4. You are given a project to sinter an alumina disk. You start with a green body made by pressing alumina powders. Before sintering, these alumina powders have spherical shape with an average diameter of 0.5 m. During sintering, densification occurs and grain boundary forms. After sintering, the original spherical powders are sintered together as cubes (not perfect cubes) with grain boundaries formed between these cubes. It is given that the molar volume of the alumina is 10 cm/mol, the free surface energy of alumina is 1 J/m, and the dihedral angle for the final system after sintering is 100. Calculate the following questions: a) Before sintering, if you start with 1 mol alumina powders, what will be the total number of spherical alumina powders? (note: diameter is 0.5 m as mentioned above) b) If one spherical alumina powder will become one alumina cube after sintering, calculate the length of the side of the cube based on mass conservation. c) Calculate the total grain boundary area (neglecting free surface) in the system after sintering. d) Calculate the total grain boundary energy of the system after sintering.