How many unique combinations of the quantum numbers l and ml are there when (a) n = 1 (b) n = 5?

검증된 단계별 안내

insert step 1> Determine the possible values of the azimuthal quantum number \( l \) for a given principal quantum number \( n = 5 \). The values of \( l \) range from 0 to \( n-1 \).

insert step 2> List the possible values of \( l \) for \( n = 5 \): \( l = 0, 1, 2, 3, 4 \).

insert step 3> For each value of \( l \), determine the possible values of the magnetic quantum number \( m_l \). The values of \( m_l \) range from \( -l \) to \( +l \).

insert step 4> Calculate the number of \( m_l \) values for each \( l \): \( l = 0 \) has 1 value, \( l = 1 \) has 3 values, \( l = 2 \) has 5 values, \( l = 3 \) has 7 values, and \( l = 4 \) has 9 values.

insert step 5> Add the number of \( m_l \) values for all \( l \) to find the total number of unique combinations of \( l \) and \( m_l \) for \( n = 5 \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

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

Quantum Numbers

Quantum numbers are sets of numerical values that describe the unique quantum state of an electron in an atom. There are four quantum numbers: principal (n), azimuthal (l), magnetic (ml), and spin (ms). Each quantum number provides specific information about the electron's energy level, shape of the orbital, orientation, and spin direction.

Principal Quantum Number (n)

The principal quantum number (n) indicates the main energy level or shell of an electron in an atom. It can take positive integer values (1, 2, 3, ...), with higher values corresponding to higher energy levels and greater distances from the nucleus. In this case, n = 5 means we are considering the fifth energy level.

Azimuthal Quantum Number (l) and Magnetic Quantum Number (ml)

The azimuthal quantum number (l) defines the shape of the orbital and can take values from 0 to n-1. For n = 5, l can be 0, 1, 2, 3, or 4, corresponding to s, p, d, f, and g orbitals, respectively. The magnetic quantum number (ml) specifies the orientation of the orbital and can take values from -l to +l, resulting in multiple ml values for each l, thus creating unique combinations.