Is it possible to have an iron-carbon alloy for which the mass fractions of total cementite and pearlite are \(0.039\) and 0.417, respectively? Why or why not?

The iron-carbon phase diagram shows the different phases and microstructures in iron-carbon alloys depending on the temperature and carbon concentration. Key phases in the diagram are austenite, ferrite, cementite, and pearlite. Cementite has a fixed carbon content of 6.67wt%; pearlite is a lamellar microstructure of alternating layers of ferrite and cementite with an overall carbon content of 0.80wt%.

To find the maximum mass fractions of cementite and pearlite in the alloy, we'll first assume that there is no other phase present besides these two. The amount of cementite is determined by the weight fraction of carbon it contains (6.67) and the weight fraction of carbon in pearlite (0.80). Let's denote the mass fraction of cementite as \(w_C\) and the mass fraction of pearlite as \(w_P\). So, \(w_C + w_P = 1\). Using the mass fractions of carbon in each phase, we can write the following equation: $$0.067w_C + 0.008w_P = x_{carbon}$$ Since pearlite has a fixed carbon content of 0.80wt%, its maximum mass fraction will occur when all of the carbon is in the pearlite, and so the alloy would be fully pearlitic. In this case, \(w_C = 0\) and \(w_P = 1\), which gives us the maximum possible mass fraction of pearlite as 1. For cementite, the situation is different since its fixed carbon content is 6.67wt%. If all of the carbon is in the cementite, the alloy would be fully cementitic, and we would have \(0.067 = x_{carbon}\). From the previous equation, we can find the maximum mass fraction of cementite: $$0.067 \cdot 1 + 0.008 \cdot 0 = x_{carbon}$$ Which implies, \(w_C = \frac{0.067}{0.067 + 0.008} = 0.893\). So, the maximum mass fraction of cementite is 0.893.

The given mass fractions are 0.039 for cementite and 0.417 for pearlite. Based on our calculations, the maximum possible mass fractions are 0.893 for cementite and 1 for pearlite. Since both the given values are lower than the maximum possible values, it is possible to have an iron-carbon alloy with these mass fractions.