Step 1: Understanding the Question:
The Effective Atomic Number (EAN) rule states that the total number of electrons surrounding the central metal ion in a complex should equal the atomic number of the next noble gas.
Step 2: Key Formula or Approach:
\[ EAN = Z - \text{O.S.} + 2 \times \text{C.N.} \]
Where $Z$ = atomic number, O.S. = oxidation state, C.N. = coordination number.
Step 3: Detailed Explanation:
1. For $[Co(NH_3)_6]^{3+$:} $Z=27$, O.S.=$+3$, C.N.=$6$.
\[ EAN = 27 - 3 + (2 \times 6) = 24 + 12 = 36 \text{ (Obeys, Kr)} \]
2. For $Cr(CO)_6$: $Z=24$, O.S.=$0$, C.N.=$6$.
\[ EAN = 24 - 0 + (2 \times 6) = 24 + 12 = 36 \text{ (Obeys, Kr)} \]
3. For $[Zn(NH_3)_4]^{2+$:} $Z=30$, O.S.=$+2$, C.N.=$4$.
\[ EAN = 30 - 2 + (2 \times 4) = 28 + 8 = 36 \text{ (Obeys, Kr)} \]
4. For $[Cu(NH_3)_4]^{2+$:} $Z=29$, O.S.=$+2$, C.N.=$4$.
\[ EAN = 29 - 2 + (2 \times 4) = 27 + 8 = 35 \text{ (Does NOT obey)} \]
Step 4: Final Answer:
The complex $[Cu(NH_3)_4]^{2+}$ has an EAN of 35, which does not match a noble gas configuration.