Given
(i) [tex]$n=5, m_i=+1$[/tex]
(ii) [tex]$n =2, \ell=1, m_{\ell}=-1, m_{ s }=-1 / 2$[/tex]
The maximum number of electron(s) in an atom that can have the quantum numbers as given in (i) and (ii) are respectively:
(a) 25 and 1
(b) 8 and 1
(c) 2 and 4
(d) 4 and 1
Answer (2)
For the quantum numbers given in case (i) with n = 5 and m l = + 1 , the maximum number of electrons is 8. For case (ii) with n = 2 , l = 1 , m l = − 1 , and m s = − 2 1 , the maximum number is 1. Therefore, the correct option is (b) 8 and 1.
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For case (i), determine the possible values of ℓ given n = 5 and m l = + 1 , and calculate the maximum number of electrons.
For case (ii), determine the number of electrons given n = 2 , ℓ = 1 , m ℓ = − 1 , m s = − 1/2 .
The maximum number of electrons for case (i) is 8, and for case (ii) is 1.
The final answer is 8 and 1 .
Explanation
Problem Analysis We are given two sets of quantum numbers and asked to find the maximum number of electrons that can have those quantum numbers. Let's analyze each case separately.
Case (i) Analysis (i) n = 5 , m l = + 1 Here, the principal quantum number n is 5, and the magnetic quantum number m l is +1. The possible values of the azimuthal quantum number ℓ range from 0 to n − 1 , so ℓ can be 0, 1, 2, 3, or 4. However, since m l = + 1 , ℓ must be at least 1. Thus, ℓ can be 1, 2, 3, or 4. For each value of ℓ , m l can range from − ℓ to + ℓ . Since m l is fixed at +1, there is only one possible value of m l for each ℓ . For each combination of n , ℓ , and m l , there are two possible values of the spin quantum number m s : +1/2 and -1/2. Therefore, for each ℓ , there can be 2 electrons. Since ℓ can be 1, 2, 3, or 4, there are 4 possible values of ℓ . So, the maximum number of electrons is 4 × 2 = 8 .
Case (ii) Analysis (ii) n = 2 , ℓ = 1 , m ℓ = − 1 , m s = − 1/2 Here, n = 2 , ℓ = 1 , m ℓ = − 1 , and m s = − 1/2 . Since all four quantum numbers are specified, there can be only one electron with these quantum numbers.
Final Answer Therefore, the maximum number of electrons for case (i) is 8, and for case (ii) is 1.
Examples
Understanding quantum numbers helps us predict the electronic configuration of atoms, which in turn determines their chemical properties. For example, knowing the possible quantum numbers for an electron in a specific energy level allows us to understand how atoms interact to form molecules and chemical bonds. This knowledge is crucial in fields like materials science, where we design new materials with specific electronic properties.
