Ionic Bonding (4) Lattice Energy and the Born Haber Cycle

Ionic Bonding: Lattice Energy and the Born Haber Cycle

Now here is a question for you: how do you measure the strength of an ionic bond?

Why are there real problems in trying to answer that kind of question?

Is there in fact such a thing as an ionic bond?

If we look closely at the structure of a typical ionic compound like sodium chloride

This diagram of the structure has the usual faults you find with these.

The whole thing is of course black and white in reality there is no colour at the ionic level.

The lines and mere construction lines they do not exist.

The circles are too small and refer to charged species which do not have “edges”.

Apart from that it’s pretty accurate!!

The highlighted blue arrangement shows that each green sodium ion has six closest chloride ion neighbours.

The reverse is true for each chloride ion, each one has six closest sodium ion neighbours.

That is what gives sodium chloride its 6:6 coordination.

It also suggests that the charge on a sodium ion influences not one but at a minimum six other chloride ions.

And the opposite is true for the chloride ions negative charge influencing six sodium ions. 

So the ratio of sodium to chloride ions might be one to one but to suggest that a singular sodium ion—chloride ion bond exists is just not on.

Here is a neat depiction of the 6:6 coordination number for sodium chloride:

The two octahedra show that each ion is surrounded by a minimum of six of the other type. 

You ought also to notice that the structure is cubic.

Here is a video of how to grow a cubic crystal of salt.

And at the centre of each face of the cube lies a single ion either positive or negative.

Hence the name for this structure is face centred cubic (FCC).

It is easier to see this fcc arrangement if we look closely at the unit cell of the structure.

Here is a diagram of the sodium chloride unit cell:

What is a unit cell you ask?

It's the smallest repeatable section of the whole structure.

If stood end to end these unit cells would generate the 3D structure of sodium chloride. 

See the youtube video here.

There is a lot to be said at this point for getting your hands on a lot of polystyrene balls lots of balls, one about half the size of the other, and feeling how the two sizes fit together and sticking them together and then cutting them up to make a unit cell.

The ionic radii of the two ions are sodium Na+  is 0.102 nm and that of the chloride ion is 0.180 nm.

Balls about twice the size of the other ought to make the structure quite well.

You’ll need some stiff glue and a sharp modelling knife.

The other point to see in the unit cell picture is that the ions are represented a space-filled spheres. 

You can see how the spherical electrical charge field influences more than one other ion of the opposite charge.

So attempts to calculate the force of attraction between one chloride ion and one sodium ion are a non—starter.

Theoretical attempts have been made to calculate this force attraction between ions in a structure.

First we have to define what we mean by this “force of attraction” between ions in an ionic structure.

Let’s think of the ions of sodium and chloride infinitely separated so that there is no force of attraction between them.

If we bring together a mole of each ion from infinity the result will be a mole of sodium chloride solid.

Here’s the equation for this process of the formation of a mole fo sodium chloride:

Na⁺ _(g)      +      Cl^— _(g)     =      NaCl_(s)

The energy released in this process is termed the Lattice energy since a sodium chloride lattice is the result. 

We’ll give this energy change the symbol ΔH_latt

For Sodium chloride the ΔH_latt  value is —786 kJ.mol^—1

You can read further into the theoretical method for calculating the value of ΔH_latt here

Lattice energies are usually calculated using an experimental approach involving a form of the Hess cycle called the Born Haber cycle.

Here’s how it works

First thing is this the lattice energy for the reaction above cannot be got at directly because we cannot put a mole of gaseous ions at infinity from each other.

So we have to find an indirect way of getting at the ΔH_latt

First we’ll atomise then ionize a mole of sodium atoms.

These are the two equations:

Atomisation forms a mole of Na(g) atoms from a mole of Na (s) atoms:

Na(s)     =      Na(g)      ΔH _at   =    + 107.3   kJ.mol^—1

Ionisation forms a mole of sodium ions from a mole of the atoms all in the gaseous state:

Na(g)      =      Na+ (g)      +      e—        Em1   =    + 496  kJ.mol^—1

Forming a mole of chloride ions requires two processes as well.

First atomization of chlorine to form a mole of chlorine atoms:

½ Cl₂ (g)      =       Cl (g)  ΔH _at   =    + 121.7  kJ.mol^—1

then the gain of a mole of electrons by this mole of chlorine atoms.  We call this the electron affinity.

Cl(g)     +    e—        =       Cl^— (g)      ΔU   =  —348.8  kJ.mol^—1

We just need one further experimental value then we can construct the Born Haber cycle and calculate the ΔH_latt

The standard enthalpy of formation of sodium chloride is given by this equation:

Na(s)    +      ½ Cl₂ (g)        =    NaCl(s)   ΔΗf  =   —411.2 kJ.mol^—1

Here is the completed Born-Haber cycle:

On the left you have the formation of the sodium ions, in the middle the formation of the chloride ions.

The lattice energy equation is laid across the top of the cycle. 

The arrow to the right stands for the enthalpy of formation of the sodium chloride. 

To calculate the lattice enthalpy from this form of the Born Haber cycle just add up all the four values to the left of the cycle and subtract the value from the enthalpy of formation of sodium chloride

So ΔH_latt [NaCl_(s)]=   ΔΗ_f [NaCl_(s)] — { ΔH_at [Na] + E_m1[Na] + ΔH_at[½ Cl_2(g)]

 + ΔU [Cl]}

ΔH_latt [NaCl_(s)]=   —411.2    — { +107.3  + +496  + 121.7  — 348.8}

ΔH_latt [NaCl_(s)]=   —787.4 kJ.mol^—1

Now what you need to do is see if you can repeat this kind of calculation for another ionic salt like magnesium chloride where the formula is different and the ion ratio is not 1:1.  

For an example like MgCl₂  you’ll need to double up the values for the chloride ions and to realize that there are two ionization energies to add up since the charge on the magnesium ion is 2+. 

You can then find the calculation done for you here on youtube.

See what you think…..