Born-Haber cycle

The enthalpy change of the red reaction path (standard enthalpy of formation) is equal to the sum of the enthalpy change of the hypothetical reaction steps (blue).

The Born-Haber cycle (also Born-Haber cycle , found in 1916 by Fritz Haber and Max Born independently of one another) links energetic quantities ( state quantities ). It is a direct consequence of Hess ' theorem , according to which the enthalpy of reaction is independent of the reaction path and depends only on the initial and final state of the products and starting materials. In such a cycle, each variable can be determined if the others are known. Instead of a reaction, a sum of hypothetical partial steps is considered, which represent an alternative and energy-equivalent reaction path. This makes it possible to determine quantities that are difficult to measure, such as the lattice energy of ion compounds, electron affinity or ionization energy using the Born-Haber cycle. All quantities used in the cycle are energetic quantities related to the metabolism (unit: kJ per mol). The cycle process is nothing more than a sum of energies. It is not possible to carry out the hypothetical partial steps in exactly this order as an overall reaction in the laboratory. However, by experimentally determining the individual values, they can be set up in a circular scheme. Like all cycle processes, the Born-Haber cycle is based on the 1st law of thermodynamics .

Thermodynamic justification

According to Hess' theorem , the total sum of the enthalpy changes must be zero.

${\displaystyle \underbrace {\sum \Delta H_{i}} _{\text{Alle Reaktionswege}}=0}$

By extracting the standard enthalpy of formation from the sum, the link between the two enthalpy balances becomes visible:

${\displaystyle \underbrace {\Delta H_{f;m}^{0}} _{\text{Standardbildungsenthalpie}}+\underbrace {\sum \Delta H_{i}} _{\text{Hypothetische Reaktionsschritte}}=0}$

By rearranging, the relationship between the standard enthalpy of formation and the remaining reaction steps becomes clear:

${\displaystyle \Delta H_{f;m}^{0}=-\sum \Delta H_{i}=\sum -(\Delta H_{i})}$

The standard enthalpy of formation therefore corresponds to the sum of all partial steps, with their signs being reversed.

The sum of all enthalpy changes of the hypothetical sub-steps must therefore be equal to the enthalpy change when the substance is formed (under standard conditions ). Once the hypothetical reaction steps have been established, they can be rearranged and the desired size can be calculated.

Establishing a scheme

Finished cycle for sodium chloride. The enthalpy change of the blue reaction path corresponds to that of the red reaction path. The lines sharply symbolize the relative enthalpy level of the intermediate steps. Dashed lines symbolize the gas phase

In the following the establishment of a cycle process for sodium chloride from solid sodium and molecular chlorine will be explained. The lattice energy of sodium chloride should be sought. While technical terms such as 1st ionization energy are largely standardized, the letters used in the literature vary significantly. Of course, only the numerical amount of energy is important. This is related to an amount of substance . The unit is therefore kilojoules per mole .

1. Draw a vertical enthalpy axis to indicate the direction of enthalpy changes . Regarding the sign convention, there are unfortunately different conceptions between physics and chemistry. In general, changes to a system must be considered. In this example, the following should apply: positive signs in front of the sub-steps cause an increase in enthalpy (direction of arrow), negative signs cause a decrease in enthalpy. Enthalpy changes should be marked with a symbol and amount of energy. ${\displaystyle \Delta H}$ ${\displaystyle \Delta H>0}$ ${\displaystyle \Delta H<0}$
2. The product is drawn on a horizontal line as far down as possible on the right to leave a lot of space above. During the entire process, attention must be paid to the states of aggregation , stoichiometry and electron configuration ! ${\displaystyle NaCl_{(s)}}$
3. The starting materials and chlorine gas (note stoichiometry!) Are applied a little higher to the left on a vertical line. A reaction arrow for the reaction product is drawn and labeled as the standard enthalpy of formation or enthalpy of reaction . The normal reaction path is thus recorded. Now it is necessary to write down the hypothetical sub-steps that would convert the starting materials (solid sodium and chlorine gas) into the same state. ${\displaystyle Na_{(s)}}$ ${\displaystyle {\frac {1}{2}}Cl_{2}(g)}$ ${\displaystyle NaCl_{(s)}}$ ${\displaystyle \Delta H_{f;m}^{0}}$ ${\displaystyle \Delta H_{R}}$
4. First the sodium has to be brought into the gas phase . For that it has to be sublimated . The enthalpy of sublimation of sodium required for this is +229 kJ per mol. The educts are now higher on the enthalpy axis and are drawn there. Sodium is now no less gas: . ${\displaystyle \Delta H_{S}}$ ${\displaystyle Na_{(g)}}$
5. Now the covalent bond of the chlorine molecule has to be split homolytically . The dissociation energy of chlorine is 242 kJ per mol. There are 2 chlorine radicals . However, the amount of energy must be halved due to the stoichiometry, since only one chlorine atom reacts with one sodium atom to form NaCl. Like the chlorine gas, these chlorine radicals are still in the gas phase. The educts are again higher up on the enthalpy axis.
6. Now sodium has to be ionized , i.e. the 3s electron has to be released from the valence shell in order to form a sodium cation . For this, the 1st ionization energy (i.e. the energy to remove the 1st electron) has to be applied by sodium. is +485 kJ per mole. The fabrics are now higher up. The released electron of sodium is still carried along. ${\displaystyle Na^{+}}$ ${\displaystyle \Delta H_{IE_{1}}}$ ${\displaystyle e^{-}}$
7. The previously released electron from sodium is now absorbed by a chlorine radical. A chloride anion is formed . The amount of energy that electron uptake is called the electron affinity of a chlorine atom. It is −349 kJ per mol. This negative amount of energy brings the substances back to a lower enthalpy level. Sodium is now as and chloride as . ${\displaystyle Na_{(g)}^{+}}$ ${\displaystyle Cl_{(g)}^{-}}$
8. The energy of the sodium cation and the chloride anion separate from solid sodium chloride only to the extent of the lattice energy of sodium chloride. This is wanted. A labeled arrow closes the cycle. ${\displaystyle \Delta U_{G}}$ ${\displaystyle NaCl_{(s)}}$

