Bohr Model Practice Answer Key

Mastering the Bohr Model: A Comprehensive Guide with Practice Problems and Solutions

The Bohr model, while a simplification of atomic structure, provides a foundational understanding of electron behavior and energy levels. It's a crucial stepping stone in learning more complex atomic models and quantum mechanics. This article provides a comprehensive guide to the Bohr model, including practice problems with detailed solutions, explanations of key concepts, and frequently asked questions. Mastering the Bohr model is key to understanding atomic spectra, ionization energy, and the quantum nature of matter.

Understanding the Bohr Model: Key Concepts

The Bohr model, proposed by Niels Bohr in 1913, revolutionized our understanding of the atom. It addressed the limitations of previous models by introducing the concept of quantized energy levels. Unlike the earlier "plum pudding" model, which depicted electrons embedded in a positive sphere, Bohr proposed a planetary model where electrons orbit the nucleus in specific, defined energy levels.

Here are the key postulates of the Bohr model:

1. Electrons orbit the nucleus in specific energy levels: These energy levels are quantized, meaning electrons can only exist in these discrete energy states and not in between. Each energy level is associated with a specific energy value. The lowest energy level is called the ground state (n=1).

2. Electrons can transition between energy levels: Electrons can absorb energy (e.g., from light) and jump to a higher energy level (excited state). When they return to a lower energy level, they emit energy in the form of light, resulting in a characteristic atomic spectrum.

3. Angular momentum is quantized: The angular momentum of an electron in a particular orbit is an integer multiple of h/2π, where h is Planck's constant. This quantization condition restricts the allowed orbits.

4. The energy of an electron is determined by its energy level: The energy of an electron in a specific energy level (n) can be calculated using the formula: E_n = -R_H(Z²/n²), where R_H is the Rydberg constant (2.18 x 10^-18 J), Z is the atomic number (number of protons), and n is the principal quantum number.

5. Electrons don't radiate energy while in a stable orbit: This is a departure from classical physics, which predicted that accelerating charged particles (like electrons orbiting a nucleus) would continuously emit radiation and spiral into the nucleus. The Bohr model postulates that electrons in stable orbits do not radiate energy.

Bohr Model Practice Problems and Solutions

Now let's apply these concepts to some practice problems. These problems will cover various aspects of the Bohr model, including calculating energy levels, transition energies, and wavelengths of emitted light.

Problem 1: Calculating the Energy of an Electron in Hydrogen

Calculate the energy of an electron in the n=3 energy level of a hydrogen atom (Z=1).

Solution:

We use the formula: E_n = -R_H(Z²/n²)

• R_H = 2.18 x 10^-18 J
• Z = 1 (for hydrogen)
• n = 3

E₃ = - (2.18 x 10^-18 J) (1²/3²) = -2.42 x 10^-19 J

The energy of the electron in the n=3 level is -2.42 x 10^-19 J. The negative sign indicates that the electron is bound to the nucleus.

Problem 2: Calculating the Energy Difference Between Energy Levels

Calculate the energy difference (ΔE) between the n=2 and n=1 energy levels of a hydrogen atom.

Solution:

First, we calculate the energy of the electron in each level:

• E₂ = -R_H(Z²/n²) = -(2.18 x 10^-18 J)(1²/2²) = -5.45 x 10^-19 J
• E₁ = -R_H(Z²/n²) = -(2.18 x 10^-18 J)(1²/1²) = -2.18 x 10^-18 J

Then, we find the difference:

ΔE = E₂ - E₁ = (-5.45 x 10^-19 J) - (-2.18 x 10^-18 J) = 1.635 x 10^-18 J

The energy difference between the n=2 and n=1 levels is 1.635 x 10^-18 J. This is the energy of the photon emitted when an electron transitions from n=2 to n=1.

Problem 3: Calculating the Wavelength of Emitted Light

What is the wavelength (λ) of the photon emitted when an electron in a hydrogen atom transitions from the n=3 to the n=2 energy level?

Solution:

1. Calculate the energy difference (ΔE): We can use the Rydberg formula to directly calculate the energy of the emitted photon:

   ΔE = R_H * Z² * (1/n₁² - 1/n₂²)

   Where:

   • R_H = 2.18 x 10⁻¹⁸ J
   • Z = 1 (for hydrogen)
   • n₁ = 2 (final energy level)
   • n₂ = 3 (initial energy level)

   ΔE = 2.18 x 10⁻¹⁸ J * 1² * (1/2² - 1/3²) = 3.02 x 10⁻¹⁹ J

2. Relate energy to wavelength using Planck's equation:

   E = hc/λ

   Where:

   • E = energy of the photon (3.02 x 10⁻¹⁹ J)
   • h = Planck's constant (6.626 x 10⁻³⁴ Js)
   • c = speed of light (3.00 x 10⁸ m/s)
   • λ = wavelength

   Solving for λ:

   λ = hc/E = (6.626 x 10⁻³⁴ Js * 3.00 x 10⁸ m/s) / (3.02 x 10⁻¹⁹ J) ≈ 6.56 x 10⁻⁷ m or 656 nm

The wavelength of the emitted photon is approximately 656 nm, which falls within the visible red region of the electromagnetic spectrum.

