Question

A hydrogen atom in its ground state absorbs a photon of energy 12.1 eV. To what energy level is the atom excited?

Short answer

The atom is excited to the third energy level (n=3).

Step-by-step solution

Understand the Problem

A hydrogen atom in its ground state absorbs a photon of energy 12.1 eV. We need to find out to which energy level the atom is excited after absorbing this energy.

Recall the Energy Level Formula

The energy of an electron in a hydrogen atom at a particular energy level \( n \) is given by the formula \( E_n = -13.6 \frac{1}{n^2} \) eV, where \( n \) is the principal quantum number.

Calculate Initial Energy Level

The ground state of a hydrogen atom corresponds to \( n = 1 \), therefore the energy is \( E_1 = -13.6 \frac{1}{1^2} = -13.6 \) eV.

Determine Final Energy Level

The final energy \( E_f \) after absorbing 12.1 eV will be \( E_f = E_1 + 12.1 \) eV. So, \( E_f = -13.6 + 12.1 = -1.5 \) eV.

Find Corresponding Principal Quantum Number

Set up the equation \( E_n = -1.5 = -13.6 \frac{1}{n^2} \). Solve for \( n \): 1.5 = 13.6 / n^2 \Rightarrow n^2 = 13.6 / 1.5 = 9.0667 \Rightarrow n = \sqrt{9.0667} \approx 3.Thus, the hydrogen atom is excited to the third energy level, \( n = 3 \).

Key concepts

Photon Energy Absorption

When a hydrogen atom absorbs a photon, it gains energy. This photon must have just the right amount of energy to move the electron from one energy level to another. This is called *quantization* of energy. For a photon to be absorbed by an atom, its energy must match the energy difference between two specific energy levels.
In our case, the photon energy is 12.1 eV. When the hydrogen atom in its ground state absorbs this, it moves the electron to a higher energy level. This means that the electron transitions from its original state (in this case, the ground state) to another state of higher energy.
Here are some key points about photon absorption by atoms:

• Each type of atom has unique energy levels, resulting in specific frequencies or wavelengths of light they can absorb.
• Only photons with the right energy can be absorbed; others pass by without interaction.
• The absorption of photons can result in electron excitation, causing the atom to move to a higher energy state.

Principal Quantum Number

The principal quantum number, denoted as \( n \), is a vital concept in quantum mechanics and helps us describe the distinct energy states of electrons in an atom. It indicates the relative size and energy of atomic orbitals. The higher the value of \( n \), the higher the energy and the further the electron is from the nucleus.
The energy levels in a hydrogen atom can be calculated using the formula:\[ E_n = -13.6 \frac{1}{n^2} \text{ eV} \]
This formula shows the energy for each level \( n \). As \( n \) increases, the energy becomes less negative, reflecting increased energy.
Here are some important aspects of the principal quantum number:

• \( n=1 \) is the ground state or lowest energy level of the electron.
• Energy levels are quantized, meaning they can only take specific values defined by \( n \).
• Transitions between these quantized levels involve absorption or emission of photons.

Understanding \( n \) helps us predict which energy levels electrons will transition to upon absorbing specific photon energies.

Electron Excitation

Electron excitation refers to the process where an electron absorbs energy and moves from a lower energy level to a higher one within an atom. In a hydrogen atom, this often involves moving from the ground state to an excited state when it absorbs a photon.
When the hydrogen atom's electron is excited from the ground state by absorbing a photon of 12.1 eV, it transitions from \( n=1 \) to \( n=3 \). The energy absorbed corresponds to the difference in energy between these two states.

• Initially, the electron is in the ground state with energy \( E_1 = -13.6 \text{ eV} \).
• After absorbing 12.1 eV, it reaches \( E_f = -1.5 \text{ eV} \).
• The new principal quantum number \( n \) is found to be 3, indicating the third energy level.

Understanding electron excitation is crucial in fields such as spectroscopy and quantum chemistry, where these energy changes help identify elements and compounds.