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Chapter 28: Problem 39

List the number of electron states in each of the subshells in the \(n=7\) shell. What is the total number of electron states in this shell?

Short Answer

Expert verified
Answer: There are a total of 98 electron states in the n=7 shell. The number of electron states in each subshell is as follows: l=0: 2 electron states l=1: 6 electron states l=2: 10 electron states l=3: 14 electron states l=4: 18 electron states l=5: 22 electron states l=6: 26 electron states

Step by step solution

01

Identify the subshells for n=7

For n=7, there will be 7 subshells, each corresponding to a unique value of the azimuthal quantum number, l. The subshells will have l values ranging from 0 to 6 (l=0,1,2,3,4,5,6).
02

Calculate the number of electron states in each subshell

For each subshell, the electron states are determined by the magnetic quantum number, m. The possible values of m range from -l to l. We can calculate the number of electron states by using the formula 2 * (2l + 1) for each subshell. l=0: 2 * (2(0)+1) = 2 l=1: 2 * (2(1)+1) = 6 l=2: 2 * (2(2)+1) = 10 l=3: 2 * (2(3)+1) = 14 l=4: 2 * (2(4)+1) = 18 l=5: 2 * (2(5)+1) = 22 l=6: 2 * (2(6)+1) = 26
03

Calculate the total number of electron states for n=7

To find the total number of electron states in the n=7 shell, we add up the number of electron states in each subshell: 2+6+10+14+18+22+26 = 98 Thus, there are a total of 98 electron states in the n=7 shell.

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Most popular questions from this chapter

A scanning electron microscope is used to look at cell structure with 10 -nm resolution. A beam of electrons from a hot filament is accelerated with a voltage of \(12 \mathrm{kV}\) and then focused to a small spot on the specimen. (a) What is the wavelength in nanometers of the beam of incoming electrons? (b) If the size of the focal spot were determined only by diffraction, and if the diameter of the electron lens is one fifth the distance from the lens to the specimen, what would be the minimum separation resolvable on the specimen? (In practice, the resolution is limited much more by aberrations in the magnetic lenses and other factors.)
The particle in a box model is often used to make rough estimates of energy level spacings. Suppose that you have a proton confined to a one-dimensional box of length equal to a nuclear diameter (about \(10^{-14} \mathrm{m}\) ). (a) What is the energy difference between the first excited state and the ground state of this proton in the box? (b) If this energy is emitted as a photon as the excited proton falls back to the ground state, what is the wavelength and frequency of the electromagnetic wave emitted? In what part of the spectrum does it lie? (c) Sketch the wave function \(\psi\) as a function of position for the proton in this box for the ground state and each of the first three excited states.
A beam of neutrons is used to study molecular structure through a series of diffraction experiments. A beam of neutrons with a wide range of de Broglie wavelengths comes from the core of a nuclear reactor. In a time-offlight technique, used to select neutrons with a small range of de Broglie wavelengths, a pulse of neutrons is allowed to escape from the reactor by opening a shutter very briefly. At a distance of \(16.4 \mathrm{m}\) downstream, a second shutter is opened very briefly 13.0 ms after the first shutter. (a) What is the speed of the neutrons selected? (b) What is the de Broglie wavelength of the neutrons? (c) If each shutter is open for 0.45 ms, estimate the range of de Broglie wavelengths selected.
(a) Show that the number of electron states in a subshell is \(4 \ell+2 .\) (b) By summing the number of states in each of the subshells, show that the number of states in a shell is \(2 n^{2} .\) [Hint: The sum of the first \(n\) odd integers, from 1 to \(2 n-1,\) is \(n^{2} .\) That comes from regrouping the sum in pairs, starting by adding the largest to the smallest: \(1+3+5+\dots+(2 n-5)+(2 n-3)+(2 n-1)\) \(=[1+(2 n-1)]+[3+(2 n-3)]+[5+(2 n-5)]+\cdots\) \(=2 n+2 n+2 n+\cdots=2 n \times \frac{n}{2}=n^{2}\)
What is the minimum kinetic energy of an electron confined to a region the size of an atomic nucleus \((1.0 \mathrm{fm}) ?\)
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