The Bohr Model of the Hydrogen Atom

Oct 13, 2025 by pinkandtrash.com 25 views

Introduction

In the context of atomic physics, the Bohr model provides a simplified explanation of the behavior of electrons in a hydrogen atom. According to this model, electrons occupy specific energy levels or shells, and they can transition from one energy level to another by emitting or absorbing energy in the form of electromagnetic radiation. In this article, we will explore the frequency of electromagnetic radiation required to promote an electron from the $n=2$ to the $n=4$ energy level in a Bohr hydrogen atom.

The Bohr Model of the Hydrogen Atom

The Bohr model of the hydrogen atom was introduced by Niels Bohr in 1913. In this model, the electron is assumed to be in a circular orbit around the nucleus, and it can only occupy specific energy levels or shells. The energy of an electron in a particular energy level is given by the equation:

E_n = -\frac{13.6 \text{ eV}}{n^2}

where $n$ is the principal quantum number, which is a positive integer that defines the energy level of the electron.

Energy Levels in the Bohr Hydrogen Atom

In the Bohr model, the energy levels of the hydrogen atom are given by the equation:

E_n = -\frac{13.6 \text{ eV}}{n^2}

The energy levels of the hydrogen atom are:

Energy Level	Principal Quantum Number (n)	Energy (E)
Ground State	1	-13.6 eV
First Excited State	2	-3.4 eV
Second Excited State	3	-1.51 eV
Third Excited State	4	-0.85 eV

Promoting an Electron from $n=2$ to $n=4$

To promote an electron from the $n=2$ to the $n=4$ energy level, the electron must absorb energy in the form of electromagnetic radiation. The energy required to promote an electron from the $n=2$ to the $n=4$ energy level can be calculated using the equation:

\Delta E = E_4 - E_2

Substituting the values of $E_4$ and $E_2$ from the table above, we get:

\Delta E = -0.85 \text{ eV} - (-3.4 \text{ eV}) = 2.55 \text{ eV}

Frequency of Electromagnetic Radiation

The frequency of electromagnetic radiation required to promote an electron from the $n=2$ to the $n=4$ energy level can be calculated using the equation:

f = \frac{\Delta E}{h}

where $h$ is the Planck constant, which is equal to $6.626 \times 10^{-34} \text{ J s}$ .

Substituting the value of $\Delta E$ from the previous equation, we get:

f = \frac{2.55 \text{ eV}}{6.626 \times 10^{-34} \text{ J s}} = 3.86 \times 10^{15} \text{ Hz}

However, this is not the correct answer. We need to convert the energy from eV to Joules and then use the equation:

f = \frac{\Delta E}{h}

to calculate the frequency.

Calculating the Frequency

To calculate the frequency, we need to convert the energy from eV to Joules. We can do this by multiplying the energy in eV by the charge of an electron, which is equal to $1.602 \times 10^{-19} \text{ C}$ .

\Delta E = 2.55 \text{ eV} \times 1.602 \times 10^{-19} \text{ C} = 4.09 \times 10^{-19} \text{ J}

Now, we can use the equation:

f = \frac{\Delta E}{h}

to calculate the frequency.

f = \frac{4.09 \times 10^{-19} \text{ J}}{6.626 \times 10^{-34} \text{ J s}} = 6.17 \times 10^{14} \text{ Hz}

Conclusion

In conclusion, the frequency of electromagnetic radiation required to promote an electron from the $n=2$ to the $n=4$ energy level in a Bohr hydrogen atom is $6.17 \times 10^{14} \text{ Hz}$ .

References

Bohr, N. (1913). On the Constitution of Atoms and Molecules. Philosophical Magazine, 26(16), 1-25.
Griffiths, D. J. (2018). Introduction to Quantum Mechanics. Pearson Education.
Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.

Discussion

Q: What is the Bohr model of the hydrogen atom?

A: The Bohr model of the hydrogen atom is a simplified explanation of the behavior of electrons in a hydrogen atom. In this model, the electron is assumed to be in a circular orbit around the nucleus, and it can only occupy specific energy levels or shells.

Q: What is the energy of an electron in a particular energy level?

A: The energy of an electron in a particular energy level is given by the equation:

E_n = -\frac{13.6 \text{ eV}}{n^2}

where $n$ is the principal quantum number, which is a positive integer that defines the energy level of the electron.

Q: What is the energy required to promote an electron from the $n=2$ to the $n=4$ energy level?

A: The energy required to promote an electron from the $n=2$ to the $n=4$ energy level can be calculated using the equation:

\Delta E = E_4 - E_2

Substituting the values of $E_4$ and $E_2$ from the table above, we get:

\Delta E = -0.85 \text{ eV} - (-3.4 \text{ eV}) = 2.55 \text{ eV}

Q: What is the frequency of electromagnetic radiation required to promote an electron from the $n=2$ to the $n=4$ energy level?

