During calcination of limestone$,$ the rate of decomposition of $CaC{O}_3$ is controlled by mass transport of $C{O}_2$ from the surface of the solid $($assume a sphere$)$ to the surrounding atmosphere. On the assumption that $C{O}_2$ diffuses through a stagnant film of air of thickness $0\mathrm{.}001m$ around the solid particle, and the concentration of $C{O}_2$ in the calcining furnace is zero, calculate the initial rate of decomposition of a $CaC{O}_3$ particle at $1000K.$ Use the concept of mass$-$transfer coefficient. Given:

i$.D_{C{O}_2-\text{A}\text{i}\text{r}}={10}^{-4}\frac{m^2}{s}$ at $1000K$

ii$.CaC{O}_3(s)=$CaO$(s)+C{O}_2(g)$;$G^0=161300-137*0T\frac{J}{g}$ mole in Kelvin