Question

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

Short answer

The angular momentum quantum number \( l \) is 4.

Step-by-step solution

Understanding the Angular Momentum Formula

The orbital angular momentum (L) of an electron is given by the formula \( L = \sqrt{l(l+1)} \cdot \hbar \), where \( l \) is the angular momentum quantum number and \( \hbar \) is the reduced Planck's constant, approximately \( 1.0545718 \times 10^{-34} \) J·s.

Substituting Known Values

We know \( L = 4.716 \times 10^{-34} \) kg·m²/s. Substitute this into the formula alongside \( \hbar = 1.0545718 \times 10^{-34} \text{ J·s} \), obtaining the equation: \( 4.716 \times 10^{-34} = \sqrt{l(l+1)} \cdot 1.0545718 \times 10^{-34} \).

Simplifying the Equation

Divide both sides of the equation by \( \hbar \) to isolate \( \sqrt{l(l+1)} \): \( \sqrt{l(l+1)} = \frac{4.716 \times 10^{-34}}{1.0545718 \times 10^{-34}} \approx 4.473 \).

Solving for Quantum Number l

Square both sides to remove the square root: \( l(l+1) = 4.473^2 \approx 20.004 \). Solve the quadratic equation \( l^2 + l - 20.004 = 0 \) using the quadratic formula, \( l = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = 1, c = -20.004 \).

Calculating Angular Momentum Quantum Number

Calculate \( l \) using the quadratic formula: \( l = \frac{-1 \pm \sqrt{1 + 4 \times 20.004}}{2} = \frac{-1 \pm \sqrt{81.016}}{2} \). The square root gives us \( \sqrt{81.016} \approx 9 \), leading to \( l = \frac{-1 + 9}{2} = 4 \) (since \( l \) must be a non-negative integer).

Key concepts

Orbital Angular Momentum

Orbital angular momentum, often shortened to L, is a fundamental concept in quantum mechanics. It represents the momentum associated with the motion of an electron in its orbit around a nucleus. This property is not like classical momentum; instead, it has quantized values due to the wave-like nature of particles at atomic scales. The orbital angular momentum is crucial in determining the shape of an electron's orbit, also known as an atomic orbital.

In quantum mechanics, L is defined by the formula \( L = \sqrt{l(l+1)} \cdot \hbar \), where \( l \) is the angular momentum quantum number and \( \hbar \) is the reduced Planck's constant. This formula shows that L is dependent on \( l \), highlighting the quantized nature of orbital momentum.

The quantum number \( l \) can take on integer values from 0 to \( n-1 \), where \( n \) is the principal quantum number. Each value of \( l \) corresponds to a specific type of orbital with unique shapes, such as s, p, d, and f orbitals, impacting the chemical and physical properties of elements.

Quantum Mechanics

Quantum mechanics is a branch of physics that deals with phenomena at microscopic scales, like atoms and subatomic particles. It's founded on principles that differ significantly from classical physics, dealing with probabilities rather than certainties.

Some key features of quantum mechanics include:

• Wave-particle duality: Particles, like electrons, exhibit both wave-like and particle-like characteristics.
• Quantization: Certain properties, like energy and angular momentum, can only take on discrete values.
• Probabilistic nature: Unlike classical mechanics, it deals with probabilities, denoted by wave functions, to describe particle states.
• Uncertainty principle: Formulated by Heisenberg, stating that certain pairs of properties, like position and momentum, cannot both be precisely measured simultaneously.

Quantum mechanics forms the basis for understanding the electronic structure of atoms, enabling us to predict atomic behaviors and reactions accurately.

Quadratic Equation Solving

Solving quadratic equations is a critical aspect of many physics problems, including determining quantum numbers in quantum mechanics. A quadratic equation follows the form \( ax^2 + bx + c = 0 \) where \( a \), \( b \), and \( c \) are constants.

In our context, solving the quadratic equation \( l^2 + l - 20.004 = 0 \) involves using the quadratic formula:

• The formula is given by \( l = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 1 \), and \( c = -20.004 \).
• This will lead to two roots; however, only the non-negative integer value is physically advantageous as a quantum number.
• The discriminant, \( \sqrt{b^2 - 4ac} \), helps to determine if the solutions are real and distinct.

Essentially, solving these equations can predict real-world physical phenomena by determining possible values of quantum numbers.

Reduced Planck's Constant

The reduced Planck's constant, \( \hbar \), is a fundamental constant in physics that is vital to quantum mechanics. It bridges the gap between classical and quantum realms and is often seen in equations involving angular momentum and energy quantization.

Defined as \( \hbar = \frac{h}{2\pi} \), where \( h \) is Planck's constant, it normalizes physical constants within the equations of quantum mechanics.

Reduced Planck's constant values the connection between the angular momentum of systems and the quantization of energy levels. The precision in these measurements is critical for accurate predictions and descriptions in quantum physics, such as calculating orbital angular momentum of particles.

The constant is instrumental in various quantum equations, ensuring the physical universe's behavior at atomic and subatomic levels aligns with observed experimental data, making it a cornerstone of modern physics.