundamental topic in the study of quantum mechanics and atomic physics.
Understanding its structure helps us comprehend more complex atoms. The electron in a hydrogen atom orbits around the nucleus, much like a planet orbiting the sun. However, instead of following a fixed path, the electron exists in quantized orbital states or "energy levels." These levels are defined by quantum numbers, particularly the principal quantum number denoted as \( n \).
When we refer to a hydrogen atom with its electron in the \( n=1 \) energy level, we are speaking of its ground state, meaning it has the least possible energy. As \( n \) increases, so does the energy level of the electron, moving it to excited states. The quantization of these energy levels is a key principle in quantum mechanics and dependent on settings specific to hydrogen, governed by factors like Planck's constant and the speed of light.

Energy Levels

Energy levels in a hydrogen atom represent the fixed energies that electrons can occupy. They're quantized, meaning electrons can only exist at specific energy states without being in between.
The concept is seriously fascinating, akin to climbing stairs; electrons can "jump" from one step to another but cannot "hang" mid-air between steps. The different energy levels are numbered as \( n=1, 2, 3, \ldots \) and so forth, with each number indicating a higher energy state.

• The ground state (\( n=1 \)) has the lowest energy and is the most stable state of the atom.
• Excited states (\( n > 1 \)) possess higher energy levels.

When an electron transitions, it moves between these quantized states. An electron absorbs energy to move to a higher level (excitation) or releases energy to fall to a lower level (emission). These transitions are central to phenomena such as spectral lines, which we observe in the hydrogen emission spectrum.

Photon Absorption

Photon absorption is critical when discussing electron transitions. When an electron in a hydrogen atom moves to a higher energy level, it absorbs a photon, which is a fundamental particle of light that carries energy.
The amount of energy required for an electron transition is directly related to the difference between the energy levels. In simple terms, the bigger the jump, the more energy is needed, or absorbed, as a photon.

• For example, transitioning from \( n=1 \) to \( n=2 \) involves absorbing a photon of lower energy compared to a jump from \( n=3 \) to \( n=9 \).
• In our earlier problem, moving from \( n=10 \) to \( n=1 \) requires the electron to absorb a large amount of energy because it's a significant drop in energy levels, making it a more energetic transition.

Photon absorption isn't just an atomic affair. It's the foundation of our understanding of how light interacts with matter, essential for technologies like lasers, solar panels, and even photosynthesis in plants.