A specially processed clay has particles that are $480$ nm thick and $7,000$ nm x $7,000$ nm wide. The specific gravity of solid particles is $2\mathrm{.}68.$ The particles lie perfectly parallel with an edge$-$to$-$edge spacing of $500$ nm $($i$.$e$.,$ they look like thin bricks stacked perfectly parallel$).$

a$)$ Initially, the predominant cation in the double layer is Na$+1,$ resulting in a face$-$to$-$face spacing of $1100$ nm$.$ How many particles per cm$3$ will there be at this pacing?

b$)$ Calculate the void ratio and water content of this soil, assuming the soil is fully saturated.

c$)$ If Na$+1$ Cations are replaced by Ca$+2$ Cations, what are the new void ratio and water content of the soil under these conditions? Assume that the edge$-$to$-$edge spacing remains $500$ nm$.$

d$)$ Answer part c$,$ if the temperature of the environment changes from $18$C to $43$C$.$

e$)$ Explain what will happen to the void ratio and water content of this soil, if the water around the particles is replaced by salty water that has a higher pH