Question

A hydrogen atom is in a state with energy -1.51 eV. In the Bohr model, what is the angular momentum of the electron in the atom, with respect to an axis at the nucleus?

Short answer

The angular momentum is $3\hslash$.

Step-by-step solution

Understanding the Energy Level

In the Bohr model, the energy levels of a hydrogen atom are given by the formula: $$E_n=\frac{-13.6\text{eV}}{n^2}$$ where $E_n$ is the energy of the electron at a particular level $n$. We know that the energy is $-1.51\text{eV}$. We need to find the principal quantum number $n$ using this formula.

Calculate the Principal Quantum Number

Set up the equation based on the known energy level:$$\frac{-13.6\text{eV}}{n^2}=-1.51\text{eV}$$Solve for $n^2$ by dividing both sides by $-13.6\text{eV}$:$$n^2=\frac{13.6}{1.51}\approx 9$$Taking the square root of both sides, we find:$$n=3$$ Thus, the electron is in the third energy level.

Find the Angular Momentum

In the Bohr model, the angular momentum $L$ of an electron is quantized and given by:$$L=n\hslash$$where $\hslash =\frac{h}{2\pi }$ is the reduced Planck's constant, and $n$ is the principal quantum number. Since we found $n=3$, the angular momentum is:$$L=3\hslash$$

Key concepts

Hydrogen Atom

The hydrogen atom, the simplest of all atoms, consists of a single electron orbiting a single proton. This simplicity makes it an excellent starting point for understanding atomic structures. In a hydrogen atom, the electron moves around the nucleus, forming various energy levels. These levels are dictated by specific rules as per Bohr's model. According to Bohr, electrons would only occupy certain orbits, called "quantized orbits," without losing energy by radiation. This concept was revolutionary because it explained the hydrogen spectrum observed.

The hydrogen atom's structure allows us to study fundamental principles in quantum mechanics. Since hydrogen holds only one electron, the interactions are minimal, providing a clean situation to interpret spectral lines. Hydrogen's spectrum consists of distinct lines, which correspond to various transitions of the electron between these energy levels.

Understanding the hydrogen atom sets the foundation for all other atomic structures, as more complex atoms can be interpreted as variations on this simplest case.

Angular Momentum

In the Bohr model, angular momentum refers to the motion of an electron around the nucleus. This concept specifies that the angular momentum of the electron is quantized, meaning it can only take specific values. The idea of quantization was pivotal as it provided a theoretical explanation for the distinct energy levels in an atom.

A key equation that represents angular momentum in Bohr's model is:

• Angular momentum, $L=n\hslash$

where $n$ is the principal quantum number and $\hslash =\frac{h}{2\pi }$, known as the reduced Planck's constant. In our example, if the principal quantum number is 3, the angular momentum of the electron is $3\hslash$.

The concept of angular momentum quantization was groundbreaking as it explained the stability of electrons in atoms, supporting why they didn't spiral into the nucleus. Instead, electrons remain in stable orbits or shells.

Principal Quantum Number

The principal quantum number, denoted as $n$, is a fundamental element in quantum mechanics. It describes the size and energy of the orbital, which tells us the distance of the electron from the nucleus. In the Bohr model, the principal quantum number is a crucial factor in determining the energy levels of an atom.

In the exercise, we used the formula for energy levels in the hydrogen atom:

• $E_n=\frac{-13.6\text{eV}}{n^2}$

We set the known energy $-1.51\text{eV}$ into the formula to solve for $n$, resulting in $n=3$. This tells us the electron resides on the third energy level or orbit. Each value of $n$ corresponds to a major energy level and is sometimes referred to as a shell.

Understanding the principal quantum number is key to knowing the distribution of electrons in an atom. It assists in explaining atomic behavior and spectra.

Energy Levels

Energy levels are core to the structure of atoms. In the Bohr model, electrons reside in specific energy levels or orbits that determine their potential energy within an atom. These energy levels are quantized, meaning electrons can jump between these levels but cannot exist in between them.

For a hydrogen atom, the energy levels can be determined by:

• $E_n=\frac{-13.6\text{eV}}{n^2}$

where $E_n$ represents the energy of a level, and $n$ is the principal quantum number. Energy levels become less negative with increasing $n$, indicating less tightly bound electrons as they move to higher energy states.

Each transition of an electron between levels emits or absorbs a photon, explaining the spectral lines seen in hydrogen's emission and absorption spectra. The quantization of these energy levels highlights the quantum nature of atomic structures.