$5$B$.2(24$ pts$)$ VACANCIES IN METALS

You are in the lab and can measure the total number of vacancies for a titanium metal bracket at room

temperature $(300K).$ However, the bracket will be used in a high temperature $(900C)$ hydrogen$-$rich gas

environment. The bracket has a mass of $2$ grams.

Boltzmann's constant is $8\mathrm{.}62E-5e\frac{V}{?}$ atom$-$K$.$ Assume a formation energy of $1\mathrm{.}8eV$ per vacancy.

$($i$)$ In terms of a ratios, how many more vacancies will there be per unit volume at $900C$

compared to room temperature? $(10$ pts