Question

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

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

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.

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.

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.

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.

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.

Key concepts

Energy Quantization

In the Bohr model, energy quantization is a cornerstone concept. Electrons circle an atom's nucleus at specific energy levels or shells. It means these energies aren't just any value. They are quantized, which is a fancy way of saying electrons only have certain set energies they can own.
As an electron moves farther from the nucleus, it settles into a higher energy level. If an electron wants to switch between energy levels, it has to jump suddenly rather than gradually moving. For instance, in climbing stairs, one can only stand on certain steps, not float between them. Electrons behave similarly in their energy levels.
This concept is essential as it explains why we see specific spectral lines in atomic emissions rather than a continuous spectrum.

• Electrons are only found in specific energy levels.
• Energy levels are distinct and separate, not continuous.
• Electrons "jump" between these levels by gaining or losing energy, not sliding.

Angular Momentum Quantization

Angular momentum quantization is another critical principle of the Bohr model. An electron's angular momentum is quantized, meaning it can have only a set number of values. This leads to the formula: \[ L = n\hbar, \] where \( n \) is the principal quantum number (a positive integer), and \( \hbar \) is the reduced Planck's constant.
This formula tells us that angular momentum comes in chunks, not continuous streams. This helps us understand that electrons can't choose just any orbit size; their paths around the nucleus are restricted to these quantized values.

• The electron's orbit size is tied to quantized angular momentum.
• Only certain orbits align with these discrete angular momentum values.
• These quantized angular momentum levels connect closely to energy levels.

Principal Quantum Number

The principal quantum number, denoted as \( n \), is a core part of the Bohr model. It's a simple integer that tells us about electron energy levels. Greater the \( n \), higher up the energy ladder the electron is.
It influences the electron's angular momentum and energy directly. An electron's energy can be expressed as: \[ E = \frac{-k}{n^2}, \] where \( k \) is a constant. This equation reveals how energy is intertwined with the principal quantum number. As \( n \) rises, the energy level gaps shrink, meaning the electron is less tightly bound to the nucleus.

• \( n \) indicates the energy level or "shell," which the electron occupies.
• Higher \( n \) values mean higher energy levels and larger orbits.
• \( n \) relates directly to both the quantized energy and angular momentum.