Problem 81 - Chemistry An Atoms First Approach

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. \(n=2, \ell=1\)

Step-by-Step Solution

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 1: Find the possible values of _ell

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

Step 2: 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

Step 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#

Step 4: Find the possible values of _ell for m_ell=+1

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

Step 5: 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#

Step 6: 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.

Step 7: 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#

Step 8: Determine the possible values of m_ell

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

Step 9: 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#

Step 10: Determine the possible values of m_ell

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

Step 11: 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#