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Chapter 34: Problem 58

(a) Find the highest possible energy for a photon emitted as the electron jumps between two adjacent energy levels in the Bohr hydrogen atom. (b) Which energy levels are involved?

Short Answer

Expert verified
The maximum energy for a photon emitted when an electron jumps between two adjacent energy levels in a hydrogen atom according to the Bohr model is 13.6 eV. The energy levels involved are \(n=2\) and \(n=1\).

Step by step solution

01

Understanding the energy levels of Bohr鈥檚 hydrogen atom

According to Bohr's model for the Hydrogen atom, the energy level of the electron is given by \(E_n = -13.6/n^2\ eV\) where \(n\) is an integer representing the energy level of the electron. The negative sign indicates that the energy is bound.
02

Analyse the transition

When an electron drops from energy level \(n\) to \(n-1\), a photon is emitted and its energy corresponds to the difference in the energy levels. This means that the photon energy \(E_{photon}\) will be \(E_{photon} = E_{n-1}-E_n\).
03

Calculating the energy difference

Calculate the energy difference. Using the formula from step 2 (\(E_{photon} = E_{n-1}-E_n\)) find the following: \(E_{photon} = -13.6/(n-1)^2 鈭� (-13.6/n^2)= -13.6[n^2-(n-1)^2] / [n^2(n-1)^2]\). Simplifying this will give the result: \(E_{photon} = 13.6*(2n-1)/[n^2*(n-1)^2]\ eV\).
04

Derive the maximum energy

To find the maximum energy photon, we need to maximize the formula derived in the previous step. Notice that as \(n\) tends to infinity, the photon's energy tends to zero. Therefore, the maximum energy occurs for the lowest possible value of \(n\), which is \(n=2\). Substitute \(n=2\) into the photon's energy formula obtained in Step 3 to derive the maximum photon energy: \(E_{photon,max} = 13.6*(2*2-1)/[2^2*(2-1)^2] = 13.6\ eV\).
05

Identify the energy levels

The largest possible photon's energy is obtained when the electron makes the transition from \(n=2\) to \(n=1\), so the energy levels involved are these two.

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Most popular questions from this chapter

Which spectral line of the hydrogen Paschen series \(\left(n_{2}=3\right)\) has wavelength 1282 nm?

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 \(0.10-\mathrm{nm}\) X rays with total power \(2.5 \mathrm{kW}.\)

Find (a) the wavelength and (b) the energy in electron volts of the photon emitted when a Rydberg hydrogen atom drops from the \(n=180\) level to the \(n=179\) level.

The converse approach is used for particles with longer lifetimes: Direct measurement of the lifetime yields, through energy-time uncertainty, a range of expected values for particle energies or masses. The longer the lifetime, a. the wider the mass range and the narrower the energy range. b. the wider the mass and energy ranges. c. the narrower the mass range and the wider the energy range. d. the narrower the mass and energy ranges.

What's the maximum wavelength of light that can ionize hydrogen in its ground state? In what spectral region is this?
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