Problem 4: Ionization Energy of Helium

Calculate the ionization energy of a helium ion (He⁺), which has only one electron.

Solution:

Ionization energy is the energy required to remove an electron from an atom or ion. In this case, we need to calculate the energy required to remove the electron from the ground state (n=1) of He⁺. We use the Bohr model equation, but remember that Z=2 for helium:

E₁ = -R_H(Z²/n²) = -(2.18 x 10^-18 J)(2²/1²) = -8.72 x 10^-18 J

The ionization energy is the absolute value of this energy, as it represents the energy required to remove the electron: 8.72 x 10^-18 J

Problem 5: Determining the Principal Quantum Number

An electron in a hydrogen atom absorbs a photon with a wavelength of 121.6 nm and transitions to a higher energy level. What is the principal quantum number (n) of the final energy level?

Solution:

1. Calculate the energy of the absorbed photon: Use E = hc/λ

   E = (6.626 x 10⁻³⁴ Js * 3.00 x 10⁸ m/s) / (121.6 x 10⁻⁹ m) ≈ 1.63 x 10⁻¹⁸ J

2. Determine the initial energy level: The electron starts in the ground state (n=1) of hydrogen:

   E₁ = -2.18 x 10⁻¹⁸ J

3. Calculate the energy of the final energy level:

   E_final = E₁ + E_photon = -2.18 x 10⁻¹⁸ J + 1.63 x 10⁻¹⁸ J = -0.55 x 10⁻¹⁸ J

4. Solve for the final principal quantum number (n): Use the Bohr model equation:

   E_n = -R_H(Z²/n²)

   -0.55 x 10⁻¹⁸ J = -(2.18 x 10⁻¹⁸ J)(1²/n²)

   Solving for n, we get n ≈ 2

Therefore, the electron transitions to the n=2 energy level.

Limitations of the Bohr Model

While the Bohr model was a significant advance, it has limitations:

• It only works well for hydrogen-like atoms: Atoms with only one electron. For multi-electron atoms, electron-electron interactions complicate the picture.
• It doesn't explain the fine structure of spectral lines: More detailed experiments revealed subtle splitting of spectral lines that the Bohr model couldn't explain.
• It doesn't account for electron wave-particle duality: The Bohr model treats electrons as particles in well-defined orbits, neglecting their wave-like nature.

These limitations led to the development of more sophisticated models, like the quantum mechanical model, which provides a more accurate and complete description of atomic structure.

Frequently Asked Questions (FAQ)

Q1: What is the Rydberg constant?

A1: The Rydberg constant (R_H) is a fundamental physical constant that appears in the formula for calculating the energy levels of hydrogen-like atoms. Its value is approximately 2.18 x 10^-18 J.

Q2: What is the difference between the ground state and an excited state?

A2: The ground state is the lowest energy level of an atom. An excited state is a higher energy level that an electron can occupy after absorbing energy.

Q3: How does the Bohr model explain atomic spectra?

A3: The Bohr model explains atomic spectra by stating that when electrons transition between energy levels, they emit or absorb photons of specific energies (and therefore wavelengths). These photons correspond to the lines observed in the atomic spectrum.

Q4: Why doesn't the Bohr model work for multi-electron atoms?

A4: The Bohr model simplifies the atom by considering only one electron interacting with the nucleus. In multi-electron atoms, electron-electron interactions significantly affect energy levels and electron behavior, making the simple Bohr model inadequate.

Q5: What replaced the Bohr model?

A5: The quantum mechanical model of the atom replaced the Bohr model. It uses the principles of quantum mechanics to accurately describe the behavior of electrons in atoms, overcoming the limitations of the Bohr model.

Conclusion

The Bohr model, despite its limitations, provides a crucial stepping stone in understanding atomic structure. It introduces fundamental concepts like quantized energy levels, electron transitions, and the relationship between energy and wavelength. By working through practice problems and grasping the key principles, you'll build a strong foundation for understanding more advanced atomic models and quantum mechanics. Remember, the Bohr model is a simplified representation, but it serves as an excellent introduction to the fascinating world of atomic physics. Continue your learning journey by exploring the quantum mechanical model to gain a more comprehensive understanding of the atom's intricate nature.