Question

What is the energy of the orbiting electron in a hydrogen atom with a quantum number of \(45 ?\)

Short answer

Answer: The energy of the orbiting electron in a hydrogen atom with a quantum number of 45 is approximately -0.0067 eV.

Step-by-step solution

Write down the given information and formula

The quantum number of the hydrogen atom is given: \(n=45\). The formula for the energy levels of the hydrogen atom is: \(E_n = -\frac{13.6 \text{eV}}{n^2}\)

Substitute the given quantum number in the formula

We need to find the energy of the orbiting electron in the hydrogen atom. Substitute the given quantum number \(n=45\) in the energy formula: \(E_{45} = -\frac{13.6 \text{eV}}{45^2}\)

Calculate the energy of the electron

Now, calculate the energy of the electron in the hydrogen atom with the given quantum number: \(E_{45} = -\frac{13.6 \text{eV}}{45^2} = -\frac{13.6 \text{eV}}{2025} = -0.006716049 \text{eV}\) The energy of the orbiting electron in a hydrogen atom with a quantum number of \(45\) is approximately \(-0.0067 \text{eV}\).

Key concepts

Quantum Number

Quantum numbers are essential in describing the specific energy levels within an atom. They are like an address system for electrons, indicating where an electron might be found in an atom and its energy level.
For any atom, the principal quantum number, denoted as \( n \), primarily determines the energy level. It takes positive integer values like 1, 2, 3, and so on.

• A higher quantum number means that the electron is in a larger orbit, farther from the nucleus, and usually at a higher energy level.
• In the given exercise, the quantum number is 45, which means the electron is quite far from the nucleus and in a very high energy orbit. Despite its large value, the actual energy calculated is small, as observed.

Understanding quantum numbers helps in predicting the arrangement of electrons in atoms and thus is crucial in quantum mechanics.

Bohr Model

The Bohr Model of the atom was a pivotal development in the understanding of atomic structure. Proposed by Niels Bohr in 1913, it introduced the concept of quantized orbits for electrons.
This model suggests that electrons move in fixed circular paths, called orbits, around the nucleus, with each orbit corresponding to a specific energy level.

• In Bohr's model, electrons transition between orbits by absorbing or emitting energy in discrete quanta.
• The specific formula used in the exercise, \( E_n = -\frac{13.6 \text{eV}}{n^2} \), comes from Bohr's work and predicts the energy levels of the hydrogen atom. It remains useful today for educational purposes, although it has been refined by newer quantum mechanical models.

The model is crucial in illustrating the quantization of energy levels and explaining phenomena such as atomic spectra.

Electron Energy Calculation

Calculating electron energy in atoms involves specific formulas derived from models like Bohr's. In the hydrogen atom, this calculation is straightforward due to its simplicity as a single-electron system.
Using the formula \( E_n = -\frac{13.6 \text{eV}}{n^2} \), we can determine the energy of an orbiting electron if the quantum number \( n \) is known.

• This equation shows that the energy is inversely proportional to the square of the quantum number, \( n^2 \). Thus, as \( n \) increases, the energy becomes less negative, approaching zero.
• For the quantum number 45, as calculated, the energy is \(-0.0067 \text{eV}\), indicating a weakly bound state slightly negative, implying an electron loosely associated with the nucleus at this high level.

Understanding electron energy calculations helps in analyzing atomic behavior and electronic transitions comprehensively, especially in hydrogen-like systems.