Step 1: Understand the concept.
The melting point of transition metals depends mainly on the strength of metallic bonding, which arises due to the participation of (n−1)d and ns electrons in bond formation. The number of unpaired d-electrons significantly affects metallic bonding strength — more unpaired electrons mean stronger metallic bonds and thus higher melting points.
Step 2: Analyze each pair.
(A) (Mn, Fe)
- Manganese (Mn) has the electronic configuration [Ar] 3d⁵ 4s², with a half-filled d-subshell. Due to its stable configuration, the metallic bonding is weaker, resulting in a relatively lower melting point.
- Iron (Fe) has [Ar] 3d⁶ 4s², allowing more delocalized electrons to participate in bonding, leading to stronger metallic bonds.
Thus, Mn < Fe.
(B) (Tc, Ru)
- Technetium (Tc) and Ruthenium (Ru) belong to the same group (Group 7 and 8 respectively) in the second transition series.
- Ruthenium (Ru) has a higher number of unpaired d-electrons available for metallic bonding than Tc, resulting in a stronger metallic bond.
Hence, Tc < Ru.
(C) (Re, Os)
- Rhenium (Re) and Osmium (Os) belong to the third transition series (5d block).
- Osmium (Os) has more unpaired d-electrons and forms stronger metallic bonds than Rhenium (Re).
Therefore, Re < Os.
Step 3: Conclusion.
The correct order of melting points based on metallic bonding strength is:
\[
\boxed{\text{Mn < Fe, Tc < Ru, and Re < Os}}
\]