The average distance between spherical particles in a colloidal suspension $($muddy river water, for example$)$ can

be described by DLVO theory, named after the four developers of the model. The free energy $G$ between

spheres balances repulsive and attractive energies and is described by the equation

$G=\frac{64n_{0}a{k}_{B}T{}^2}{{}^2}$exp$(-H)-\frac{A_{121}a}{12H}$

The parameters and variables are defined as

$n_0$ salt concentration $(mo\frac{l}{v}$olume$)$

a particle size. Assume $a=0\mathrm{.}1m.$

$k_b$ Boltzmann constant. $In$ molar units, $it$ equals the gas constant $R=8\mathrm{.}314\frac{J}{m}ol-K.$

$Tabsolute$ temperature $(K)$

$$ inverse $of$ the size $(called$ the Debye length$)ofan$ electrical double layer that surrounds

$$ each particle. Assume $=\frac{1}{5nm}.$

$A_{121}$ a parameter based $on$ the electrical potential $at$ the surface. Assume $=0\mathrm{.}750$

$H$ separation between particles.

The equilibrium separation occurs where the free energy $G$ is minimized with respect to particle separation

H$,$

$\frac{dG}{dH}=f(H)=0=-\frac{64n_{0}a{k}_{B}T{}^2}{}$exp$(-H)+\frac{A_{121}a}{12H^2}$

Use fixed point iteration to solve equation $2$ for the equilibrium separations $H$ at salt concentrations $n_0$ of

${10}^{-4}mo\frac{l}{L}$ and ${10}^{-2}mo\frac{l}{L}$ at $T=300K.$ What happens to $H$ as $n_0$ changes?

Hint: Be careful with units! You'll need to do some conversions, particularly with $n_0,a,$ and $.$