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Chapter 38: Problem 59

What is the energy of the orbiting electron in a hydrogen atom with a quantum number of \(45 ?\)

Short Answer

Expert verified
Answer: The energy of the orbiting electron in a hydrogen atom with a quantum number of 45 is approximately -0.0067 eV.

Step by step solution

01

Write down the given information and formula

The quantum number of the hydrogen atom is given: \(n=45\). The formula for the energy levels of the hydrogen atom is: \(E_n = -\frac{13.6 \text{eV}}{n^2}\)
02

Substitute the given quantum number in the formula

We need to find the energy of the orbiting electron in the hydrogen atom. Substitute the given quantum number \(n=45\) in the energy formula: \(E_{45} = -\frac{13.6 \text{eV}}{45^2}\)
03

Calculate the energy of the electron

Now, calculate the energy of the electron in the hydrogen atom with the given quantum number: \(E_{45} = -\frac{13.6 \text{eV}}{45^2} = -\frac{13.6 \text{eV}}{2025} = -0.006716049 \text{eV}\) The energy of the orbiting electron in a hydrogen atom with a quantum number of \(45\) is approximately \(-0.0067 \text{eV}\).

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Most popular questions from this chapter

The muon has the same charge as an electron but a mass that is 207 times greater. The negatively charged muon can bind to a proton to form a new type of hydrogen atom. How does the binding energy \(E_{\mathrm{B} \mu}\) of the muon in the ground state of a muonic hydrogen atom compare with the binding energy \(E_{\mathrm{Be}}\) of an electron in the ground state of a conventional hydrogen atom? a) \(\left|E_{\mathrm{B} \mu}\right| \approx\left|E_{\mathrm{Be}}\right|\) d) \(\left|E_{\mathrm{B} \mu}\right| \approx 200 \mid E_{\mathrm{Be}}\) b) \(\left|E_{\mathrm{B} \mu}\right| \approx 100\left|E_{\mathrm{Be}}\right|\) e) \(\left|E_{\mathrm{B} \mu}\right| \approx\left|E_{\mathrm{Be}}\right| / 200\) c) \(\left|E_{\mathrm{B} \mu}\right| \approx\left|E_{\mathrm{Be}}\right| / 100\)

What is the energy of a transition capable of producing light of wavelength \(10.6 \mu \mathrm{m}\) ? (This is the wavelength of light associated with a commonly available infrared laser.)

An excited hydrogen atom emits a photon with an energy of \(1.133 \mathrm{eV}\). What were the initial and final states of the hydrogen atom before and after emitting the photon?

Show that the number of different electron states possible for a given value of \(n\) is \(2 n^{2}\).

Following the steps outlined in our treatment of the hydrogen atom, apply the Bohr model of the atom to derive an expression for a) the radius of the \(n\) th orbit, b) the speed of the electron in the \(n\) th orbit, and c) the energy levels in a hydrogen-like ionized atom of charge number \(Z\) that has lost all of its electrons except for one electron. Compare the results with the corresponding ones for the hydrogen atom.
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