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Consider an electron in the \(N\) shell. (a) What is the smallest orbital angular momentum it could have? (b) What is the largest orbital angular momentum it could have? Express your answers in terms of \(\hslash\) and in SI units. (c) What is the largest orbital angular momentum this electron could have in any chosen direction? Express your answers in terms of \(\hslash\) and in SI units. (d) What is the largest spin angular momentum this electron could have in any chosen direction? Express your answers in terms of \(\hslash\) and in SI units. (e) For the electron in part (c), what is the ratio of its spin angular momentum in the z-direction to its orbital angular momentum in the z-direction?

Short Answer

Expert verified
(a) 0, (b) \(2\sqrt{3}\hslash\), (c) \(3\hslash\), (d) \(\frac{1}{2}\hslash\), (e) \(\frac{1}{6}\)

Step by step solution

01

Determine Quantum Number for N shell

For the N shell, the principal quantum number \( n = 4 \). This means the possible azimuthal quantum numbers \( l \) can range from 0 up to \( n - 1 \), so \( l = 0, 1, 2, 3 \).
02

Calculate Smallest Orbital Angular Momentum

The orbital angular momentum is given by \( L = \sqrt{l(l+1)}\hslash \). For the smallest \( l = 0 \), so the smallest angular momentum is \( L = \sqrt{0(0+1)}\hslash = 0 \).
03

Calculate Largest Orbital Angular Momentum

The largest possible value of \( l \) is 3. Thus, the largest orbital angular momentum is \( L = \sqrt{3(3+1)}\hslash = \sqrt{12}\hslash = 2\sqrt{3}\hslash \).
04

Calculate Largest Orbital Angular Momentum in a Direction

The maximum orbital angular momentum in a given direction is defined by \( L_z = m_l\hslash \). For \( l = 3 \), \( m_l \) ranges from -3 to +3, so the largest value is when \( m_l = 3 \). Hence, \( L_z = 3\hslash \).
05

Determine Largest Spin Angular Momentum in a Direction

Electrons have a spin quantum number of \( s = \frac{1}{2} \). The largest spin angular momentum in any direction is thus \( S_z = m_s\hslash = \frac{1}{2}\hslash \), where \( m_s \) can be \( +\frac{1}{2} \) or \( -\frac{1}{2} \).
06

Calculate Ratio of Spin to Orbital Angular Momentum in z-Direction

Using the largest values from previous steps, the ratio of spin angular momentum \( S_z \) to orbital angular momentum \( L_z \) in the z-direction is \( \frac{\frac{1}{2}\hslash}{3\hslash} = \frac{1}{6} \).

