Energy Levels in Hydrogen Atoms

Oct 13, 2025 by talynt.org 79 views

Introduction

In chemistry, understanding the energy levels of atoms is crucial for explaining various phenomena, such as atomic spectra and chemical reactivity. The hydrogen atom, being the simplest atom, is an ideal system to study energy transitions. In this article, we will calculate the energy difference $(\Delta E)$ for the transition from $n=2$ to $n=5$ for 1 mol of hydrogen atoms.

Fundamental Constants

Before we dive into the calculation, let's recall some fundamental constants:

Planck's constant ( $h$ ): 6.62607015 × 10^-34 J s
Speed of light ( $c$ ): 299792458 m/s
Elementary charge ( $e$ ): 1.60217662 × 10^-19 C
Mass of an electron ( $m_e$ ): 9.10938356 × 10^-31 kg
Bohr radius ( $a_0$ ): 5.29177209 × 10^-11 m

Energy Levels in Hydrogen Atoms

The energy levels of a hydrogen atom are given by the formula:

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

where $n$ is the principal quantum number. For $n=2$ and $n=5$ , the energies are:

E_2 = -\frac{13.6 \text{ eV}}{2^2} = -3.4 \text{ eV}

E_5 = -\frac{13.6 \text{ eV}}{5^2} = -0.544 \text{ eV}

Calculating Energy Differences

The energy difference $(\Delta E)$ for the transition from $n=2$ to $n=5$ is given by:

\Delta E = E_5 - E_2 = -0.544 \text{ eV} - (-3.4 \text{ eV}) = 2.856 \text{ eV}

Converting to Joules

To convert the energy difference from electronvolts (eV) to joules (J), we use the conversion factor:

1 \text{ eV} = 1.60217662 × 10^{-19} \text{ J}

Therefore, the energy difference in joules is:

\Delta E = 2.856 \text{ eV} × 1.60217662 × 10^{-19} \text{ J/eV} = 4.572 × 10^{-19} \text{ J}

Calculating for 1 Mol of Hydrogen Atoms

To calculate the energy difference for 1 mol of hydrogen atoms, we need to multiply the energy difference by Avogadro's number ( $N_A$ ):

N_A = 6.02214076 × 10^{23} \text{ mol}^{-1}

Therefore, the energy difference for 1 mol of hydrogen atoms is:

\Delta E = 4.572 × 10^{-19} \text{ J} × 6.02214076 × 10^{23} \text{ mol}^{-1} = 276.5 \text{ J/mol}

Rounding to 4 Significant Figures

Finally, we round the energy difference to 4 significant figures:

\Delta E = 276.5 \text{ J/mol}

Conclusion

In this article, we calculated the energy difference $(\Delta E)$ for the transition from $n=2$ to $n=5$ for 1 mol of hydrogen atoms. We used the fundamental constants and the energy level formula to derive the energy difference in joules and then converted it to joules per mole. The final answer is:

276.5 J/mol

Frequently Asked Questions

In this article, we will address some common questions related to calculating energy differences in hydrogen atoms.

Q: What is the significance of calculating energy differences in hydrogen atoms?

A: Calculating energy differences in hydrogen atoms is crucial for understanding various phenomena in chemistry and physics, such as atomic spectra and chemical reactivity. It helps us to explain how atoms interact with each other and how they respond to different energies.

Q: What is the formula for calculating energy levels in hydrogen atoms?

A: The energy levels in hydrogen atoms are given by the formula:

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

where $n$ is the principal quantum number.

Q: How do I calculate the energy difference between two energy levels?

A: To calculate the energy difference between two energy levels, you need to subtract the energy of the lower energy level from the energy of the higher energy level. For example, to calculate the energy difference between $n=2$ and $n=5$ , you would use the formula:

\Delta E = E_5 - E_2 = -\frac{13.6 \text{ eV}}{5^2} - \left(-\frac{13.6 \text{ eV}}{2^2}\right)

Q: What is the conversion factor between electronvolts (eV) and joules (J)?

A: The conversion factor between electronvolts (eV) and joules (J) is:

1 \text{ eV} = 1.60217662 × 10^{-19} \text{ J}

Q: How do I calculate the energy difference for 1 mol of hydrogen atoms?

A: To calculate the energy difference for 1 mol of hydrogen atoms, you need to multiply the energy difference by Avogadro's number ( $N_A$ ):

N_A = 6.02214076 × 10^{23} \text{ mol}^{-1}

Q: What is the final answer for the energy difference between $n=2$ and $n=5$ for 1 mol of hydrogen atoms?

A: The final answer for the energy difference between $n=2$ and $n=5$ for 1 mol of hydrogen atoms is:

276.5 J/mol

Common Mistakes to Avoid

When calculating energy differences in hydrogen atoms, it's essential to avoid common mistakes. Here are a few:

Incorrect calculation of energy levels: Make sure to use the correct formula for calculating energy levels.
Incorrect conversion between units: Be careful when converting between electronvolts (eV) and joules (J).
Incorrect calculation of energy differences: Double-check your calculations to ensure that you're getting the correct energy difference.

Conclusion

Calculating energy differences in hydrogen atoms is a fundamental concept in chemistry and physics. By understanding how to calculate energy differences, you can gain insights into various phenomena, such as atomic spectra and chemical reactivity. Remember to avoid common mistakes and use the correct formulas and conversion factors to ensure accurate results.