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Chapter 36: Problem 32

The \(4 f \rightarrow 3 p\) transition in sodium produces a spectral line at \(567.0 \mathrm{nm} .\) Find the energy difference between these two levels.

Short Answer

Expert verified
Calculate the energy in Joules by evaluating the above expression. Convert the energy found into electron volts using the relation \(1 \, \mathrm{eV} = 1.6 \times 10^{-19} \, \mathrm{J}\). That will give the result.

Step by step solution

01

Convert wavelength to meters

The wavelength provided is \(567.0 \, \mathrm{nm}\), but the equations typically use the unit meters. Therefore, we need to convert the wavelength to meters using the relation \(1 \, \mathrm{m} = 10^{9} \, \mathrm{nm}\). This gives \(\lambda = 567.0 \times 10^{-9} \, \mathrm{m}\).
02

Calculation of energy

Now use the energy-wavelength relationship using Planck's equation \(E = \frac{h \cdot c}{\lambda}\) where \(h\) is Planck's constant (\(6.62607 \times 10^{-34} \, \mathrm{J \cdot s}\)), \(c\) is the speed of light (\(3.00 \times 10^{8} \, \mathrm{m/s}\)) and \(\lambda\) is the wavelength in meters. Substituting the values, the energy is calculated as \(E = \frac{6.62607 \times 10^{-34} \, \mathrm{J \cdot s} \times 3.00 \times 10^{8} \, \mathrm{m/s}}{567.0 \times 10^{-9} \, \mathrm{m}}\). Evaluate this expression to find the energy in Joules.
03

Convert Energy to Electronvolt

The usual custom in atomic physics is to quote energies in electron volts rather than Joules. Use the relation \(1 \, \mathrm{eV} = 1.6 \times 10^{-19} \, \mathrm{J}\) to convert your result into electron volts.

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Most popular questions from this chapter

Which of the following is not a possible value for the magnitude of the orbital angular momentum in hydrogen: (a) \(\sqrt{12} \hbar\) (b) \(\sqrt{20} \hbar ;\) (c) \(\sqrt{30} \hbar ;\) (d) \(\sqrt{40} \hbar ;\) (e) \(\sqrt{56} \hbar ?\)

A hydrogen atom is in the \(6 f\) state. Find (a) its energy and (b) the magnitude of its orbital angular momentum.

The \(4 p \rightarrow 3 s\) transition in sodium produces a double spectral line at 330.2 and \(330.3 \mathrm{nm} .\) What's the energy splitting of the \(4 p\) level?

A selection rule for the infinite square well allows only those transitions in which \(n\) changes by an odd number. Suppose an infinite square well of width \(0.200 \mathrm{nm}\) contains an electron in the \(n=4\) state. (a) Draw an energy-level diagram showing all allowed transitions that could occur as this electron drops toward the ground state, including transitions from lower levels that could be reached from \(n=4 .\) (b) Find all the possible photon energies emitted in these transitions.

What's the most orbital angular momentum that could be added to an atomic electron initially in the \(6 d\) state without changing its principal quantum number? What would be the new state?
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