Question

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

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

Step-by-step solution

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 $l=0,1,2,3$.

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 $L=\sqrt{0(0+1)}\hslash =0$.

Calculate Largest Orbital Angular Momentum

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

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, $L_z=3\hslash$.

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 $-\frac{1}{2}$.

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 $\frac{\frac{1}{2}\hslash }{3\hslash }=\frac{1}{6}$.

Key concepts

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.