Question

For the same electronic transition in the following atom or ion, the frequency of the emitted radiation will be maximum for (a) H-atom (b) D-atom (c) \(\mathrm{He}^{+}\) ion (d) \(\mathrm{Li}^{2+}\) ion

Short answer

The frequency of the emitted radiation will be maximum for the \(\mathrm{Li}^{2+}\) ion.

Step-by-step solution

Understanding the Rydberg Formula

The frequency of the emitted radiation during an electronic transition can be calculated using the Rydberg formula for hydrogen-like atoms: \( f = RZ^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \), where \( R \) is the Rydberg constant, \( Z \) is the atomic number of the atom/ion, \( n_1 \) and \( n_2 \) are the principal quantum numbers of the lower and higher energy levels, respectively. Since the transitions are the same, the difference in \( \frac{1}{n_1^2} - \frac{1}{n_2^2} \) is constant for all cases.

Determining the Effect of Atomic Number

As the Rydberg formula indicates, the frequency is directly proportional to the square of the atomic number \( Z \). Thus, we need to compare the atomic numbers of the given atoms and ions to determine which has the highest frequency of emitted radiation.

Comparing Atomic Numbers

The atomic numbers for each species are as follows: H-atom (Z=1), D-atom (Z=1, D or deuterium is an isotope of hydrogen with the same atomic number), \( \mathrm{He}^{+} \) ion (Z=2), and \( \mathrm{Li}^{2+} \) ion (Z=3).

Determining the Maximum Frequency

Since frequency increases with the square of the atomic number, and \( \mathrm{Li}^{2+} \) has the highest atomic number (Z=3), the emitted radiation's frequency will be maximum for the \( \mathrm{Li}^{2+} \) ion.

Key concepts

Electronic Transition

Understanding electronic transitions is crucial when studying the behavior of electrons in atoms. An electronic transition occurs when an electron moves between energy levels within an atom. This movement is energy-dependent; an electron must absorb or release energy to transition between these discreet levels. These energy changes correspond to specific wavelengths of light emitted or absorbed. For instance, when an electron in a hydrogen atom transitions from a higher to a lower energy level, a photon of light is emitted.

The energy levels, designated by principal quantum numbers (1, 2, 3...), are integral to predicting the electron behavior during these transitions. Each transition also has a unique fingerprint in terms of radiation frequency – the number of times a wave completes a cycle per second, commonly measured in Hertz (Hz). Understanding these principles helps us to comprehend the spectrums of light emitted or absorbed by atoms, essentially allowing us to 'read' the information encoded in the light they emit or absorb.

Radiation Frequency

Radiation frequency is directly connected to electronic transitions. It refers to the number of waves that pass a fixed point per second and is essential in identifying the type of radiation emitted during an electronic transition. Higher frequency radiation has more energy and is associated with larger energy transitions within an atom. The unit of frequency is Hertz (Hz), which is one cycle per second.

When an electron drops from a higher to a lower energy level, it emits a photon with a frequency directly related to the energy difference between these levels. The relationship between frequency () and energy (E) is defined by the equation e = h, where h is Planck's constant. This is why the frequency of emitted radiation is a key aspect in examining the nature of an electronic transition and why as educators, we stress the importance of understanding the direct correlation between energy changes and radiation frequency.

Atomic Number

The atomic number, denoted by Z, is fundamental to the properties of an atom. It is equal to the number of protons in an atom's nucleus and determines the chemical behavior of the element. In the context of electronic transitions and radiation frequency, the atomic number also plays a crucial role. According to the modified Rydberg formula, the frequency of radiation emitted from an atom during an electronic transition is affected by the square of the atomic number: f = RZ^2 left( frac{1}{n_1^2} - frac{1}{n_2^2} right)Here, R represents the Rydberg constant. This formula indicates a direct proportionality between the square of the atomic number and the frequency. Therefore, as the atomic number increases, the frequency of the emitted radiation also increases, provided the transition is the same, as in the textbook's example comparing hydrogen, deuterium, helium, and lithium ions. It is why the lithium ion, with the highest atomic number among the options, emits radiation with the highest frequency.