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Key Concepts

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

Quantum Numbers
Quantum numbers are essential to understanding the behavior and characteristics of electrons in an atom. They include four different numbers:
  • Principal Quantum Number ( n ): Determines the electron shell or energy level of an electron. It can be any positive integer.
  • Azimuthal Quantum Number ( l ): Defines the shape of the orbital and can take values from 0 to ( n-1 ).
  • Magnetic Quantum Number ( m_l ): Indicates the orientation of the orbital in space with possible values ranging from ( -l ) to ( +l ).
  • Spin Quantum Number ( s ): Describes the intrinsic spin of the electron, usually +/- 1/2.
In the context of the N shell, the principal quantum number ( n ) is 4. This allows for azimuthal numbers from 0 to 3, permitting a variety of orbital shapes and orientations.
Electron Shell
Electron shells are the designated regions surrounding an atom where electrons are likely to be found. Each shell corresponds to a certain energy level determined by the principal quantum number ( n ). For example, for the N shell, n = 4, which represents a specific energy range and spatial distribution:
  • The higher the value of n , the further the electron is from the nucleus and the higher its energy.
  • Each shell contains sub-shells corresponding to different azimuthal quantum numbers ( l ).
  • For n = 4, the possible values of l are 0, 1, 2, and 3, representing sub-shells named s, p, d, and f respectively.
Understanding the structure of electron shells provides insight into how electrons are arranged, which affects an atom's chemical properties.
Spin Angular Momentum
Spin angular momentum is a fundamental property of electrons, representing the internal rotation of the electron on its axis. It is characterized by the spin quantum number ( s ), which determines the two possible spin states of an electron:
  • The value of s is often \( \pm \frac{1}{2} \).
  • These values represent the two possible orientations of spin: "up" or "down".
  • The actual spin angular momentum is calculated as \( S = \sqrt{s(s+1)}\hslash \), but in a specific direction (like z), it becomes \( S_z = m_s\hslash \).
In any chosen direction, the largest possible value for an electron's spin angular momentum is \( \frac{1}{2}\hslash \), a crucial component in understanding magnetic and quantum behaviors.
Orbital Angular Momentum
Orbital angular momentum involves the movement of an electron around the nucleus in its orbital path, analogous to how planets orbit the sun. This angular momentum is quantized and associated with the azimuthal quantum number ( l ):
  • For any value of l , orbital angular momentum L is calculated as \( L = \sqrt{l(l+1)}\hslash \).
  • The smallest possible l value is 0, which means the electron has minimal movement around the nucleus.
  • The largest l value for the N shell is 3, yielding a maximum orbital angular momentum of \( 2\sqrt{3}\hslash \).
  • In a specific direction, such as z, the maximum directional component of orbital angular momentum is described by \( L_z = m_l\hslash \), with m_l ranging from -l to +l .
These quantization principles allow prediction of electron behavior in magnetic fields and other interactions.

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Most popular questions from this chapter

A hydrogen atom undergoes a transition from a 2\(p\) state to the 1\(s\) ground state. In the absence of a magnetic field, the energy of the photon emitted is 122 nm. The atom is then placed in a strong magnetic field in the z-direction. Ignore spin effects; consider only the interaction of the magnetic field with the atom's orbital magnetic moment. (a) How many different photon wavelengths are observed for the 2p \(\rightarrow\) 1s transition? What are the \(m$$_l\) values for the initial and final states for the transition that leads to each photon wavelength? (b) One observed wavelength is exactly the same with the magnetic field as without. What are the initial and final \(m$$_l\) values for the transition that produces a photon of this wavelength? (c) One observed wavelength with the field is longer than the wavelength without the field. What are the initial and final \(m$$_l\) values for the transition that produces a photon of this wavelength? (d) Repeat part (c) for the wavelength that is shorter than the wavelength in the absence of the field.

(a) Show that the total number of atomic states (including different spin states) in a shell of principal quantum number \(n\) is \(2n^2\). \([Hint\): 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?

An electron is in the hydrogen atom with \(n\) = 5. (a) Find the possible values of \(L\) and \(L$$_z\) for this electron, in units of \(\hslash\). (b) For each value of \(L\), find all the possible angles between \(\vec{L}\) and the z-axis. (c) What are the maximum and minimum values of the magnitude of the angle between \(L\) S and the z-axis?

A hydrogen atom initially in an \(n\) = \(3,\) \(l\) = 1 state makes a transition to the \(n\) = \(2\), \(l\) = \(0\), \(j\) = \\(\frac{1}{2}\\) state. Find the difference in wavelength between the following two photons: one emitted in a transition that starts in the \(n\) = \(3\), \(l\) = \(1\), \(j\) = \\(\frac{3}{2}\\) state and one that starts instead in the \(n\) = \(3\), \(l\) = \(1\), \(j\) = \\(\frac{1}{2}\\) state. Which photon has the longer wavelength?

An electron is in a three-dimensional box with side lengths \(L_X =\) 0.600 nm and \(L_Y = L_Z = 2L_X\). What are the quantum numbers \(n_X, n_Y,\) and \(n_Z\) and the energies, in eV, for the four lowest energy levels? What is the degeneracy of each (including the degeneracy due to spin)?

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