Problem 95 - A Complete Resource Book in Chemistry for JEE Main

Question

For the electronic transition from \(\mathrm{n}=2 \rightarrow \mathrm{n}=1\), which of the following will produce shortest wave length? (a) \(\mathrm{Li}^{2+}\) ion (b) D atom (c) \(\mathrm{He}^{+}\)ion (d) \(\mathrm{H}\) atom

Step-by-Step Solution

Answer

(a) \( \mathrm{Li}^{2+} \) ion.

Step 1: Understanding the Energy Transition

The electronic transition from higher energy level \( n = 2 \) to lower energy level \( n = 1 \) involves the release of energy. The emitted photon's energy determines its wavelength: higher energy corresponds to shorter wavelength.

Step 2: Using the Formula for Wavelength

The formula for the wavelength of light emitted during an electronic transition is given by \[ \dfrac{1}{\lambda} = RZ^2 \left( \dfrac{1}{n_1^2} - \dfrac{1}{n_2^2} \right) \]where \( R \) is Rydberg constant, \( Z \) is the atomic number, \( n_1 \) and \( n_2 \) are the lower and higher energy levels, respectively.

Step 3: Identifying Atomic Numbers

For each given species, the atomic numbers are:- \( \mathrm{Li}^{2+} \): \( Z = 3 \)- \( \mathrm{D} \) atom (deuterium, an isotope of hydrogen): \( Z = 1 \)- \( \mathrm{He}^{+} \): \( Z = 2 \)- \( \mathrm{H} \) atom: \( Z = 1 \)

Step 4: Calculating Relative Wavelengths

The shortest wavelength corresponds to the highest value of \( Z^2 \). Substitute the values for \( Z \) into the formula:- \( \mathrm{Li}^{2+} \): \( (3)^2 = 9 \)- \( \mathrm{D} \) atom: \( (1)^2 = 1 \)- \( \mathrm{He}^{+} \): \( (2)^2 = 4 \)- \( \mathrm{H} \) atom: \( (1)^2 = 1 \)Thus, the \( \mathrm{Li}^{2+} \) ion will produce the shortest wavelength.

Key Concepts

Electronic Transitions

An electronic transition occurs when an electron in an atom moves between energy levels. These levels are denoted by the principal quantum number, typically represented by the letter \( n \). When an electron drops from a higher energy level (higher \( n \)) to a lower energy level (lower \( n \)), it releases energy in the form of a photon. The energy of this photon corresponds to the difference between the two energy levels.

The energy released during a transition is key to determining the wavelength of the emitted photon. The basic rule is simple: the greater the energy released, the shorter the emitted wavelength. Thus, electronic transitions are crucial in explaining phenomena like spectral lines observed in atomic spectra.

Rydberg Formula

The Rydberg Formula is used to calculate the wavelengths of photons emitted or absorbed during electronic transitions. The formula is expressed as: \[ \frac{1}{\lambda} = RZ^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \] where:

• \( \lambda \) is the wavelength of the emitted or absorbed photon.
• \( R \) is the Rydberg constant, a fundamental constant in atomic physics.
• \( Z \) is the atomic number of the atom.
• \( n_1 \) and \( n_2 \) are the principal quantum numbers of the lower and higher energy levels involved.

This formula indicates that the wavelength is inversely proportional to the square of the atomic number and the difference between the reciprocals of the squares of the energy levels. Thus, larger atomic numbers result in photons with shorter wavelengths, assuming identical transitions.

Energy Levels

Energy levels in an atom are like rungs on a ladder that electrons can move between. Each energy level is associated with a specific amount of energy. The principal quantum number \( n \) describes these levels. For example, \( n=1 \) is the lowest energy level, and higher numbers correspond to higher energy levels.

In simple atoms like hydrogen, energy levels are well-defined, and the energy difference between levels can be precisely calculated. The higher the electron climbs in energy levels, the more energy it has.

• When an electron falls to a lower energy level, it releases this energy.
• This release is what causes the emission of photons detectable as spectral lines.

Understanding these levels and transitions helps in identifying elements and isotopes through their spectral lines.

Atomic Number

The atomic number \( Z \) of an element is the number of protons in its nucleus. It is this number that determines the element's identity. Each element on the periodic table has a unique atomic number. For example, hydrogen has an \( Z \) of 1, while helium has an \( Z \) of 2.

The atomic number is also crucial when determining the wavelengths of emitted light via the Rydberg Formula. As \( Z \) increases, the potential energy well of the atom becomes deeper, meaning more energy is released during electron transitions from a specific initial energy level to a lower one. This explains why elements with higher atomic numbers, like \( \mathrm{Li}^{2+} \) with \( Z = 3 \), have smaller wavelength emissions for the same transition compared to elements with lower atomic numbers.