Calculate the Madelung Constant

[abstracted6]

Homework Statement

Calculate the Madelung constant for an octahedral arrangement of doubly charged anions about a doubly charged cation. Use r0 for the anion−cation distance. (Hint: Remember that the Madelung constant considers the sum of Coulomb interactions over all atom/ion pairs).

Homework Equations

Possibly helpful Equations
E_A=-(Mke²)/r₀

α=2M/((n₁+n₂)|Z₁Z₂|)

E_A=-(ke²α(n₁+n₂(|Z₁Z₂)/2r₀

The Attempt at a Solution

So we have a theoretical bond pair X⁺²Y^-2 with a coordination number of 6.

I believe the next step should be to calculate E_A, but I fail to see how to do that without α.

 

[Liquidxlax]

Your Madelung equation doesn't seem familiar to me, what textbook are you using?

 

[abstracted6]

Here is a screenshot of my professors notes I took these equations from.

This is our book:

https://www.amazon.com/dp/0521651514/?tag=pfamazon01-20

What equations would you have used?

 

[Borek]

None of these. Madelung constant is calculated just by summing electrostatic interactions between ions.

 

[paranormal]

To calculate the Madelung constant, we need to consider the sum of Coulomb interactions between all atom/ion pairs in the octahedral arrangement. This includes interactions between the doubly charged cation and the six doubly charged anions surrounding it. We can use the equation EA=-(Mke2)/r0 to calculate the energy of interaction between the cation and a single anion, where M is the Madelung constant, k is the Coulomb constant, e is the elementary charge, and r0 is the anion-cation distance.

To find the value of α, we can use the equation α=2M/((n1+n2)|Z1Z2|), where n1 and n2 are the number of anions and cations, respectively, and Z1 and Z2 are their respective charges. In this case, n1=6, n2=1, Z1=2, and Z2=2, giving us α=2M/((6+1)|2(2)|)=M/14.

Substituting this value of α into the equation EA=-(ke2α(n1+n2(|Z1Z2)/2r0, we get EA=-(ke2(M/14)(6+1)|2(2)|)/2r0.

Finally, to calculate the Madelung constant, we can rearrange this equation to get M=-(EA)2r0/((ke2(n1+n2)|Z1Z2|)/14). Plugging in the values for EA, r0, k, e, n1, n2, Z1, and Z2, we can calculate the Madelung constant for this octahedral arrangement of doubly charged anions and cation.