Chapter 2: Problem 42
Consider the spectral lines resulting from the transition \(n=2\) to \(n=1\), in the atoms and ions given below, the shortest wavelength is produced by (1) IIydrogen atom (2) Deuterium atom (3) Singly ionised lithium (4) Doubly ionised lithium
Short Answer
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Singly ionised lithium or Doubly ionised lithium.
Step by step solution
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
energy levels
In atoms, electrons reside in specific regions called energy levels or electron shells. These levels are denoted by quantum numbers such as n=1, n=2, and so on. The energy associated with each level increases as we move further from the nucleus.
When electrons transition between these levels, they either absorb or emit energy in the form of photons. The energy difference between two levels determines the energy of the emitted or absorbed photon, given by the equation: \[ \text{Energy Difference} = E_{n_2} - E_{n_1} \]
Where \[E_n = -13.6 \times Z^2 / n^2 \] (in electron volts, where Z is the atomic number).
Thus, understanding energy levels helps in predicting the wavelengths of the spectral lines generated during these transitions.
When electrons transition between these levels, they either absorb or emit energy in the form of photons. The energy difference between two levels determines the energy of the emitted or absorbed photon, given by the equation: \[ \text{Energy Difference} = E_{n_2} - E_{n_1} \]
Where \[E_n = -13.6 \times Z^2 / n^2 \] (in electron volts, where Z is the atomic number).
Thus, understanding energy levels helps in predicting the wavelengths of the spectral lines generated during these transitions.
wavelength calculation
The wavelength of light emitted during an electronic transition is directly influenced by the energy difference between the initial and final energy levels. According to the Planck-Einstein relation, the energy of the photon is inversely proportional to its wavelength, which is given by: \[ E = \frac{hc}{\text{Wavelength}} \]
Here, \[ h \] is Planck's constant (\[6.626 \times 10^{-34} \] J·s), and \[ c \] is the speed of light (\[3.00 \times 10^8 \] m/s). By rearranging the equation, we can find the wavelength: \[ \text{Wavelength} = \frac{hc}{E} \]
Therefore, a larger energy difference results in a shorter wavelength, and vice versa. This principle is key to interpreting atomic spectra.
Here, \[ h \] is Planck's constant (\[6.626 \times 10^{-34} \] J·s), and \[ c \] is the speed of light (\[3.00 \times 10^8 \] m/s). By rearranging the equation, we can find the wavelength: \[ \text{Wavelength} = \frac{hc}{E} \]
Therefore, a larger energy difference results in a shorter wavelength, and vice versa. This principle is key to interpreting atomic spectra.
hydrogen-like atoms
Hydrogen-like atoms are a class of atoms or ions that contain only one electron, similar to a hydrogen atom. Examples include singly ionized helium (He+) and doubly ionized lithium (Li2+).
The energy levels in these hydrogen-like atoms can be calculated using a modified version of the energy level equation for hydrogen: \[ E_n = -13.6 \times Z^2 / n^2 \]
Here, \[ Z \] is the atomic number. In hydrogen-like atoms, the value of \[ Z \] changes, resulting in different energy levels and differing spectral lines from hydrogen.
In our exercise, we compare hydrogen (\[ Z=1 \]) and different ionized states of lithium (\[ Z=3 \]), which demonstrates that increasing \[ Z \] significantly changes the energy level differences and thus the wavelength of emitted light.
The energy levels in these hydrogen-like atoms can be calculated using a modified version of the energy level equation for hydrogen: \[ E_n = -13.6 \times Z^2 / n^2 \]
Here, \[ Z \] is the atomic number. In hydrogen-like atoms, the value of \[ Z \] changes, resulting in different energy levels and differing spectral lines from hydrogen.
In our exercise, we compare hydrogen (\[ Z=1 \]) and different ionized states of lithium (\[ Z=3 \]), which demonstrates that increasing \[ Z \] significantly changes the energy level differences and thus the wavelength of emitted light.
spectral lines
Spectral lines are unique patterns of dark or bright lines seen in the spectrum of a light source, caused by the transition of electrons between energy levels in atoms. When an electron drops from a higher energy level to a lower one, it emits a photon with a specific wavelength. This results in an emission line in the spectrum.
Similarly, absorption lines occur when electrons absorb photons and jump to higher energy levels. The wavelength of these lines provides vital information about the elements present in the light source and their electronic structures.
Spectral lines are a fundamental tool in disciplines such as astronomy, where they are used to identify the compositions and properties of distant stars and galaxies. In our exercise, the spectral lines compare the wavelengths resulting from electron transitions in different atoms and ions, helping us determine which produces the shortest wavelength.
Similarly, absorption lines occur when electrons absorb photons and jump to higher energy levels. The wavelength of these lines provides vital information about the elements present in the light source and their electronic structures.
Spectral lines are a fundamental tool in disciplines such as astronomy, where they are used to identify the compositions and properties of distant stars and galaxies. In our exercise, the spectral lines compare the wavelengths resulting from electron transitions in different atoms and ions, helping us determine which produces the shortest wavelength.
