Question

For your work in a mass spectrometry lab, you are investigating the absorption spectrum of one-electron ions. To maintain the atoms in an ionized state, you hold them at low density in an ion trap, a device that uses a configuration of electric fields to confine ions. The majority of the ions are in their ground state, so that is the initial state for the absorption transitions that you observe. (a) If the longest wavelength that you observe in the absorption spectrum is 13.56 nm, what is the atomic number Z for the ions? (b) What is the next shorter wavelength that the ions will absorb? (c) When one of the ions absorbs a photon of wavelength 6.78 nm, a free electron is produced. What is the kinetic energy (in electron volts) of the electron?

Short answer

(a) Calculate \( Z \) using the Rydberg formula. (b) Use the next transition formula for \( Z^2 \) found. (c) Photon energy minus ionization energy gives electron kinetic energy.

Step-by-step solution

Understand the Ion Configuration

Since we are dealing with one-electron ions, we should note that the spectrum is similar to that of the hydrogen atom but with adjustments depending on the atomic number \( Z \). The longest wavelength corresponds to the transition from the first excited state (\( n=2 \)) to the ground state (\( n=1 \)).

Calculate Atomic Number (\( Z \))

Using the Rydberg formula \( \frac{1}{\lambda} = R Z^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \) for the longest wavelength at \( \lambda = 13.56 \text{ nm} = 13.56 \times 10^{-9} \text{ m} \), \( n_1 = 1 \), \( n_2 = 2 \), and the Rydberg constant \( R = 1.097 \times 10^7 \text{ m}^{-1} \), solve for \( Z \):\[ Z^2 = \frac{1}{R \lambda} \frac{4}{3} \]Calculate \( Z \).

Determine the Next Shorter Wavelength

The next shorter wavelength will correspond to a transition from \( n_2=3 \) to \( n_1=1 \). Use \[ \frac{1}{\lambda} = RZ^2 \left( \frac{1}{1^2} - \frac{1}{3^2} \right) \] to find this wavelength.\[ Z^2 \] is already calculated from the previous step.

Convert Wavelength to Energy for Ionization

For (c), calculate the energy of a photon with wavelength \( 6.78 \text{ nm} \) using \( E = \frac{hc}{\lambda} \), where \( h = 6.626 \times 10^{-34} \text{ Js} \) and \( c = 3 \times 10^8 \text{ m/s} \). Convert this energy to electron volts by dividing by \( eV, 1.6 \times 10^{-19} \text{ J/eV} \).

Calculate Kinetic Energy of the Electron

Subtract the ionization energy corresponding to the ground state to free electron (i.e., full ionization with \( n_1=1, n_2=\infty \)) from the previously calculated photon energy to find the kinetic energy of the electron.

Key concepts

Understanding Mass Spectrometry

Mass spectrometry is a powerful tool used to analyze ions based on their mass-to-charge ratio. It can provide detailed information about the composition and abundance of ions. In this exercise, we're looking at one-electron ions confined in an ion trap to maintain them in an ionized state.

• Ion Trap: An ion trap uses electric fields to keep ions stable and in place, which is crucial for precise measurements in mass spectrometry.
• Mass-to-Charge Ratio: The primary information obtained from mass spectrometry, critical for distinguishing between different ions.

By understanding mass spectrometry in this context, we gain insight into how the ions are analyzed as they transition between energy states.

Exploring Absorption Spectrum

The absorption spectrum of an element shows the specific wavelengths of light absorbed by the electrons of its atoms, which corresponds to transitions between energy levels. In the case of one-electron ions, these spectra resemble the hydrogen spectrum but are adjusted based on the atomic number.

• Ground State: Most of the ions are in this lowest energy state initially, requiring energy absorption to transition to a higher state.
• Longest Wavelength: Represents the smallest energy difference, typically seen in transitions from the first excited state to the ground state.

This spectrum aids in calculating the atomic number and understanding electron transitions for the ions in question.

Explaining the Rydberg Formula

The Rydberg formula helps calculate the wavelengths of spectral lines of one-electron ions, adjusting for the atomic number. It’s expressed as:\[ \frac{1}{\lambda} = R Z^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]

• Rydberg Constant (\( R \)): A fundamental constant in physics, typically \( 1.097 \times 10^7 \text{ m}^{-1} \).
• Quantum Numbers: \( n_1 \) and \( n_2 \) represent the initial and final energy levels of the electron transition, respectively.

Using this formula, we can calculate important characteristics like the atomic number or other photon interactions with the ions.

Performing Atomic Number Calculation

Calculating the atomic number \( Z \) of an ion involves analyzing the energy transitions within its electron configuration. For one-electron ions, knowing the wavelength of absorbed light allows for \( Z \) determination using the Rydberg formula.

• Variables Involved: Knowing the wavelengths and applying the appropriate energy levels (\( n_1, n_2 \)).
• Solving for \( Z \): By rearranging the Rydberg formula to isolate \( Z \), we can calculate its value accurately using the known line spectrum data.

This precise calculation is foundational to understanding the ion's energy transitions and its behavior in such an experiment.