# Calculate the lattice energy of ammonium carbonate

The lattice energy is a measure of the stability of a compound and I need this data for the ammonium carbonate ($$\ce{(NH4)2CO3}$$) but I don't know how to calculate and I can't find it anywhere. I know that it can be calculated with the Born-Lande equation but I don't know how to apply this equation to solve my problem.

I wonder if anybody knows a database where I can find this value, I need it for an important chemistry assessment.

The lattice energy is not directly measurable, and it is difficult to calculate. In a paper about estimating lattice energies, the estimation for $$\ce{Na2CO3}$$ is 30% off from the reference value (which is from CRC Handbook of Chemistry and Physics). I mention sodium carbonate because ammonium carbonate is not one of their examples. The introduction explains why it is not possible to directly measure the lattice energy:

Lattice potential energy (UPOT) is a dominant term in the thermodynamic analysis of the existence and stability of ionic solids. Direct experimental determination is generally not possible since, in practice, the crystalline solid dissociates into atoms and not into gaseous ions, as is required in the lattice energy evaluation. Therefore, its indirect experimental determination, computation, or estimation is of considerable interest in modern materials science; indeed, whenever the energetics of condensed-state materials are studied, the chemical processes under consideration may be rationalized if the appropriate lattice energy steps can be incorporated into the thermochemical cycle.

They go one to discuss how you can obtain the lattice energy from experimental or theoretical data:

A variety of estimation methods for lattice energies is vavailable. These include the Born-Haber-Fajans thermochemical cycle (which requires ancillary thermodynamic data) and modern computational methods (which generally require knowledge of the lattice constants and the coordinates of the ions as well as needing an established force field). The computational methods range from direct energy calculational procedures through to programs that produce lattice energies in the course of their modeling of the solid. Quantum mechanical procedures are also available, but are highly computationally expensive and are generally applied only to the simpler systems.

Then they talk about cheaper methods that are less accurate. The paper is about a new quick estimation methods.

By contrast, a few exceedingly rapid estimation methods based on ionic radii or volume per formula unit (“molecular”volume) have been developed. Originally only applicable to the simpler binary ionic solids (such as MX (with a 1:1 charge ratio)), the volume-based approach has recently been extended by us to include MX2(2:1) and M2X (1:2)) salts.

So check the CRC Handbook of Chemistry and Physics.

• Just to comment that in the 20 years since the paper computers and computational methods have come on considerably - it should be better to do a somewhat better job nowadays. This is easily in the range of ab initio packages, though I admit don;t know how to extract the lattice energy term from the calculated total energy. Mar 16, 2020 at 10:36
• @IanBush So you could calculate it with periodic boundary conditions, but if you use a spherical cut-off for the electrostatic term (assuming you don't do ab initio for the entire infinite lattice), you might end up with an answer that depends chaotically on the cutoff radius, see Madelung constant.
– Karsten
Mar 16, 2020 at 12:56
• Yes, I know about Ewald summations (e.g.sciencedirect.com/science/article/pii/S0010465506001901) Any respectable condensed matter code will do this correctly, be it ab initio or not, as otherwise the answers are wrong - as you say. Mar 16, 2020 at 12:59

The preview of this paper, DECOMPOSITION OF AMMONIUM CARBONATE AND AMMONIUM BICARBONATE AND PROTON AFFINITIES OF THE ANIONS, by J. E. House, Jr. cites the use of lattice energies for ammonium carbonate as an aid to determining proton affinities.

The full work may provide some data.