Find the value of wave number (v̅) in terms of Rydberg's constant, when transition of electron t

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Find the value of wave number (v̅) in terms of Rydberg's...

Find the value of wave number $(\bar{v})$ in terms of Rydberg's constant, when transition of electron takes place between two levels of $\mathrm{He}^{+}$ ion whose sum is 4 and difference is $2 .$
(a) $\frac{8 R}{9}$
(b) $\frac{32 R}{9}$
(c) $\frac{3 R}{4}$
(d) none of these

Key Concepts

Rydberg Constant and Rydberg Formula

The Rydberg constant is a fundamental physical constant that appears in the formula used to calculate the wavelengths (or wave numbers) of the spectral lines of hydrogen-like atoms. The Rydberg formula relates the inverse wavelength (wave number) to the difference between the inverses of the squares of the principal quantum numbers of the initial and final energy levels. This relationship is central in spectroscopic analysis of atomic transitions.

Electron Transitions in Hydrogen-like Systems

Electron transitions refer to the movement of an electron between discrete energy levels in an atom. In hydrogen-like systems, such as a He+ ion, the electron transitions can be described using a simple quantum mechanics model, with energy differences determined by the principal quantum numbers. The Rydberg formula is adjusted for these systems by scaling with the square of the nuclear charge, which is crucial for accurately predicting the spectral lines.

Quantum Numbers and Their Relationships

Quantum numbers are used to characterize the quantized energy levels of electrons within an atom. In problems involving transitions, sums and differences of the quantum numbers (or related indices) often dictate which energy levels are involved in the transition. Understanding how to interpret sums and differences of quantum numbers is key to deriving the correct initial and final energy states involved in the transition.

Scaling in Hydrogen-like Ions

Hydrogen-like ions, such as He+ ions, have only one electron but possess a higher nuclear charge compared to hydrogen. This increased charge alters the energy levels by a factor proportional to the square of the nuclear charge (Z²). Recognizing this scaling effect is vital when applying the Rydberg formula to non-hydrogen atoms, ensuring that the calculated wavelengths or wave numbers reflect the correct energetic differences.

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