Adapted from problem $4\mathrm{.}23(40$ marks$)$

a$.$ Given an alloy of elements A and B with mass fractions $f_A$ and $f_B,$ with molar masses $M_A$ and $M_B,$

and densities ${}_A$ and ${}_B$ prove the following equations true:

$1$:$,M_{\text{a}\text{v}\text{g}}=\frac{1}{\frac{f_A}{M_A}+\frac{f_B}{M_B}}$

$2$:$,{}_{\text{a}\text{v}\text{g}}=\frac{1}{\frac{f_A}{{}_A}+\frac{f_B}{{}_B}}$

b$.$ Calculate the unit cell edge length for an $85wt\%Fe-15wt\%V$ alloy. All of the vanadium is in solid

solution, and, at room temperature the crystal structure for this alloy is BCC$.$