Bohr Model: Videos & Practice Problems

In the Bohr model of the atom, electrons orbit the nucleus in defined paths known as shells, each with a specific energy level denoted by the principal quantum number (n). The potential energy of an electron in a shell is given by the equation:

$E_n=-R(Z^2/n^2)$

where R is the Rydberg constant (2.178 x 10^-18 J), Z is the atomic number, and n is the shell number. Electrons can transition between shells by absorbing or emitting energy, with absorption moving an electron to a higher energy shell and emission returning it to a lower one. The energy required for these transitions depends on the distance between shells; the greater the distance, the more energy needed. As shell numbers increase, the distance between them decreases, requiring less energy for an electron to move between higher shells. This concept is critical for understanding atomic structure and electron behavior.

Bohr Model of the Atom

In the Bohr model of the atom, electrons are depicted as traveling in circular orbits, known as shells, around the nucleus, which contains protons and neutrons. The shells are designated by the variable n, representing both the shell number and the energy level of the electrons. Each shell corresponds to a specific potential energy, which is influenced by the distance of the electron from the nucleus.

Central to the Bohr model is the Rydberg constant, which quantifies the energy levels of electrons. When expressed in joules, the Rydberg constant is valued at \( R = 2.178 \times 10^{-18} \) joules. The nucleus, depicted in orange, contains positively charged protons and neutral neutrons, while negatively charged electrons occupy the surrounding shells. The first shell corresponds to n = 1, the second shell to n = 2, and so forth.

The potential energy of an electron in a specific shell can be calculated using the formula:

\( \Delta E = E_n = -R \cdot \frac{Z^2}{n^2} \)

In this equation, \( \Delta E \) represents the change in energy, \( E_n \) is the potential energy of the electron, \( R \) is the Rydberg constant, \( Z \) is the atomic number of the element, and \( n \) is the shell number. For instance, hydrogen, the first element in the periodic table, has an atomic number of \( Z = 1 \). The formula illustrates that the energy of an electron is inversely proportional to the square of its shell number, indicating that electrons in lower shells (closer to the nucleus) have lower potential energy compared to those in higher shells.

Understanding this model allows for the determination of the potential energy associated with any electron in an atom, emphasizing the relationship between an electron's position and its energy within the atomic structure.

Bohr Model Example

To calculate the energy of an electron in the second shell of a hydrogen atom, we focus on its potential energy, which can be expressed using the formula:

$\Delta E = -R \cdot \frac{Z^2}{n^2}$

In this equation, $\Delta E$ represents the change in energy, $R$ is the Rydberg constant, $Z$ is the atomic number, and $n$ is the principal quantum number corresponding to the energy level or shell number.

For hydrogen, the atomic number $Z$ is 1. The Rydberg constant $R$ is approximately $-2.178 \times 10^{-18}$ joules. Since we are interested in the second shell, we set $n = 2$. Plugging these values into the formula gives:

$\Delta E = -(-2.178 \times 10^{-18}) \cdot \frac{1^2}{2^2}$

Calculating $1^2$ yields 1, and $2^2$ yields 4. Thus, the equation simplifies to:

$\Delta E = -2.178 \times 10^{-18} \cdot \frac{1}{4}$

This results in:

$\Delta E = -5.445 \times 10^{-19}$ joules

When reporting this value, it is important to consider significant figures. In this case, using four significant figures is appropriate, leading to the final energy of the electron in the second shell of the hydrogen atom being approximately:

$\Delta E \approx -5.445 \times 10^{-19}$ joules

Electronic Transitions

In an atom, electrons occupy specific energy levels known as orbits or shells. These shells are numbered, with lower numbers indicating closer proximity to the nucleus. Electrons can transition between these shells through processes called absorption and emission. Absorption occurs when an electron gains energy, allowing it to move from a lower shell to a higher shell. For instance, when an electron in the first shell absorbs energy from an external source, such as a photon, it can jump to a higher energy state, referred to as the excited state. In this scenario, the electron might move from shell 1 to shell 3.

