Step 1: Understanding the Concept:
A fractional oxidation state is typically an average value that arises when a compound contains multiple atoms of the same element existing in different discrete, integer oxidation states. We need to calculate the oxidation state of the central metal atom in each compound.
Step 2: Key Formula or Approach:
Approach: Use the standard rules for assigning oxidation numbers (e.g., Oxygen is usually $-2$, Alkali metals are $+1$) to calculate the average oxidation state of the designated metal in each option.
Step 3: Detailed Explanation:
Let's evaluate the average oxidation states in each given compound:
- (A) $\text{Fe_3\text{O}_4$:} This is a mixed oxide physically consisting of $\text{FeO}$ and $\text{Fe}_2\text{O}_3$. The average oxidation state of the metal iron (Fe) is calculated as: $3x + 4(-2) = 0 \implies 3x = 8 \implies x = +8/3$ (Fractional).
- (B) $\text{Mn_3\text{O}_4$:} Similar to the iron compound, this is a mixed oxide ($\text{MnO} \cdot \text{Mn}_2\text{O}_3$). The average oxidation state of the metal manganese (Mn) is $3x + 4(-2) = 0 \implies x = +8/3$ (Fractional).
- (C) $\text{Pb_3\text{O}_4$:} Commonly known as red lead, this is also a mixed oxide ($2\text{PbO} \cdot \text{PbO}_2$). The average oxidation state of the metal lead (Pb) is calculated as $x = +8/3$ (Fractional).
- (D) $\text{Na_2\text{S}_4\text{O}_6$:} In sodium tetrathionate, the central non-metal sulfur atoms have different integer oxidation states (two are $+5$ and two are $0$), giving an average fractional oxidation state for sulfur of $10/4 = +2.5$. However, the question specifically asks about the metal. The only metal in this compound is Sodium ($\text{Na}$). Alkali metals invariably have an oxidation state of $+1$ in all their compounds. Thus, the oxidation state of the metal $\text{Na}$ is exactly $+1$, which is a whole integer.
Step 4: Final Answer:
The metal sodium in $\text{Na}_2\text{S}_4\text{O}_6$ is present in a $+1$ (non-fractional) oxidation state.