NaCl Binding Energy in Water: Calculation & Analysis

[v_pino]

Homework Statement

It is well known that common salt (NaCl) is easily dissolved in water. In the solution Na and Cl atoms are present as positive and negative ("solvated") ions. Show that, due to the high dielectric constant of water and the resulting screening of the Coulomb potential, the binding energy of a NaCl crystal in water is smaller than the mean thermal energy of the free ions. Calculate the equilibrium separation of the ions of a hypothetical NaCl crystal in water and show that this separation is larger than the van der Waals radius of a water molecule, thereby justifying the approximate approach used here to discuss the solubility of NaCl.

Homework Equations

$$U(r)=N\left ( \frac{e^2}{4 \pi \epsilon_0 r}A + \frac{B}{r^n} \sum \frac{1}{P^n_{ij}} \right )$$

The Attempt at a Solution

Is this the right equation to use to find the binding energy?: differentiate U and equate it to zero.

What is the bulk modulus of NaCl in water? Is it the same as that in free space?

 

[NateD]

The equilibrium separation can be calculated using the Debye-Huckel equation: r_eq = (A/B)^{1/(n-2)}The van der Waals radius of a water molecule is 0.3 nm. Therefore, the equilibrium separation of the ions must be larger than this for the approximate approach to be valid.