Conversely, emission is the process where an electron releases energy and transitions from a higher shell back to a lower shell. After a period of time in the excited state, the electron will emit the excess energy and return to its original position, known as the ground state. For example, an electron may fall from shell 3 back down to shell 1.

The energy required for these transitions varies depending on the shells involved. As the shell number increases, the distance between the shells decreases. This means that moving between lower-numbered shells requires more energy than moving between higher-numbered shells. For instance, the energy needed to transition from shell 1 to shell 2 is greater than that required to move from shell 3 to shell 4. Therefore, the relationship between distance and energy is crucial: the greater the distance an electron must travel between shells, the more energy is needed for that transition. Consequently, it is easier for an electron to move from shell 6 to shell 7 than from shell 1 to shell 2.

Which of the electron transitions represents absorption with the greatest frequency?

• A. n = 5 to n = 3
• B. n = 1 to n =3
• C. n = 2 to n = 4
• D. n = 6 to n = 7
• E. n = 4 to n = 5

Answer: (B)

Which of the following transitions (in a hydrogen atom) represent emission of the shortest wavelength?

• A. n = 3 to n = 1
• B. n = 2 to n = 4
• C. n = 1 to n = 4
• D. n = 5 to n = 3
• E. n = 2 to n = 5

Answer: (A)

If the energy of an electron within the boron atom was calculated as –6.0556 x 10^-18 J, at what energy level would it reside?

• A. 2
• B. 3
• C. 4
• D. 5

Answer: (B)

Here’s what students ask on this topic:

What is the Bohr model?

The Bohr model, proposed by Niels Bohr in 1913, is a theory for the hydrogen atom based on quantum theory that energy is transferred only in certain well-defined quantities. It posits that electrons orbit the nucleus in discrete, stable orbits with specific, fixed energies. This was a significant advancement from the earlier Rutherford model, which couldn't explain why atoms emitted light in specific wavelengths or why electrons weren't pulled into the nucleus.

Bohr suggested that as long as an electron remains in its orbit, it does not radiate energy. However, it can jump from one energy level to another, and when it does, it either absorbs or emits a photon with energy equal to the difference between the two levels. This model introduced the concept of quantization in the atomic structure, which was a fundamental step towards the development of quantum mechanics. However, the Bohr model only accurately describes the hydrogen atom and is oversimplified for more complex atoms.

How did Niels Bohr describe electrons in his atomic model?

Niels Bohr, building upon Rutherford's atomic model, proposed that electrons orbit the nucleus in distinct energy levels, which he referred to as "shells" or "orbits." According to Bohr's atomic model, these electrons could only occupy certain allowed orbits where they would not emit radiation. However, they could jump from one energy level to another, a process accompanied by the absorption or emission of a quantum of electromagnetic radiation, which we observe as light or other forms of electromagnetic waves.

Bohr's model was revolutionary because it introduced the concept of quantization in the structure of the atom, meaning that the electrons' energy was quantized, existing in discrete levels rather than a continuous spectrum. This model successfully explained the spectral lines of hydrogen but had limitations when applied to more complex atoms. Despite its limitations, Bohr's model was a significant step forward in the development of quantum mechanics and our understanding of atomic structure.

How do you draw a Bohr model?

To draw a Bohr model for an atom, follow these steps:

1. Determine the Number of Electrons: Find out how many electrons the atom has. This is equal to the atomic number of the element, which you can find on the periodic table.
2. Draw the Nucleus: In the center of your paper, draw a circle to represent the nucleus. Inside it, write the symbol of the element you're drawing.
3. Add Electron Shells: Around the nucleus, draw concentric circles to represent the electron shells. Each shell can hold a certain number of electrons: the first can hold up to 2, the second up to 8, the third up to 18, and so on.
4. Place Electrons: Starting with the innermost shell, place dots around the circles to represent electrons. Fill each shell according to its capacity before moving to the next one.
5. Check Your Work: Ensure that the total number of electrons equals the atomic number of the element. The Bohr model is a simplified representation and works well for explaining atoms up to calcium (with atomic number 20). For elements with more electrons, the model becomes less accurate due to the complexities of subshell filling and electron repulsion.