DATA For your work in a mass spectrometry lab, you are...

DATA 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 $13.56 \mathrm{nm},$ 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 $6.78 \mathrm{nm},$ a free electron is produced. What is the kinetic energy (in electron volts) of the electron?

Key Concepts

Photon Absorption and Spectral Transitions

When a photon is absorbed by an atom or ion, its energy is used to promote an electron from a lower energy level to a higher one or to ionize the electron if the energy surpasses the ionization threshold. The specific wavelengths at which absorption occurs are directly linked to the differences in energy levels. This concept is key in spectroscopy for interpreting absorption spectra and understanding how atomic structure affects the wavelengths of absorbed light.

Ionization Energy

Ionization energy is the amount of energy required to remove an electron completely from an atom or ion, transitioning it from a bound state to a free state. For hydrogen-like ions, this energy scales as Z² times that of hydrogen's ionization energy. Understanding ionization energy is crucial for determining the kinetic energy of electrons that are freed after photon absorption when the photon energy exceeds the binding energy.

Rydberg Formula

The Rydberg formula relates the wavelengths of spectral lines to transitions between energy levels in one?electron systems. It is expressed as 1/? = R Z² (1/n?² - 1/n?²), where R is the Rydberg constant, Z is the atomic number, and n? and n? are the principal quantum numbers of the lower and upper energy levels. This formula is essential for calculating both the absorption wavelengths and the atomic number from spectral data.

Hydrogenic Energy Levels

This concept refers to the discrete energy levels of one?electron ions that are similar to the Hydrogen atom. According to the Bohr model, the energy of an electron in a given level is determined by the atomic number (Z) and the principal quantum number (n). These quantized energy levels are fundamental to predicting the wavelengths at which the ion can absorb or emit photons during transitions.

Transcript

00:01 Okay, here we're talking about the energy levels of electrons that are changing over the levels.

00:10 So we're told that we have some ion and most of its electrons are in the ground state, so i'll take an initial to be equal to one.

00:24 And we're told that the longest wavelength that we observe for absorption is the wavelength have written up there 13 .56 nanometers.

00:36 So what that's telling us is that the photon emitted, or i should say, absorbed when the electron moves from one energy levels to next, has this wavelength.

00:51 We know that it's this particular transition from n equals 1 to n equals 2 because we're told that this is the longest wavelength.

00:59 And remember that energy is inversely correlated with, or actually it's proportional to wavelength.

01:09 So if this is the longest wavelengths, then this has got to be the smallest energy transition.

01:16 The smallest transition from one is obviously to two.

01:20 All right, so we know our initial n is equal to one and our final is equal to two.

01:25 So write that down and i equals one and f equals two.

01:30 All right, so what we're asked to find in this question, part a, is the z value of this ion.

01:40 All right, so we're going to solve this using our formula for the wavelength and transitions, and that formula says that 1 over lambda is equal to z squared times r, which is the redbird constant, over 1 over n .i squared minus 1 over n.

02:00 F squared.

02:02 All right, so we're going to be solving for z...

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