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The ionization energy of sodium is $\pu{5.139eV}$. This is the energy absorbed when a neutral sodium atom is stripped of its outermost electron. The electron affinity of chlorine is $\pu{3.62eV}$. This is the energy released when an electron is added to a neutral chlorine atom to form the chlorine anion.

Now suppose we have a system containing exactly one neutral sodium atom and exactly one neutral chlorine atom infinitely far apart so that the systems potential energy is zero. If we wish to make a $\ce{NaCl}$ molecule from our system, we need to put $\pu{5.139eV}$ of energy into the system to ionize the sodium atom. Afterwards, we place the electron we have just pulled off the sodium atom onto the chlorine atom upon which we get back only $\pu{3.62eV}$ of energy. So the total energy of the system has increased by $\Delta E =\pu{5.139eV -3.62eV= 1.519eV}$.

If the total energy of the system has increased after performing the previous supposition, then why does the $\ce{NaCl}$ ionic bond occur? Surely it is less stable than having the two neutral atoms separate from each other? My understanding up till now has been that molecules form when the total energy of the system is lower than the total energy of the sum of its constituents?

Is this all because an individual $\ce{NaCl}$ bond is not stable on its own but it is stable when its within a crystal lattice?

Any help on this issue would be most appreciated!

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    $\begingroup$ Sodium chloride molecule is not much of a thing at all. Then again, you might want to consider that the ions are not infinitely far apart. $\endgroup$ May 5 at 19:57
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    $\begingroup$ A similar thing with potassium and fluorine: chemistry.stackexchange.com/a/128771/79678. If you had two well separated atoms of K and F, an energy input would be required to get potassium ion and fluoride ion. But that is just one piece of the whole energetics. Same for sodium and chlorine. If you boil NaCl, there are some actual NaCl molecules in the vapor, for what it is worth. $\endgroup$
    – Ed V
    May 5 at 22:10
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    $\begingroup$ As you can see from Maurice's answer, you've underestimated the amount of energy required to produce NaCl molecules from pure Na and Cl2. But essentially, you're on the right track: What you've shown is that molecular NaCl is unstable...which is why NaCl's equilibrium state is not the molecular form, but the ionic form. Once you account for the energy you get from forming the ionic crystal, the net energy change is highly negative. $\endgroup$
    – theorist
    May 5 at 23:29
  • $\begingroup$ In short you are right just did not considered that when the ions get togheter the potential energy is lowered. Every X+Y- is more stable than the two taken with separation. Though it can be crucial the energy required for the two be formed,as per the answer of Maurice. But hold on the idea that molecules form when the total energy of the system is lower than the sum...... It is correct. So normally its is the lattice formation, though NaCl is certainly a possible molecular thing. $\endgroup$
    – Alchimista
    May 6 at 9:38
  • $\begingroup$ The lattice energy numbers for NaCl (a whole mole) were addressed in a recent answer to another question: chemistry.stackexchange.com/questions/138455/… The "lattice energy" for a single pair of ions would be significantly less. $\endgroup$ Jun 4 at 14:01
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What you have calculated is the energy required to make two separated ions out of two separated $\ce{Na}$ ad $\ce{Cl}$ atoms. When they are separated, these ions do not make a molecule or a crystal of salt. Separated ions attract one another and are able to release a huge amount of Coulomb energy when approaching each other.

Anyhow your calculation should still take into account the fact that chlorine atoms do not actually exist. They must be produced by breaking a $\ce{Cl2}$ molecule. And this requires some energy. You may also introduce the sublimation energy of sodium atoms

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