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Chapter 41: Problem 47

(a) Show that the total number of atomic states (including different spin states) in a shell of principal quantum number \(n\) is 2\(n^{2} .[\)Hint : The sum of the first \(N\) integers \(1+2+3+\cdots+N\) is equal to \(N(N+1) / 2 .\) (b) Which shell has 50 states?

Short Answer

Expert verified
The total number of states in a shell is \(2n^2\), and the shell with 50 states is \(n=5\).

Step by step solution

01

Understand Quantum Numbers

Each electron in an atom is described by four quantum numbers: the principal quantum number \( n \), the azimuthal (or angular momentum) quantum number \( l \), the magnetic quantum number \( m_l \), and the spin quantum number \( m_s \). For a given shell defined by \( n \), it contains subshells defined by \( l = 0, 1, 2, \ldots, n-1 \).
02

Count States for Each Subshell

Each subshell with a given \( l \) accommodates magnetic states characterized by \( m_l = -l, -(l-1), \ldots, l-1, l \). Therefore, there are \( 2l + 1 \) possible values of \( m_l \) for each \( l \).
03

Account for Spin States

Each magnetic state can have two possible spin states \( m_s = \pm \frac{1}{2} \). Thus, each magnetic state \( m_l \) contributes 2 states when spin is considered.
04

Calculate Total States in a Shell

To find the total number of states for a given shell \( n \), sum over all possible \( l \) values:\[\text{Total states} = \sum_{l=0}^{n-1}(2l+1) \times 2 = 2 \sum_{l=0}^{n-1}(2l+1).\]The sum of the first \( x \) integers \( 1 + 2 + \ldots + x \) is \( \frac{x(x+1)}{2} \), so:\[\sum_{l=0}^{n-1}(2l+1) = n^2.\]Thus,\[\text{Total states} = 2n^2.\]
05

Determine Shell with 50 States

Set up the equation for a shell with 50 states: \( 2n^2 = 50 \). Solve for \( n \):\[n^2 = 25 \implies n = 5.\]Therefore, the shell with 50 states is \( n = 5 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Atomic States
Atomic states refer to the distinct states in which an electron can exist within an atom. Every electron in an atom is associated with a unique set of quantum numbers, allowing for different combinations that define these states. When considering these atomic states, it's important to remember that they include not just the position and energy of the electron, but also its angular momentum and spin configurations.
  • Each shell in an atom, characterized by a principal quantum number \( n \), encompasses subshells and further atomic states.
  • The total atomic states within a shell are the combinations of position (given by quantum numbers \( l \) and \( m_l \)) and spin states (given by \( m_s \)).
  • The maximum number of atomic states possible in a shell is described by the formula \( 2n^2 \), which accounts for all positional and spin variations within a shell.
These variations fully encompass how electrons can be configured in a given energy level of an atom, allowing scientists to predict electron arrangements in more complex systems.
Principal Quantum Number
The principal quantum number, denoted by \( n \), is one of the four quantum numbers used to characterize every electron in an atom. It indicates the main energy level or shell of an electron, hence dictating the electron's distance from the nucleus.
  • \( n \) can take any positive integer value starting from 1: \( n = 1, 2, 3, \ldots \)
  • A higher \( n \) value implies that the electron is further away from the nucleus and is generally associated with higher energy levels.
  • Each shell's capacity (number of atomic states) is directly related to \( n \) as described by \( 2n^2 \) states.
  • The principal quantum number is crucial in determining the "shell" an electron belongs to, which subsequently affects the determination of its possible subshells and associated states.
The principal quantum number provides the foundational classification that helps define an atom's electron configuration.
Shell and Subshell in Quantum Mechanics
In quantum mechanics, the concept of shells and subshells helps to organize electrons within atoms based on their energy levels and spatial configurations. These structures are crucial for understanding atomic habitations.
  • A "shell" is associated with a particular principal quantum number \( n \), and marks a group of electron states that have roughly the same energy level.
  • Within each shell, there are "subshells" which are defined by the azimuthal quantum number \( l \). For a given \( n \), \( l \) can take values from 0 to \( n-1 \).
  • Each subshell \( l \) contains "orbitals" characterized by the magnetic quantum number \( m_l \), with the number of orbitals dictated by \( 2l + 1 \).
  • Electrons in the same orbital have two possible "spin" states indicated by the spin quantum number \( m_s = \pm \frac{1}{2} \), effectively doubling the capacity of each subshell orbit.
These classifications help in visualizing and predicting how electrons populate the various energy levels within an atom, allowing for a deeper understanding of atomic structure and electron dynamics.

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Most popular questions from this chapter

The orbital angular momentum of an electron has a magnitude of \(4.716 \times 10^{-34} \mathrm{kg} \cdot \mathrm{m}^{2} / \mathrm{s}\) . What is the angular- momentum quantum number \(/\) for this electron?

Consider states with angular-momentum quantum number \(l=2 .\) (a) In units of \(\hbar,\) what is the largest possible value of \(L_{z} ?\) (b) In units of \(\hbar,\) what is the value of \(L ?\) Which is larger: \(L\) or the maximum possible \(L_{z} ?\) (c) For each allowed value of \(L_{z}\) , what angle does the vector \(\vec{\boldsymbol{L}}\) make with the \(+z\) -axis? How does the minimum angle for \(l=2\) compare to the minimum angle for \(l=3\) calculated in Example 41.3\(?\)

CALC A particle is described by the normalized wave function \(\psi(x, y, z)=A x e^{-\alpha x^{2}} e^{-\beta y^{2}} e^{-\gamma y^{2}},\) where \(A, \alpha, \beta,\) and \(\gamma\) are all real, positive constants. The probability that the particle will be found in the infinitesimal volume \(d x d y d z\) centered at the point \(\left(x_{0}, y_{0}, z_{0}\right)\) is \(\left|\psi\left(x_{0}, y_{0}, z_{0}\right)\right|^{2} d x d y d z z\) (a) At what value of \(x_{0}\) is the particle most likely to be found? (b) Are there values of \(x_{0}\) for which the probability of the particle being found is zero? If so, at what \(x_{0} ?\)

(a) Write out the ground-state electron configuration \(\left(1 s^{2},\right.\) \(2 s^{2}, \ldots .\) for the beryllium atom. (b) What element of next-larger \(Z\) has chemical properties similar to those of beryllium? Give the ground- state electron configuration of this element. (c) Use the procedure of part (b) to predict what element of next-larger \(Z\) than in (b) will have chemical properties similar to those of the element you found in part (b), and give its ground-state electron configuration.

Electron Spin Resonance. Electrons in the lower of two spin states in a magnetic field can absorb a photon of the right frequency and move to the higher state. (a) Find the magnetic-field magnitude \(B\) required for this transition in a hydrogen atom with \(n=1\) and \(l=0\) to be induced by microwaves with wavelength \(\lambda\) . (b) Calculate the value of \(B\) for a wavelength of 3.50 \(\mathrm{cm} .\)
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