The water droplet with radius R and spherical form suspended in the stagnant and large$-$volume air environment first evaporates and then this vapor spreads in the stagnant air environment. The temperature of the air and water droplet is $40\backslash$deg C and the ambient pressure is constant at $101\mathrm{.}32$ kPa. Under these conditions, the density of water is $993\mathrm{.}69$ kg$/$m$^3$ and the vapor pressure of pure substance A is $7\mathrm{.}384$ kPa. Calculate the time by deriving the equation expressing the elapsed time for the case where the liquid droplet radius is R$/2.$

Data:

At conditions of $25\backslash$deg C and $101\mathrm{.}32$ kPa, the diffusivity of water vapor in the air is $2\mathrm{.}60$x$105$ m$^2/$s$.$ Ideal gas constant R$=8314$ m$^3$ Pa$/$kmol K

The initial radius of the water droplet is $5$ mm and the molecular mass of water is $18$