What is the proeutectoid phase for an iron-carbon alloy in which the mass fractions of total ferrite and total cementite are \(0.86\) and \(0.14\), respectively? Why?

For iron-carbon alloys, the phase diagram shows phase fields of various phases depending on temperature and carbon content. The most important phases for us are ferrite (alpha), austenite (gamma), and cementite (Fe3C). The eutectoid reaction takes place at a temperature of 727°C and at 0.76% carbon content. In this reaction, austenite (gamma) transforms to a combination of ferrite (alpha) and cementite (Fe3C).

We are given the mass fractions of total ferrite and total cementite as 0.86 and 0.14, respectively. Here, we need to decide which phase to pick based on the mass fractions given in an alloy.

The eutectoid composition for iron-carbon alloys has a carbon content of 0.76%. For the current alloy, we have the mass fractions of ferrite (0.86) and cementite (0.14). When deciding between these two options, we are trying to find which phase forms before the eutectoid reaction as it is the proeutectoid phase. Since ferrite (alpha) has a lower carbon content (0.022% at eutectoid temperature) compared to cementite (Fe3C - 6.69% carbon content), it is more likely to form before the eutectoid reaction when the given alloy has a lower overall carbon content. We know that the eutectoid composition is 0.76% carbon, so we need to check if the alloy given can have lesser carbon content than the eutectoid composition, while still having the given mass fractions of ferrite and cementite. Let the carbon content in ferrite be x%. Therefore, the carbon content in cementite is (6.69 - x)%. We can, thus, write the equation for the total carbon content: 0.86 * x + 0.14 * (6.69 - x) = Total carbon in the alloy Let’s solve the equation to find x: 0.86x + 0.9346 - 0.14x = Total carbon in the alloy 0.72x = Total carbon in the alloy - 0.9346 Now we know the lowest carbon content can be achieved when the alloy is all ferrite (x=0.022%): 0.72*0.022 = Total carbon in the alloy - 0.9346 Total carbon in the alloy= 0.7250% Since the total carbon content in the alloy can be lower than the eutectoid composition, it is confirmed that the ferrite (alpha) phase forms before the eutectoid reaction.

Based on the analysis, the proeutectoid phase for this iron-carbon alloy, with the given mass fractions of ferrite and cementite, is ferrite (alpha phase). This is because ferrite forms at a lower carbon content than cementite and can form at a lower carbon content than the eutectoid composition (0.76%) in the given alloy.