Question

Give the maximum number of electrons in an atom that can have these quantum numbers: a. \(n=4\) b. \(n=5, m_{\ell}=+1\) c. \(n=5, m_{s}=+\frac{1}{2}\) d. \(n=3, \ell=2\) e. \(n=2, \ell=1\)

Short answer

a. 32 electrons b. 2 electrons c. 25 electrons d. 10 electrons e. 6 electrons

Step-by-step solution

Calculate total number of orbitals

For n = 4, we can have: ℓ = 0, 1, 2, 3 For ℓ = 0, m_ℓ = 0, m_s = +1/2, -1/2 (one s-orbital) For ℓ = 1, m_ℓ = -1, 0, 1, m_s = +1/2, -1/2 (three p-orbitals) For ℓ = 2, m_ℓ = -2, -1, 0, 1, 2, m_s = +1/2, -1/2 (five d-orbitals) For ℓ = 3, m_ℓ = -3, -2, -1, 0, 1, 2, 3, m_s = +1/2, -1/2 (seven f-orbitals)

Count maximum electrons

Each orbital can accommodate two electrons (m_s = +1/2 and -1/2). So: 1s: 2 electrons 3p: 6 electrons 5d: 10 electrons 7f: 14 electrons Total number of electrons = 2 + 6 + 10 + 14 = 32 electrons #b. n = 5, m_ℓ = +1#

Determine possible values of ℓ and m_s

Given m_ℓ = 1, ℓ can only be 1 (p-orbital), as values for m_ℓ are from -ℓ to +ℓ. m_s has two possible values: +1/2 and -1/2

Count maximum electrons

As ℓ = 1, it means that we are referring to a p-orbital. A p-orbital can hold a maximum of 2 electrons with different spin quantum numbers (m_s = +1/2 and -1/2). Total number of electrons = 2 electrons #c. n = 5, m_s = +1/2#

Determine total number of orbitals

For n = 5, we can have: ℓ = 0, 1, 2, 3, 4 Since m_s value is constant, all orbitals can accommodate only one electron with a given m_s value.

Count maximum electrons

For each value of ℓ, we can count the number of electrons: ℓ = 0 (s): 1 electron ℓ = 1 (p): 3 electrons ℓ = 2 (d): 5 electrons ℓ = 3 (f): 7 electrons ℓ = 4 (g): 9 electrons Total number of electrons: 1 + 3 + 5 + 7 + 9 = 25 electrons #d. n = 3, ℓ = 2#

Determine possible values of m_ℓ and m_s

Since ℓ = 2, we'll have m_ℓ values from -2 to 2. m_ℓ = -2, -1, 0, 1, 2 m_s has two possible values: +1/2 and -1/2

Count maximum electrons

As ℓ = 2, it means that we are referring to a d-orbital. A d-orbital can hold a maximum of 10 electrons (2 electrons for each m_ℓ value). Total number of electrons = 10 electrons #e. n = 2, ℓ = 1#

Determine possible values of m_ℓ and m_s

Since ℓ = 1, we'll have m_ℓ values from -1 to 1. m_ℓ = -1, 0, 1 m_s has two possible values: +1/2 and -1/2

Count maximum electrons

As ℓ = 1, it means that we are referring to a p-orbital. A p-orbital can hold a maximum of 6 electrons (2 electrons for each m_ℓ value). Total number of electrons = 6 electrons