$7\mathrm{.}6$ A silicate glass is crystallizing following the Johnson$-$Mehl$-$Avrami equation, where the fraction crystallized is given by $f_{\text{t}\text{r}\text{a}\text{n}\text{s}\text{f}\text{o}\text{r}\text{m}\text{e}\text{d}}=1-e^{-A{t}^4}.$ The following temperature and times are observed for $50\%$ crystallization: $2$ seconds at $600C,7$ seconds at $500C,30$ seconds at $700C,200$ seconds at $750C,400$ seconds at $300C,$ and $300$ seconds at $800C.$

a$.$ Calculate the value of $"$A$"$ for each of these temperatures and times.

b$.$ From these data make a table of $0\mathrm{.}1,0\mathrm{.}5,$ and $0\mathrm{.}9$ transformed as a function of temperature

and also include the rate of transformation $(s^{-1})$ for $50\%$ transformed by taking $50/$

time$($s$).$

c$.$ From the data, make a plot of temperature vertically and rate of transformation horizon$-$

tally for the rate calculated for $50\%$ crystallized.

d$.$ Make a TTT diagram for this crystallization by plotting the points for $0\mathrm{.}1,0\mathrm{.}5,$ and $0\mathrm{.}9$ crys$-$

tallized with a vertical temperature axis and a horizontal axis of $lo{g}_{10}($time$,$ seconds$).$