Calculate the Madelung Constant

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In summary, the Madelung constant for an octahedral arrangement of doubly charged anions about a doubly charged cation can be calculated by summing the electrostatic interactions between ions and using the given equations.
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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
EA=-(Mke2)/r0

α=2M/((n1+n2)|Z1Z2|)

EA=-(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 EA, but I fail to see how to do that without α.
 
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  • #2
Your Madelung equation doesn't seem familiar to me, what textbook are you using?
 
  • #3
Here is a screenshot of my professors notes I took these equations from.

Notes_zps8f083b95.jpg


This is our book:

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

What equations would you have used?
 
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  • #4
None of these. Madelung constant is calculated just by summing electrostatic interactions between ions.
 
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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.
 

Related to Calculate the Madelung Constant

1. What is the Madelung constant?

The Madelung constant is a mathematical constant that is used to calculate the electrostatic potential of a crystal lattice. It takes into account the charges and positions of all the ions in the lattice.

2. How is the Madelung constant calculated?

The Madelung constant is calculated by summing up the contributions from all the ions in a crystal lattice using the Madelung formula. This formula takes into account the charge and position of each ion, as well as the distance between the ions.

3. Why is the Madelung constant important?

The Madelung constant is important because it helps us understand the electrostatic interactions between ions in a crystal lattice. This knowledge is crucial in fields such as materials science and solid state physics.

4. Can the Madelung constant be negative?

Yes, the Madelung constant can be negative. This occurs when the crystal lattice has a net negative charge due to the arrangement of ions. A negative Madelung constant indicates that the electrostatic potential is lower at the center of the lattice compared to infinity.

5. How is the Madelung constant used in real-world applications?

The Madelung constant is used in various real-world applications, such as predicting the properties of ionic compounds, designing high-performance materials, and understanding the stability of crystal structures. It is also used in computer simulations to model the behavior of complex systems.

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