A: The frequency of electromagnetic radiation required to promote an electron from the $n=2$ to the $n=4$ energy level can be calculated using the equation:

f = \frac{\Delta E}{h}

where $h$ is the Planck constant, which is equal to $6.626 \times 10^{-34} \text{ J s}$ .

Substituting the value of $\Delta E$ from the previous equation, we get:

f = \frac{2.55 \text{ eV}}{6.626 \times 10^{-34} \text{ J s}} = 3.86 \times 10^{15} \text{ Hz}

However, this is not the correct answer. We need to convert the energy from eV to Joules and then use the equation:

f = \frac{\Delta E}{h}

to calculate the frequency.

Q: How do I calculate the frequency of electromagnetic radiation required to promote an electron from the $n=2$ to the $n=4$ energy level?

A: To calculate the frequency, you need to convert the energy from eV to Joules by multiplying the energy in eV by the charge of an electron, which is equal to $1.602 \times 10^{-19} \text{ C}$ .

\Delta E = 2.55 \text{ eV} \times 1.602 \times 10^{-19} \text{ C} = 4.09 \times 10^{-19} \text{ J}

Now, you can use the equation:

f = \frac{\Delta E}{h}

to calculate the frequency.

f = \frac{4.09 \times 10^{-19} \text{ J}}{6.626 \times 10^{-34} \text{ J s}} = 6.17 \times 10^{14} \text{ Hz}

Q: What is the significance of the frequency of electromagnetic radiation required to promote an electron from the $n=2$ to the $n=4$ energy level?

A: The frequency of electromagnetic radiation required to promote an electron from the $n=2$ to the $n=4$ energy level is significant because it is a fundamental aspect of atomic physics. Understanding the frequency of electromagnetic radiation required to promote an electron from one energy level to another is crucial in understanding the behavior of electrons in atoms.

Q: Can you provide more information about the Bohr model of the hydrogen atom?

A: Yes, the Bohr model of the hydrogen atom is a simplified explanation of the behavior of electrons in a hydrogen atom. In this model, the electron is assumed to be in a circular orbit around the nucleus, and it can only occupy specific energy levels or shells. The energy of an electron in a particular energy level is given by the equation:

E_n = -\frac{13.6 \text{ eV}}{n^2}

where $n$ is the principal quantum number, which is a positive integer that defines the energy level of the electron.

Q: Can you provide more information about the energy levels of the hydrogen atom?

A: Yes, the energy levels of the hydrogen atom are given by the equation:

E_n = -\frac{13.6 \text{ eV}}{n^2}

The energy levels of the hydrogen atom are:

Energy Level	Principal Quantum Number (n)	Energy (E)
Ground State	1	-13.6 eV
First Excited State	2	-3.4 eV
Second Excited State	3	-1.51 eV
Third Excited State	4	-0.85 eV

Q: Can you provide more information about the frequency of electromagnetic radiation required to promote an electron from the $n=2$ to the $n=4$ energy level?

A: Yes, the frequency of electromagnetic radiation required to promote an electron from the $n=2$ to the $n=4$ energy level is given by the equation:

f = \frac{\Delta E}{h}

where $h$ is the Planck constant, which is equal to $6.626 \times 10^{-34} \text{ J s}$ .

Substituting the value of $\Delta E$ from the previous equation, we get:

f = \frac{2.55 \text{ eV}}{6.626 \times 10^{-34} \text{ J s}} = 3.86 \times 10^{15} \text{ Hz}

However, this is not the correct answer. We need to convert the energy from eV to Joules and then use the equation:

f = \frac{\Delta E}{h}

to calculate the frequency.

Q: Can you provide more information about the significance of the frequency of electromagnetic radiation required to promote an electron from the $n=2$ to the $n=4$ energy level?

A: Yes, the frequency of electromagnetic radiation required to promote an electron from the $n=2$ to the $n=4$ energy level is significant because it is a fundamental aspect of atomic physics. Understanding the frequency of electromagnetic radiation required to promote an electron from one energy level to another is crucial in understanding the behavior of electrons in atoms.

Conclusion

In conclusion, the frequency of electromagnetic radiation required to promote an electron from the $n=2$ to the $n=4$ energy level in a Bohr hydrogen atom is $6.17 \times 10^{14} \text{ Hz}$ . This frequency is significant because it is a fundamental aspect of atomic physics, and understanding it is crucial in understanding the behavior of electrons in atoms.

References

Bohr, N. (1913). On the Constitution of Atoms and Molecules. Philosophical Magazine, 26(16), 1-25.
Griffiths, D. J. (2018). Introduction to Quantum Mechanics. Pearson Education.
Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.