Problem 1 - College Physics

Question

\(\cdot\) 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?

Step-by-Step Solution

Answer

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

Step 1: Understand Orbital Angular Momentum

The orbital angular momentum of an electron is given by the formula: \[ L = \sqrt{l(l+1)} \hbar \] where \( L \) is the magnitude of angular momentum, \( l \) is the angular-momentum quantum number, and \( \hbar \) is the reduced Planck's constant \( \hbar = 1.055 \times 10^{-34} \text{ J s} \). Our task is to solve for \( l \).

Step 2: Calculate Angular Momentum in Terms of Quantum Number

We know that: \[ L = \sqrt{l(l+1)} \hbar \] Plug in the given value of \( L = 4.716 \times 10^{-34} \text{ kg m}^2 / \text{s} \):\[ 4.716 \times 10^{-34} = \sqrt{l(l+1)} \times 1.055 \times 10^{-34} \]

Step 3: Isolate the Quantum Number Expression

Rearrange the equation to solve for \( l(l+1) \):\[ \sqrt{l(l+1)} = \frac{4.716 \times 10^{-34}}{1.055 \times 10^{-34}} \]Calculate the quotient:\[ \sqrt{l(l+1)} \approx 4.47 \]

Step 4: Remove the Square Root

Square both sides to get rid of the square root:\[ l(l+1) = (4.47)^2 \] \[ l(l+1) = 19.9809 \]

Step 5: Solve for Quantum Number \( l \)

Now we need to find integer \( l \) that satisfies the equation \( l(l+1) \approx 19.9809 \). The value of \( l \) that makes this close to equal is \( l = 4 \), since:\[ 4(4+1) = 20 \].This value is closest to our calculated result.

Key Concepts

orbital angular momentum

In the realm of quantum mechanics, the term "orbital angular momentum" describes the motion of an electron around the nucleus. This concept is somewhat similar to classical angular momentum but aligns with quantum rules. It is a crucial part of an electron's behavior in an atom, as it helps define the electron's 'orbit' or path.

Unlike the traditional orbits of planets, an electron's path is described by a probability distribution. Here, the magnitude of orbital angular momentum is calculated through the formula:

• \( L = \sqrt{l(l+1)} \hbar \)

where \( L \) is the angular momentum, \( l \) is the angular-momentum quantum number, and \( \hbar \) is the reduced Planck's constant. This formula shows that angular momentum is quantized, only taking on specific discrete values. This quintessential feature of quantum mechanics illustrates how certain properties, like angular momentum, can't take just any value but must fall within specific, quantized levels.

The quantum number \( l \) plays a vital role in determining these quantized levels, as it takes only integer values starting from 0. Understanding this quantization helps explain the electron's limited paths or 'allowed energies', essential for predicting chemical behavior and properties.

electron quantum states

Electron quantum states describe the possible states or configurations that an electron can occupy in an atom. These states are defined by four quantum numbers, giving a complete description of an electron's position and energy.

The quantum numbers are:

• Principal quantum number \( n \): Determines the shell or energy level. Higher \( n \) values mean higher energy levels.
• Angular-momentum quantum number \( l \): Associated with orbital angular momentum. It can be any integer from 0 to \( n-1 \).
• Magnetic quantum number \( m_l \): Indicates the orientation of the orbital in space, ranging from \( -l \) to \( +l \).
• Spin quantum number \( m_s \): Represents electron spin, with values of \( +\frac{1}{2} \) or \( -\frac{1}{2} \).

These quantum numbers together describe the precise quantum state of an electron, allowing chemists and physicists to predict the electron's behavior, how it will interact with other particles, and its role in chemical bonding.

The multitude of combinations of these quantum numbers gives rise to different electron configurations, crucial for understanding chemical reactivity and the periodic table structure.

quantum mechanics

Quantum mechanics is the branch of physics dealing with the behavior of very small particles, like electrons. It provides a mathematical framework to describe the intricacies of atomic and subatomic particles.

In classical physics, particles have well-defined paths. However, quantum mechanics portraits particles as having probabilistic 'clouds' around them. This perspective helps explain phenomena that have no explanation under classical physics, such as electron orbitals.

Key principles of quantum mechanics include:

• Wave-particle duality: Particles, like electrons, exhibit properties of both waves and particles.
• Uncertainty principle: Precisely measuring one property of a particle can come at the expense of uncertainty in another.
• Quantum entanglement: Particles can become entwined, meaning the state of one instantly influences the state of another, regardless of distance.

These principles emphasize how quantum mechanics differs profoundly from classical physics, providing insights into the micro-world's complex behavior. Understanding these concepts is vital for advances in technology, like semiconductors and quantum computing.

Planck's constant

Planck's constant, symbolized by \( h \), is a fundamental constant in quantum mechanics. It describes the quantization of physical action in nature. This tiny constant is pivotal in equations governing quantum phenomena, linking the energy of photons to their frequency.

The reduced Planck’s constant, \( \hbar = \frac{h}{2\pi} \), frequently appears in quantum equations, such as the formula for angular momentum:

• \( L = \sqrt{l(l+1)} \hbar \)

The value of \( h \) is approximately \( 6.626 \times 10^{-34} \) Js, emphasizing the very small scales at which quantum effects operate.

Planck’s constant is a cornerstone in understanding quantization. It signifies the smallest possible energy exchange unit, helping describe electromagnetic radiation or photon properties.

This constant led to the development of quantum theory itself, altering our perception of energy levels and particle behavior. From defining energy levels in atoms to its role in cutting-edge technologies, Planck's constant remains a fundamental entity in physics.