In Fe-Fe3C phase diagram, the eutectoid composition is 0.8 weight % of carbon at 725 °C. The maximum solubility of carbon in αferrite phase is 0.025 weight % of carbon. A steel sample, having no other alloying element except 0.5 weight % of carbon, is slowly cooled from 1000 °C to room temperature. The fraction of pro-eutectoid αferrite in the above steel sample at room temperature is

The correct option is 0.387.

To find the fraction of pro-eutectoid $\alpha$-ferrite in a steel sample containing 0.5 weight % carbon when it is slowly cooled to room temperature, we apply the lever rule in the iron-carbon (Fe-Fe₃C) phase diagram just above the eutectoid temperature (approx. 725 °C).

Let us identify the key compositions from the given data:
- Carbon content of the steel alloy, $C_0=0.5\text{ wt\% C}$
- Eutectoid composition (composition of austenite $\gamma$ just before eutectoid reaction), $C_{\gamma }=0.8\text{ wt\% C}$
- Maximum solubility of carbon in $\alpha$-ferrite phase, $C_{\alpha }=0.025\text{ wt\% C}$

Since the overall carbon content of the steel ($0.5\text{ wt\% C}$) is less than the eutectoid composition ($0.8\text{ wt\% C}$), this is a hypo-eutectoid steel. Upon slow cooling from 1000 °C, the austenite phase ($\gamma$) enters the $\gamma +\alpha$ two-phase region, and pro-eutectoid ferrite ($\alpha$) begins to precipitate. Just above the eutectoid temperature of 725 °C, the microstructural constituents are pro-eutectoid $\alpha$-ferrite and austenite ($\gamma$).

The weight fraction of pro-eutectoid $\alpha$-ferrite ($W_{\text{pro-}\alpha }$) is calculated using the lever rule on the tie-line between $C_{\alpha }$ and $C_{\gamma }$:
$W_{\text{pro-}\alpha }=\frac{C_{\gamma }-C_0}{C_{\gamma }-C_{\alpha }}$

Substituting the given values into the formula:
$W_{\text{pro-}\alpha }=\frac{0.8-0.5}{0.8-0.025}$

Let's simplify the numerator and denominator:
$W_{\text{pro-}\alpha }=\frac{0.3}{0.775}$

Performing the division yields:
$W_{\text{pro-}\alpha }\approx 0.387096$

Rounding to three decimal places, the fraction of pro-eutectoid $\alpha$-ferrite in the steel sample is 0.387.