Calculating the Total Energy of a lattice w/ the Madelung constant

In summary, the conversation discusses the contribution of a single ion to the electric potential energy in an ionic lattice. The energy can be expressed as a sum of various terms and is represented by the Madelung constant, which takes into account the ionic spacing and charges of the ions. The book then presents a formula for the total energy of the lattice per mole, which may be double counting interactions. The individual is questioning this formula and believes the correct formula should have a factor of half. However, they later realize that the book is referring to one mole of each type of ion, while their calculation is for one mole of total ions, leading to the missing factor of two.
  • #1
etotheipi
For an ionic lattice, the contribution to the electric potential energy from a single ion will be ##U_i = \sum_{j\neq i} U_{ij}##, which can be expressed as$$\begin{align*}U_i &= -6 \left( \frac{z_+ z_- e^2}{4\pi \varepsilon_0 r_0} \right) + 12 \left( \frac{z_+ z_- e^2}{4\pi \varepsilon_0 \sqrt{2} r_0} \right) - 8 \left( \frac{z_+ z_- e^2}{4\pi \varepsilon_0 \sqrt{3} r_0} \right) + \dots \\ \\

&= -\frac{z_+ z_- e^2}{4\pi \varepsilon_0 r_0} \left(6 - \frac{12}{\sqrt{2}} + \frac{8}{\sqrt{3}} + \dots \right) := -\frac{z_+ z_- e^2}{4\pi \varepsilon_0 r_0} M

\end{align*}$$where ##M## is the Madelung constant, ##r_0## the ionic spacing, and ##q_+ = z_+ e##, ##q_- = -z_- e##. The next line in the book ("why chemical reactions happen", Keeler/Wothers - two very distinguished chemists!) states that the total energy, per mole, of the lattice can be obtained with $$U = \sum_i U_i = -\frac{z_+ z_- e^2}{4\pi \varepsilon_0 r_0} M N_A$$However, I don't think this is right. Specifically, it seems like they are double counting interactions, when in fact we should only be counting one out of ##U_{ij}## and ##U_{ji}## each time. I would have thought the total energy be half of this$$U = \frac{1}{2} \sum_i \sum_{j \neq i} U_{ij} = -\frac{z_+ z_- e^2}{8\pi \varepsilon_0 r_0} M N_A$$The thing is, this factor of a half is absent from other references too (like here).

So I wondered if anyone else would agree, or if I'm missing something (N.B. for now, I'm ignoring the repulsive ##\frac{B}{r^n}## contribution to the energy). Thanks!
 
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  • #2
Oh hang on, I suspect they're talking about ##N_A## of ##M^+## and ##N_A## of ##A^-## (i.e. one mole of ##MA##), whilst I'm talking about ##N_A/2## of ##M^+## and ##N_A/2## of ##A^-## (i.e. one mole of total ions). There's the implicit factor of two. Whoops :doh:
 

Related to Calculating the Total Energy of a lattice w/ the Madelung constant

1. What is the Madelung constant and why is it important in calculating total energy of a lattice?

The Madelung constant is a numerical value that represents the sum of the Coulombic interactions between all the particles in a lattice. It is important in calculating the total energy of a lattice because it takes into account the long-range interactions between particles, which cannot be accurately calculated using other methods.

2. How is the Madelung constant calculated?

The Madelung constant is calculated by summing the Coulombic interactions between a lattice point and all of its neighboring lattice points, taking into account the distance between them and the charges of the particles. This process is repeated for all lattice points and the final sum is divided by the number of lattice points.

3. Can the Madelung constant be negative?

Yes, the Madelung constant can be negative. This occurs when the lattice has a net negative charge, such as in the case of an ionic crystal with more negative ions than positive ions.

4. How does the Madelung constant affect the total energy of a lattice?

The Madelung constant is directly related to the total energy of a lattice. As the Madelung constant increases, the total energy of the lattice also increases. This is because a higher Madelung constant indicates stronger Coulombic interactions between particles, leading to a higher energy state.

5. Are there any limitations to using the Madelung constant in calculating total energy of a lattice?

One limitation of using the Madelung constant is that it assumes a perfect, infinite lattice structure. In reality, most lattices have defects and imperfections which can affect the accuracy of the calculation. Additionally, the Madelung constant does not take into account other types of interactions, such as van der Waals forces, which may also contribute to the total energy of a lattice.

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