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. Maximum number of electrons: 32 b. Maximum number of electrons: 4 c. Maximum number of electrons: 25 d. Maximum number of electrons: 10 e. Maximum number of electrons: 6

Step-by-step solution

Find the possible values of _ell

Since the principal quantum number n=4, the possible values of _ell are 0, 1, 2, and 3.

Find the possible values of m_ell for each _ell

For each value of _ell, the possible values of m_ell are as follows: - _ell = 0: m_ell = 0 - _ell = 1: m_ell = -1, 0, +1 - _ell = 2: m_ell = -2, -1, 0, +1, +2 - _ell = 3: m_ell = -3, -2, -1, 0, +1, +2, +3

Find the total number of electrons

For each value of m_ell, there are two possible values of m_s. Thus we have: - _ell = 0: 2 electrons - _ell = 1: 6 electrons - _ell = 2: 10 electrons - _ell = 3: 14 electrons So, for n=4, the total number of electrons is $2+6+10+14=32$. #a. Maximum number of electrons: 32# #b. n=5, m_ell=+1#

Find the possible values of _ell for m_ell=+1

Possible values of _ell for m_ell=+1 are _ell=1 and _ell=2.

Find the total number of electrons for each _ell

For each value of m_ell, there are two possible values of m_s: - _ell = 1: 2 electrons - _ell = 2: 2 electrons So, for n=5 and m_ell=+1, the total number of electrons is $2+2=4$. #b. Maximum number of electrons: 4# #c. n=5, m_s=+1/2#

Calculate possible values of _ell and m_ell

For n=5, possible values of _ell are 0, 1, 2, 3, 4, and for each _ell, we can have their respective m_ell values.

Find the total number of electrons

For all of these cases, m_s=+1/2, so there will be only 1 electron for each of m_ell. Counting the possible combinations, we have a total of: 1 + 3 + 5 + 7 +9 = 25 electrons. #c. Maximum number of electrons: 25# #d. n=3, _ell=2#

Determine the possible values of m_ell

For _ell=2, the possible values of m_ell are: -2, -1, 0, +1, +2.

Find the total number of electrons

For each value of m_ell, there are two possible values of m_s. Thus we have 5 × 2 = 10 electrons. #d. Maximum number of electrons: 10# #e. n=2, _ell=1#

Determine the possible values of m_ell

For _ell=1, the possible values of m_ell are: -1, 0, +1.

Find the total number of electrons

For each value of m_ell, there are two possible values of m_s. Thus we have 3 × 2 = 6 electrons. #e. Maximum number of electrons: 6#