A solution contains charged, globular $($i$.$e$.,$ approximately spherical$)$ proteins of radius $5$ nm

dispersed in an aqueous solution containing $0\mathrm{.}1$ mM NaCl at $298$ K$.$

$($a$)$ Compute and plot the average center$-$to$-$center and surface$-$to$-$surface separation

distance between proteins for volume fractions up to $60$ vol$\%.$

$($b$)$ Comparing the values in $($a$)$ to the Debye length under these conditions, at what

concentration do the particles begin to experience strong electrostatic forces between

neighboring particles?

$($c$)$ How would you expect electrostatic interactions to change the concentration at which

the proteins can form colloidal crystals?