Interaction Energies and Debye Length

[burns96]

Homework Statement

(c) (i) Neglecting hydrogen-bonding, calculate the interaction energy between (i) H₃O⁺ and H₂O and (ii) H₃O⁺ and H₃O⁺ , if each pair is separated by 0.3 nm and assuming that the aqueous solvent can be treated as a medium with constant relative permittivity. Using your result, comment on the likelihood of ideal behaviour by H₃O⁺ at high molality.

(ii) Using the Debye-Hückel Limiting Law, calculate the activity coefficient of H₃O⁺ in a solution of 1 mol kg^–1 HCl.

What is the Debye length of the H₃O⁺ ion in this solution and what does this tell us about the ideality of the solution?

Homework Equations

Ion-dipole interaction:
V=−qμ/(4πϵ0)r²

Dipole-dipole interaction:
V=−2/3 μ₁²μ₂²/(4πϵ0)²r⁶ 1/K_BT

Dipole moment:
μ=qr

Debye length:
λD =√εT_eff/n₀e²

The Attempt at a Solution

Are these the right equations to use?
So for (i) I would use the first and for (ii) the second equation?
What do I use to work out the dipole moment of H₃O⁺? I'm assuming not the 0.3nm value and the charge is 1?

For Debye length: ε = ε0(permittivity of a vacuum) x εr (relative permittivity).
Would I assume room temperature so T = 298K ?
I've got that n₀ = constant volume density but not sure what that means, and e² - are these both constants?
Sorry for all the questions, really confused.

 

[_coyote_]

For part (i), you can use the ion-dipole interaction equation to calculate the interaction energy between H3O+ and H2O, assuming that each pair is separated by 0.3 nm and the aqueous solvent can be treated as a medium with constant relative permittivity. The dipole moment of H3O+ can be calculated using the equation μ=qr, where q is the charge of the ion (1 in this case) and r is the distance between the two charges.For part (ii), you can use the Debye-Hückel Limiting Law to calculate the activity coefficient of H3O+ in a solution of 1 mol kg–1 HCl. The Debye length of the H3O+ ion in the solution can be calculated using the equation λD =√εTeff/n0e2, where ε is the relative permittivity, T is the temperature, n0 is the constant volume density, and e2 is the square of the charge of the ion. The Debye length tells us about the ideality of the solution, as it indicates the range of distances at which ions interact. If the Debye length is small, then the ions interact over a shorter range, leading to more ideal behaviour in the solution.