Question

Hydrogen atoms are bombarded with 13.1-eV electrons. Determine the shortest wavelength line the atom will emit.

Short answer

Answer: The shortest wavelength line is 121.6 nm.

Step-by-step solution

Calculate the Energy of an Electron

First, we need to convert the energy of the electron in electron volts (eV) to Joules (J) using the relation 1 eV = 1.6 x 10^-19 J. This will allow us to have the energy in a more convenient unit for the calculations. Energy of electron in eV = 13.1 eV Energy of electron in Joules (J) = 13.1 eV * (1.6 * 10^-19 J/eV) = 2.096 x 10^-18 J.

Calculate the Maximum Change in Energy Levels

Since we need to find the shortest wavelength, we need to consider the maximum change in energy levels. The initial energy level (n_initial) is given by the energy of the bombarded electron. The hydrogen atom energy levels are given by the Rydberg formula: E_n = -13.6 eV/n^2, where E_n is the energy of the level n. Now we need to determine the highest possible energy level (n_final) when the electron absorbs the bombarded electron's energy. E_final = E_initial + 13.1 eV, where E_initial = -13.6 eV/n_initial^2, and E_final = -13.6 eV/n_final^2.

Solve for the Initial and Final Energy Levels

Solve the equation for n_final in terms of n_initial: -13.6 eV/n_final^2 = -13.6 eV/n_initial^2 + 13.1 eV. Now, we need to find the values of n_initial and n_final for which the energy difference is the highest: max[(-13.6 eV/n_initial^2) - (-13.6 eV/n_final^2)] = 13.1 eV, by using trial and error, we find n_initial = 1 and n_final = 3.

Calculate the Wavelength using the Rydberg Formula

Now that we have the initial and final energy levels, we can use the Rydberg formula for the hydrogen atom to calculate the wavelength: 1/λ = R_H * (1/n_initial^2 - 1/n_final^2), where λ is the wavelength and R_H is the Rydberg constant for hydrogen (1.097 x 10^7 m^-1). Plug in the values for n_initial = 1 and n_final = 3: 1/λ = 1.097 x 10^7 m^-1 * (1/1^2 - 1/3^2) = 1.097 x 10^7 m^-1 * (1 - 1/9) = 1.097 x 10^7 m^-1 * (8/9).

Calculate the Shortest Wavelength

Finally, calculate the shortest wavelength: λ = 1 / [1.097 x 10^7 m^-1 * (8/9)] = 1.216 x 10^-7 m. Hence, the shortest wavelength line that the hydrogen atom will emit when bombarded by 13.1-eV electrons is 1.216 x 10^-7 m or 121.6 nm.

Key concepts

Rydberg Formula

The Rydberg Formula helps us understand the relationship between energy levels and the emission or absorption of specific wavelengths of light in hydrogen atoms. It is expressed as:\[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_{initial}^2} - \frac{1}{n_{final}^2} \right) \]where:

• \( R_H \) is the Rydberg constant, approximately \( 1.097 \times 10^7 \text{ m}^{-1} \)
• \( n_{initial} \) and \( n_{final} \) are the principal quantum numbers of the initial and final energy levels
• \( \lambda \) is the wavelength of light emitted or absorbed

For the hydrogen atom to emit light of the shortest possible wavelength, the transition must involve the largest possible drop in energy levels.
In the given exercise, going from an energy level with \( n_{initial} = 1 \) to \( n_{final} = 3 \) provides us this maximum energy difference.
Inserting these values into the Rydberg Formula, we can determine the wavelength of light emitted.This concept is crucial in spectral analysis, where scientists use these calculations to identify elements in stars and galaxies based on their emitted light.

Electron Energy Conversion

Energy conversion is vital for determining how much energy an electron gains or loses when moving between energy levels in an atom.
When we talk in terms of electrons, their energy is often measured in electron volts (eV).However, for scientific calculations, it is often necessary to convert this energy to Joules (J).

• For conversion: \( 1 \text{ eV} = 1.6 \times 10^{-19} \text{ J} \)

In the original problem, the electron is bombarded with \( 13.1 \text{ eV} \).
This energy needs to be converted to Joules to be compatible with other values in calculations.
So, the converted energy becomes \( 2.096 \times 10^{-18} \text{ J} \).This energy helps us understand how high an electron can be excited before it releases its energy as light, moving between energy levels.

Energy Level Transitions

Energy level transitions refer to the changes in the electron's state within an atom when it absorbs or releases energy.
In hydrogen, these transitions are particularly important because each level corresponds to a specific energy.

• When an electron absorbs energy, it moves to a higher energy level \((n_{final} > n_{initial})\).
• Alternatively, when it emits energy, it moves to a lower energy level \((n_{final} < n_{initial})\).

In our exercise, we track an electron transitioning from initial level \( n_{initial} = 1 \) to \( n_{final} = 3 \).This maximum transition signifies the release of energy which results in the emission of light with the shortest wavelength.
The energy that is emitted is calculated using the difference between these energy levels.
Energy level transitions are the basis behind the spectral lines observed and are extensively used for analyzing atomic spectra and identifying elements.