Given. Bulk composition $C_0=1.00$ wt% C; eutectoid composition $C_{eutectoid}=0.77$ wt% C; cementite composition $C_{Fe_3C}=6.70$ wt% C (33.3 at% C); steel is slow (equilibrium) cooled from the single-phase austenite field.
Find. The weight fraction of pearlite, $W_{pearlite}$, in the room-temperature microstructure.
Fig. 2.1 — tie line just below 727 °C between the eutectoid composition and cementite, with $C_0=1.00$ wt% C marked. The lever rule uses the arm OPPOSITE each phase.
Approach. This is a hypereutectoid alloy ($C_{eutectoid}
Set up the lever rule. With the tie line endpoints at $C_{eutectoid}=0.77$ and $C_{Fe_3C}=6.70$, and $C_0=1.00$ wt% C in between:
$$W_{pearlite}=\dfrac{C_{Fe_3C}-C_0}{C_{Fe_3C}-C_{eutectoid}}$$
Substitute and evaluate.
$$W_{pearlite}=\dfrac{6.70-1.00}{6.70-0.77}=\dfrac{5.70}{5.93}=\boxed{0.961=96.1\%}$$
Check with the complementary phase. The proeutectoid cementite fraction is
$$W_{Fe_3C,\,proeutectoid}=\dfrac{C_0-C_{eutectoid}}{C_{Fe_3C}-C_{eutectoid}}=\dfrac{1.00-0.77}{5.93}=0.039=3.9\%$$
and $0.961+0.039=1.000$, confirming internal consistency. (Using the alternative textbook value $C_{Fe_3C}=6.67$ wt% C changes the result by under 0.1 percentage point.)
The lever-rule value above is a WEIGHT (equivalently, at nearly equal densities, volume) fraction computed from bulk composition — it is the theoretically "true" fraction. Quantitative metallography, by contrast, does not weigh the phases; it measures an AREA fraction of cementite on a two-dimensional polished-and-etched section, by point counting or image analysis, and then invokes the (statistically valid, for a random section through a homogeneous structure) equivalence of area fraction to volume fraction. The systematic shortfall arises from how thin the eutectoid cementite lamellae actually are: within pearlite, the ferrite:cementite lamellar thickness ratio is set by the SAME lever rule applied between pure ferrite ($\approx$0.022 wt% C) and cementite (6.70 wt% C) at the eutectoid composition (0.77 wt% C), giving cementite only about 11–12% of the pearlite itself by weight — and because cementite is denser than ferrite, the cementite lamellae are geometrically even thinner, relative to the ferrite lamellae, than that weight ratio alone suggests. These very thin plates approach, and in fine (rapidly cooled or high-temperature-annealed) pearlite can fall below, the resolving power of the optical microscope; etching, plate overlap along the sectioning direction, and the finite width of the etched groove at each ferrite/cementite interface all further erode the apparent width of the cementite lamella that is actually resolved and counted. The net effect is a SYSTEMATIC undercount of the cementite phase's area fraction relative to its true lever-rule weight fraction — not a random measurement scatter, and not evidence that the phase diagram or the lever rule is wrong.
Check: this is a stated, reproducible metallographic bias (optical-resolution limitation on very thin lamellae), not a suggestion that the sample's actual bulk composition differs from 1.00 wt% C.
