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Chapter 34: Problem 3

Imagine an atom that, unlike hydrogen, had only three energy levels. If these levels were evenly spaced, how many spectral lines would result? How would their wavelengths compare?

Short Answer

Expert verified
The atom with three evenly spaced energy levels will produce three spectral lines, and as the energy differences are equal, the wavelengths of these lines will be identical.

Step by step solution

01

Identify the energy levels

For a given atom, there are 3 available energy levels: 1, 2 and 3.
02

Calculate the possible transitions

Spectral lines are created due to the transitions in levels by electrons. Let's take each level and see the possible transitions. \n1. From level 3, electron can transition to level 2 and level 1 (two possibilities)\n2. From level 2, electron can transition only to level 1 (one possibility)\nThe total spectral lines would therefore be the sum of these possibilities, which is 3.
03

Compare Wavelengths

The wavelength of the spectral lines would depend on the difference in energy levels. Since the energy levels are evenly spaced, the energy difference between levels is constant. Wavelength \( \lambda \) is related to energy \( E \) through the equation: \( E = h*c/\lambda \), where \( h \) is the Planck's constant and \( c \) is the speed of light. Since the energy transitions are the same, the wavelengths of the spectral lines are identical.

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Most popular questions from this chapter

Show that in the Bohr model, the frequency of a photon emitted in a transition between levels \(n+1\) and \(n,\) in the limit of large \(n\) is equal to the electron's orbital frequency. (This is an example of Bohr's correspondence principle.)

How are the uncertainty principle and wave-particle duality related?

The radiance of a blackbody peaks at \(660 \mathrm{nm}\). (a) What's its temperature? (b) How does its radiance at 400 nm compare with that at \(700 \mathrm{nm} ?\)

What would the constant in Equation \(34.2 \mathrm{a}\) be if blackbody radiance were defined for fixed intervals of frequency rather than wavelength? (Hint: Use \(\lambda=\) c/f to express the radiance as \(R(f, T),\) then differentiate to find the maximum, and solve the resulting relation numerically. Express your answer in a form like Equations \(34.2 \mathrm{a}\) and \(\mathrm{b} .\) )

Find the rate of photon production by (a) a radio antenna broadcasting \(1.0 \mathrm{kW}\) at \(89.5 \mathrm{MHz},\) (b) a laser producing \(1.0 \mathrm{mW}\) of 633-nm light, and (c) an X-ray machine producing 0.10-nm X rays with total power \(2.5 \mathrm{kW}\)
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