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Chapter 4: Problem 63

Use the Bohr model of the atom to explain why saying that an electron can be only at certain distances from the nucleus is the same thing as saying that the electron can have only certain energies.

Short Answer

Expert verified
In the Bohr model, electrons orbit around the nucleus at specific distances or energy levels, which are quantized. This means that electrons cannot exist between these levels. The angular momentum quantization rule (L = nℏ) and the energy formula (E = -k/n²) further establish a connection between electron positions and energy levels. Therefore, stating that electrons can only be at certain distances from the nucleus is equivalent to stating that electrons can only have specific energy levels.

Step by step solution

01

Introduce the Bohr model

The Bohr model, proposed by Niels Bohr in 1913, is a simplified representation of an atom where electrons orbit around the nucleus at specific distances, called energy levels.
02

Explain energy quantization

In the Bohr model, electrons can only exist at certain energy levels. These levels are quantized, meaning that electrons can only occupy specific, discrete levels and do not exist between these levels. Thus, there is a direct relationship between energy and distance from the nucleus: the farther the electron is from the nucleus, the higher its energy level.
03

Define the angular momentum quantization

Bohr's model also included the angular momentum quantization rule, which states that the angular momentum (L) of an orbiting electron can only take certain discrete values given by the formula: \[ L = n\hbar, \] where n is a positive integer (called the principal quantum number), and \(\hbar\) (h-bar) is the reduced Planck constant.
04

Relate electron energy to angular momentum

Since the angular momentum is quantized and depends on the distance of the electron from the nucleus, the associated energy is also quantized. The energy (E) of an electron is given by the formula: \[ E = \frac{-k}{n^2}, \] where k is a constant and n is the principal quantum number. The energy of an electron is inversely proportional to the square of the quantum number n. As n increases, the energy level separation decreases and the electron becomes less tightly bound to the nucleus.
05

Summarize the relationship between electron distance and energy

In conclusion, according to the Bohr model, electrons can only be located at specific distances from the nucleus, called energy levels. Their energy is directly related to their distance from the nucleus, with higher energy levels corresponding to greater distances. This model's quantization of energy levels, coupled with the quantization of angular momentum, demonstrates that saying an electron can be only at certain distances from the nucleus is the same thing as saying the electron can have only certain energies.

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Most popular questions from this chapter

Predict the formula of the compound formed by the reaction between: (a) \(\mathrm{Ca}\) and \(\mathrm{Br}\) (b) \(\mathrm{K}\) and \(\mathrm{N}\) (c) \(\mathrm{Al}\) and \(\mathrm{S}\) (d) \(\mathrm{Na}\) and \(\mathrm{I}\) (e) \(\mathrm{Mg}\) and \(\mathrm{O}\)

Which atom is larger, lithium (Li) or sodium (Na)? Explain your answer.

What gave Schrödinger justification to think of an electron in an atom as a nebulous cloud?

Predict the formula for the compound aluminum nitride made from the elements aluminum and nitrogen, and explain how you made your prediction.

When a hydrogen atom is excited in a flame, a line of blue-violet light is emitted. This happens when an electron makes a transition between the \(n=5\) and the \(n=2\) orbit of the atom. If the energy of this light is \(2.9 \mathrm{eVs}\) and the energy of an electron in the \(n=2\) hydrogen orbit is \(11.2 \mathrm{eVs}\), what must be the energy of the \(n=5\) orbit?
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