Consider an electron in the $N$ shell. What is the largest orbital angular momentum it could have? Express your answers in terms of $\hslash$ and in SI units.

검증된 단계별 안내

Identify the principal quantum number (n) for the N shell. The N shell corresponds to n = 4.

Recall that the orbital angular momentum quantum number (l) can take integer values from 0 to (n - 1). For n = 4, the possible values of l are 0, 1, 2, and 3. The largest value of l is 3.

The orbital angular momentum (L) is given by the formula:

L = √(l(l + 1))ℏ

, where ℏ is the reduced Planck's constant (

ℏ = 1.0545718 × 10

^-34 \, $\text{J·s}$).

Substitute the largest value of l (l = 3) into the formula:

L = √(3(3 + 1))ℏ = √(12)ℏ

Express the result in terms of ℏ (U) and in SI units (J·s). The largest orbital angular momentum is

L = √(12)ℏ

, where ℏ is the reduced Planck's constant.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Orbital Angular Momentum

Orbital angular momentum is a measure of the rotational motion of an electron around the nucleus in an atom. It is quantized and can be expressed using the formula L = mvr, where L is the angular momentum, m is the mass of the electron, v is its velocity, and r is the radius of its orbit. For electrons in atomic shells, the angular momentum is also related to the principal quantum number n, with the maximum value given by L = nħ, where ħ is the reduced Planck's constant.

Principal Quantum Number (n)

The principal quantum number (n) indicates the energy level of an electron in an atom and determines the size and energy of the orbital. It can take positive integer values (1, 2, 3, ...), with higher values corresponding to higher energy levels and larger orbitals. For an electron in the N shell, n equals 4, which influences the maximum possible angular momentum it can possess.

SI Units and Conversion

SI units are the standard units of measurement used in science, including physics. Angular momentum is typically expressed in joule-seconds (J·s) in SI units. To express the angular momentum of an electron in the N shell in SI units, one must convert the quantized values derived from quantum mechanics into these standard units, ensuring consistency in calculations and comparisons.