Question

Energy levels \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) of a certain atom corresponding values of energy i.e. \(E_{A}

Short answer

The correct relationship between the wavelengths is: \( \lambda_{3} = \frac{\lambda_{1} \lambda_{2}}{\lambda_{1} + \lambda_{2}} \).

Step-by-step solution

Write down the energy differences

The energy differences for the transitions are: - ΔE₁: from C to B: EB - EC - ΔE₂: from B to A: EA - EB - ΔE₃: from C to A: EA - EC

Relate energy differences to corresponding wavelengths

Using the Planck-Einstein relation, we can write the energy differences in terms of the corresponding wavelengths: - ΔE₁ = h * (c/λ₁) - ΔE₂ = h * (c/λ₂) - ΔE₃ = h * (c/λ₃)

Use energy conservation to find the relationship

Since energy is conserved during these transitions, we can write: ΔE₁ + ΔE₂ = ΔE₃ Substituting the energy differences from Step 2, we get: \( h\frac{c}{\lambda_{1}} + h\frac{c}{\lambda_{2}} = h\frac{c}{\lambda_{3}} \)

Simplify the equation and isolate λ₃

Dividing both sides by Planck's constant (h) and the speed of light (c), we simplify the equation to: \( \frac{1}{\lambda_{1}} + \frac{1}{\lambda_{2}} = \frac{1}{\lambda_{3}} \) Taking the reciprocal of all terms, we obtain the relationship between the wavelengths: \( \lambda_{3} = \frac{\lambda_{1} \lambda_{2}}{\lambda_{1} + \lambda_{2}} \) Comparing this result with the given options, we find that the correct answer is: (C) \( \lambda_{3} = \left[\left(\lambda_{1} \lambda_{2}\right) / \left(\lambda_{1}+\lambda_{2}\right)\right] \)

Key concepts

Planck-Einstein relation

The Planck-Einstein relation forms the foundation for understanding quantum mechanics and the behavior of electromagnetic radiation. This formula, given by \( E = hu \), connects the energy \( E \) of a photon to its frequency \( u \), where \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \text{ Js} \)). This relation is crucial for linking the worlds of classical and quantum physics.

• The energy of photons features prominently in calculating energy transitions in atomic structures.
• Frequency \( u \) can also be expressed in terms of the wavelength \( \lambda \) using the speed of light \( c \): \( u = \frac{c}{\lambda} \).

Using these relationships, we substitute frequency in the Planck-Einstein formula, obtaining \( E = \frac{hc}{\lambda} \), effectively linking energy with wavelength. This connection allows scientists to predict phenomena in atomic transitions, such as the emission of light at specific wavelengths.
Thus, when atoms transition between energy levels, they do so by absorbing or emitting photons. Each photon's energy corresponds to the difference in energy between the initial and final states of the atom, quantified neatly by the Planck-Einstein relation.

Wavelength of radiation

Wavelengths determine the color and type of energy we perceive from electromagnetic waves. Each transition of energy levels within an atom involves the emission or absorption of electromagnetic radiation, characterized by its wavelength \( \lambda \).

• Short wavelengths correspond to higher energy photons, while longer wavelengths are associated with lower energy photons.
• In the context of multiple transitions, each transition in an atom (e.g., \( C \to B \), \( B \to A \), \( C \to A \)) relates to a unique wavelength.

Understanding how these wavelengths relate to one another during transitions is vital for solving problems like the one in the exercise. The resulting equation from the exercise shows \( \lambda_{3} = \frac{\lambda_{1}\lambda_{2}}{\lambda_{1}+\lambda_{2}} \).
This equation illustrates how the wavelength of a photon emitted or absorbed during a straight-line transition from \( C \) to \( A \) can be expressed in terms of multiple, individual transitions \( C \to B \to A \). The relationships of these wavelengths are essential for various applications, including the study of atomic spectra and quantum mechanics simulations.

Energy conservation in transitions

In physics, particularly in the study of atomic energy levels, energy conservation plays a pivotal role. When an electron transitions between energy states in an atom, the total energy before and after the transition must remain constant. This principle is mathematically represented as \( \Delta E_1 + \Delta E_2 = \Delta E_3 \) in our problem.

• Each difference \( \Delta E \) is derived from the Planck-Einstein relation: \( \Delta E = \frac{hc}{\lambda} \).
• The value \( \Delta E_1 \) (for transition \( C \to B \)) plus \( \Delta E_2 \) (\( B \to A \)) equals \( \Delta E_3 \) (\( C \to A \)).

Applying the conservation law ensures any photon emitted or absorbed during these processes conforms to energy conservation.
This consistency not only highlights the logical integration of quantum laws but also allows the calculation of any unknown variable present in the system. Properly applying energy conservation leads us to the correct relationship among the wavelengths of photons in complex systems.