The energy sum results from the cycle process set out above:

${\displaystyle \Delta H_{f,m}^{0}(NaCl)=\Delta H_{subl.}(Na)+\Delta H_{IE_{1}}(Na)+{\frac {1}{2}}\Delta H_{Diss.}(Cl_{2})+\Delta H_{EAff.}(Cl)+\Delta U_{G}(NaCl)}$

By rearranging according to the lattice energy sought one obtains

${\displaystyle \Delta U_{G}(NaCl)=\Delta H_{f,m}^{0}(NaCl)-\Delta H_{subl.}(Na)-\Delta H_{IE_{1}}(Na)-{\frac {1}{2}}\Delta H_{Diss.}(Cl_{2})-\Delta H_{EAff.}(Cl)}$

Pay attention to the signs when moving and inserting! Inserting values ​​from the literature gives:

${\displaystyle \Delta U_{G}(NaCl)=-410\,\mathrm {\tfrac {kJ}{mol}} -\left(+108\,\mathrm {\tfrac {kJ}{mol}} \right)-\left(+496\,\mathrm {\tfrac {kJ}{mol}} \right)-\left(+{\frac {1}{2}}\,\mathrm {244{\tfrac {kJ}{mol}}} \right)-\left(-349\,\mathrm {\tfrac {kJ}{mol}} \right)}$

The lattice energy calculated in this way is:

${\displaystyle \Delta U_{G}(NaCl)=-787\,\mathrm {\tfrac {kJ}{mol}} }$

which comes close to the literature values ​​between to very close. ${\displaystyle -766\,\mathrm {\tfrac {kJ}{mol}} }$ ${\displaystyle -780\,\mathrm {\tfrac {kJ}{mol}} }$

Alternative representations

A complete cycle of all reaction pathways of magnesium (II) chloride. The enthalpy changes are relative, but true to scale. It should be noted that magnesium must be ionized twice. Both the first and the second ionization require an extremely large amount of energy compared to other sub-steps.

As long as the principle of Hess's theorem and energy conservation is not broken, other forms of representation are also permitted. Some representations prefer the absence of an enthalpy axis. Other forms represent all possible reaction paths at once by specifying all starting materials in all aggregate states and other states.

Quality of expression

One can also make predictions about the stabilities of previously unknown compounds , for example those with high oxidation numbers . Even the noble gas compounds , which were still unknown at the time, were predicted to be stable before their synthesis by means of stability assessments with the help of the Born-Haber cycle process .

criticism

The accuracy of calculated values ​​depends on the measurement accuracy of all other values. A certain uncertainty is based on the fact that some values ​​are difficult to determine precisely with average equipment. However, a large number of values ​​are tabulated and can be found in the literature. The lattice energy can usually be determined more precisely with the Born-Landé equation .

See also

• Born-Landé equation
• Kapustinskii equation

Web links

• Cycle process using the example of sodium chloride formation

Individual evidence

1. ^ E. Müller, Ulrich Mortimer: "Chemistry - The basic knowledge of chemistry" (10th edition), page 98, ISBN 978-3-13-484310-1 .
2. ^ Erwin Riedel , Christoph Janik: "Riedel - Inorganische Chemie" (8th edition), page 92, publisher: De Gruyter Studium. ISBN 978-3-11-022566-2 .
3. ↑ Lesley E. Smart, Elaine A. Moore. "Solid State Chemistry". Third Edition (2005), page 97, ISBN 0-203-49635-3 .
4. ↑ http://www.chemieunterricht.de/dc2/tip/04_00.htm
5. ↑ Michael Binnewies, Manfred Jäckel, Helge Willner, Georg Rayner-Canham. "General and Inorganic Chemistry". Spectrum Akademischer Verlag Heidelberg - Berlin, 1st edition 2004, page 128, ISBN 3-8274-0208-5 .
6. ↑ http://www.chemgapedia.de/vsengine/vlu/vsc/de/ch/11/aac/vorlesung/kap_4/vlu/gitterenergie.vlu/Page/vsc/de/ch/11/aac/vorlesung/kap_4 /kap4_4/kap44_2/kap442_1.vscml.html
7. ↑ Archive link ( Memento from November 15, 2016 in the Internet Archive )