Problem 123 - A Complete Resource Book in Chemistry for JEE Main

Question

Energy of level \(1,2,3\) of a certain atom corresponds to increasing value of energy \(\mathrm{E}_{1}<\mathrm{E}_{2}<\mathrm{E}_{3}\). If \(\lambda_{1}, \lambda_{2}\) and \(\lambda\) are the wavelength of radiation corresponding to transition \(3 \rightarrow 2,2 \rightarrow 1\) and \(3 \rightarrow 1\) respectively. Which of following statement is/are correct? (a) \(\frac{1}{\lambda_{2}}=\frac{1}{\lambda_{1}}+\frac{1}{\lambda_{3}}\). (b) \(\lambda_{2}=\frac{\lambda_{1} \lambda_{3}}{\lambda_{1}+\lambda_{2}}\). (c) \(\frac{1}{\lambda_{3}}=\frac{1}{\lambda_{1}}+\frac{1}{\lambda_{2}}\). (d) \(\lambda_{3}=\frac{\lambda_{1} \lambda_{2}}{\lambda_{1}+\lambda_{2}}\).

Step-by-Step Solution

Answer

Statement (c) is correct: \( \frac{1}{\lambda_{3}}=\frac{1}{\lambda_{1}}+\frac{1}{\lambda_{2}} \).

Step 1: Understand the Energy Transitions

For atomic energy levels, the energy difference between two states where a particle jumps from a higher energy state ( _E_3 , _E_2_, or _E_1_ ) to a lower one corresponds to a photon's energy. The energy transition changes 3→2 (λ_1), 2→1 (λ_2), and 3→1 (λ_3).

Step 2: Establish the Energy-Wavelength Relationship

The energy of a photon is inversely proportional to its wavelength: E = h c/λ, where E is the energy of the transition, h is Planck's constant, and c is the speed of light. This means that for a transition 3→2, E_3 - E_2 = hc/λ_1, similarly for 2→1, E_2 - E_1 = hc/λ_2 and finally for 3→1, E_3 - E_1 = hc/λ_3.

Step 3: Analyze the Combined Transition

A transition from level 3 to 1 can be considered the sum of two transitions: 3→2 and 2→1. Thus, E_3 - E_1 = (E_3 - E_2) + (E_2 - E_1). Using the energy-wavelength relationship, this test transforms into hc/λ_3 = hc/λ_1 + hc/λ_2.

Step 4: Simplify the Equation Using Wavelength

Cancel out the common factors (hc) in the equation from step 3: 1/λ_3 = 1/λ_1 + 1/λ_2. This derived formula checks which of the given statements is correct.

Key Concepts

Photon Energy

Understanding photon energy is key to grasping how atomic energy levels interact. Photons are tiny particles that carry electromagnetic energy, and their energy is directly linked with the atomic transitions they emerge from. When an electron in an atom moves from a higher energy level to a lower one, it emits a photon. This photon's energy is equal to the difference in energy between these two levels.
The energy of a photon can be calculated using the formula:\[E = h u\]Here, \(E\) is the energy of the photon, \(h\) is Planck's constant, and \(u\) is the frequency of the photon. Often though, it is more practical to use the wavelength in calculations, as:\[E = \frac{hc}{\lambda}\]where \(c\) is the speed of light and \(\lambda\) is the wavelength. In our exercise, each wavelength \(\lambda_1\), \(\lambda_2\), and \(\lambda_3\) corresponds to a specific energy transition within the atom. This concept plays a central role in connecting energy transitions with the photon's observable properties.

Energy Transitions

Energy transitions occur within an atom when an electron moves between defined energy levels. Each transition results in the emission or absorption of a photon, and this photon's energy is directly tied to how far the electron "jumps" between levels. In the scenario depicted in the exercise, the transitions are shown as:

• From level 3 to 2, releasing a photon corresponding to wavelength \(\lambda_1\).
• From level 2 to 1, releasing a photon corresponding to wavelength \(\lambda_2\).
• From level 3 directly to 1, releasing a photon corresponding to wavelength \(\lambda_3\).

Each transition reflects a drop in energy level, and as such, the accompanying photon reflects this energy change.
It's crucial to note that in our setup, the energy released in the combined transition from 3 to 1 is the sum of the energy changes from 3 to 2 and then from 2 to 1. This brings about the relationship:\[E_3 - E_1 = (E_3 - E_2) + (E_2 - E_1)\]This relationship helps us derive important equations that verify specific properties of the wavelengths and photon energies involved.

Energy-Wavelength Relationship

The energy-wavelength relationship builds a bridge between the photon energy and the spectral line emitted during an atomic transition. According to the principle:\[E = \frac{hc}{\lambda}\]we see energy \(E\) is inversely proportional to wavelength \(\lambda\). So, as the energy difference between two atomic levels becomes greater, the wavelength of the emitted photon becomes shorter. This relation helps in understanding which transitions contribute more energetically, as in our exercise where transitions are:

• \(E_3 - E_2\) contributing to \(\lambda_1\)
• \(E_2 - E_1\) relating to \(\lambda_2\)
• \(E_3 - E_1\) relating to \(\lambda_3\)

By examining the combined transition from 3 to 1, it transforms into:\[\frac{1}{\lambda_3} = \frac{1}{\lambda_1} + \frac{1}{\lambda_2}\]This clearly illustrates how separate transitions' wavelengths add up reciprocally to form the overall wavelength of a direct transition, ultimately impacting our ability to predict and understand the behavior of atomic systems.