Example 6.7: Working with Shells and Subshells
Indicate the number of subshells, the number of orbitals in each subshell, and the values of
\(l\) and
\(m_{\mathrm{ℓ}}\) for the orbitals in the
\(n = 4\) shell of an atom.
Solution
For
\(n = 4\),
\(ℓ\) can have values of
0,
1,
2, and
3. Thus,
\(s\) ,
\(p\) ,
\(d\) , and
\(f\) subshells are found in the
\(n = 4\) shell of an atom. For
\(ℓ = Failed to interpret math QQQ 0 (the s_ subshellQQQ\)),
\(m_{\mathrm{ℓ}}\) can only be
0. Thus, there is only one [4s] orbital. For
\(ℓ = Failed to interpret math QQQ 1 (p_ -typeQQQ\) orbitals),
\(m_{\mathrm{ℓ}}\) can have values of
\(–_{\mathrm{1}}\),
0,
+1, so we find three
\(4p\) orbitals. For
\(ℓ = Failed to interpret math QQQ 2 (d_ -typeQQQ\) orbitals),
\(m_{\mathrm{ℓ}}\) can have values of
-2,
-1,
0,
+1,
+2, so we have five
\(4d\) orbitals. When
\(ℓ = Failed to interpret math QQQ 3 (f_ -typeQQQ\) orbitals),
\(m_{\mathrm{ℓ}}\) can have values of
-3,
-2,
-1,
0,
+1,
+2,
+3, and we can have seven
\(4f\) orbitals. Thus, we find a total of
16 orbitals in the
\(n = 4\) shell of an atom.