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Chapter 18: Problem 2465

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

Short Answer

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

Step by step solution

01

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
02

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)
03

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: \( h\frac{c}{\lambda_{1}} + h\frac{c}{\lambda_{2}} = h\frac{c}{\lambda_{3}} \)
04

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) \( \lambda_{3} = \left[\left(\lambda_{1} \lambda_{2}\right) / \left(\lambda_{1}+\lambda_{2}\right)\right] \)

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Most popular questions from this chapter

The wave length of the first line of Lyman series for hydrogen atom is equal to that of hydrogen atom is equal to that of second line of Balmar series for a hydrogen like ion. The atomic number \(\mathrm{Z}\) of hydrogen like ion is (A) 1 (B) 2 (C) 3 (D) 4

A nucleus of \({ }^{210}{ }_{84}\) Po originally at rest emits \(\alpha\) -particle with speed \(\mathrm{v}\) what will be the recoil speed of the daughter nucleus (A) \([\mathrm{v} /(214)]\) (B) \([(4 \mathrm{v}) /(214)]\) (C) \([(4 \mathrm{v}) /(206)]\) (D) \([\mathrm{v} /(206)]\)

when \({ }_{3}^{7}\) Li nuclear are bombarded by Proton and the resultant nuclei are \({ }^{8}{ }_{4}\) Be, the emitted particle will be (A) neutron (B) gamma (C) alpha (D) Beta

which of the following cannot be emitted in radioactive decay of the substance? (A) Helium-nucleus (B) Electrons (C) Neutrinos (D) Proton.

The activity of a radioactive sample is measured as \(\mathrm{N}_{0}\) counts per minute at \(\mathrm{t}=0\) and \(\left(\mathrm{N}_{0} / \mathrm{e}\right)\) counts Per minute at \(\mathrm{t}=5 \mathrm{~min} .\) The time (in min) at which activity reduces to half its value is (A) log e \((2 / 5)\) (B) \(5 \log _{10} 2\) (C) \(5 \log _{\mathrm{e}} 2\) (D) \(\log _{10}^{(2 / 5)}\)
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