In the 3d series of transition elements, the element more likely to exhibit +1 oxidation state is:

Cu

The correct answer is option 3. Cu.

Transition metals often exhibit multiple oxidation states due to the involvement of both their \(s\) and \(d\) electrons in bonding. However, the stability of these oxidation states can vary based on the electronic configuration and the specific metal.

Oxidation States of Each Element

1. Iron (Fe)

Iron primarily exhibits \(+2\) and \(+3\) oxidation states. The electronic configurations are:

Fe (neutral): [Ar] 3d\(^6\) 4s\(^2\)

Fe\(^{2+}\): [Ar] 3d\(^6\)

Fe\(^{3+}\): [Ar] 3d\(^5\)

The \(+1\) oxidation state for iron is less common and less stable due to the loss of one electron from the 4s orbital. The more stable \(+2\) and \(+3\) states are favored due to their ability to achieve more stable configurations or due to their greater stability in various compounds.

2. Manganese (Mn)

Manganese exhibits a wide range of oxidation states, including \(+2\), \(+3\), \(+4\), \(+6\), and \(+7\). The electronic configurations are:

Mn (neutral): [Ar] 3d\(^5\) 4s\(^2\)

Mn\(^{2+}\): [Ar] 3d\(^5\)

Mn\(^{4+}\): [Ar] 3d\(^3\)

Mn\(^{7+}\): [Ar] 3d\(^0\) (in MnO\(_4^{-}\))

The \(+1\) oxidation state is rare for manganese and is not commonly observed. Manganese tends to form more stable compounds in its higher oxidation states due to the stability provided by half-filled or fully filled d-orbitals.

3. Copper (Cu)

Copper shows significant stability in both \(+1\) and \(+2\) oxidation states. The electronic configurations are:

Cu (neutral): [Ar] 3d\(^9\) 4s\(^1\)

Cu\(^{+}\): [Ar] 3d\(^10\) (d^10 configuration is stable)

Cu\(^{2+}\): [Ar] 3d\(^9\)

The \(+1\) oxidation state (cuprous) is stable for copper, and copper(I) compounds, such as CuCl, are well-known. This stability is due to the fully filled d-orbital in the \(+1\) state, which provides electronic stability.

4. Silver (Ag)

Silver primarily exhibits the \(+1\) oxidation state (argentous), and it rarely exhibits the \(+2\) state. The electronic configurations are:

Ag (neutral): [Kr] 4d\(^10\) 5s\(^1\)

Ag\(^{+}\): [Kr] 4d\(^10\) (full d-orbital)

The \(+1\) oxidation state is very stable for silver due to the completely filled d-orbital. However, silver is not a member of the 3d series; it belongs to the 5d series of transition elements.

Conclusion

Among the elements in the 3d series (Fe, Mn, Cu) and considering the commonness and stability of the oxidation states:

Iron and Manganese predominantly exhibit \(+2\) and \(+3\) states, with \(+1\) being less common and less stable for these elements.

Copper is well-known for its stable \(+1\) oxidation state, which is significant in various compounds.

Silver primarily exhibits the \(+1\) oxidation state, but since it's not in the 3d series, it does not fit the criteria of the question.

Thus, among the 3d series elements, copper (Cu) is most commonly associated with the \(+1\) oxidation state.

Hence, the correct answer is: 3. Cu