If we look at solubility of salts in water, we are told that they disassociate, because the positive $\ce{Na}$ ion is attracted to the partially negative oxygen in water and the negative $\ce{Cl}$ is attracted to the partially positive hydrogen. But why does this happen? I mean, the couloumb force should be much stronger between the $+1$ and $-1$ charges on the $\ce{Na}$ and $\ce{Cl}$ than the partial charges on the polar water molecule. Why does $\ce{NaCl}$, then, disassociate? It would make much more sense for the two ions to stick together.
Also, when you cut $\ce{NaCl}$, shouldn't it stick to its pieces? Look at salt crystals: they don't have any force of attraction between them. But, if you have positive sodium ions and negative chloride ions, then they should stick together.
Answer
As it happens, the enthalpy of solution of $\ce{NaCl}$ in water (that is, the energy change associated with the dissolution of sodium chloride crystals in water) at standard conditions is very slightly positive, i.e., it is an endothermic process. At a constant temperature and pressure, these kinds of thermodynamic processes are dictated by the change in Gibbs free energy, described by the equation
$$ \Delta G = \Delta H - T\Delta S $$
Where $\Delta G < 0$ is a necessary criterion for a spontaneous process. Given that $\Delta H$ is positive, $\Delta S$ must be positive as well, otherwise the process wouldn't occur spontaneously (which is to say, not without input of work from the surroundings, or coupling to some other strongly favorable reaction, neither of which is the case for the dissolution of salt). In other words, this is a process that is driven by the increase in entropy, which is entirely to be expected when moving from a highly ordered state (i.e., a crystalline solid) to a less ordered liquid solution. Indeed, in ideal solutions, where intermolecular forces of attraction are taken to be equal between all components, enthalpy change is necessarily zero, and entropy is always positive, so that the process of mixing in ideal solutions is always spontaneous. Of course, in real solutions, this is not the case.
Your intuition that the Coulombic forces between ions should be stronger is correct in this particular instance, as indicated by the positive enthalpy (meaning, the breaking of ionic bonds in the crystal lattice, as well as intermolecular bonds between solvent molecules, requires more energy than is released in the forming of ion-dipole bonds). The input of energy required for this process comes in the form of heat, drawn from the solvent. However, there are numerous examples of salts for which the enthalpy of solution is negative under a wide range of conditions.
The statement "[l]ook at salt crystals: they don't have any force of attraction between them" is inaccurate, though. In fact, there is strong ionic bonding. The very fact that the crystals remain solid except at very high temperatures is sufficient evidence of this, and the strength of the bonding in the crystal lattice can be quantified by calculating the lattice energy. It's a mistake to consider any substance in isolation. When you talk about "cutting" $\ce{NaCl}$, what you're talking about is applying physical force to the crystals, exposing them to the complex mixtures of gasses and mositure in the air, and bringing them in contact with, say, the metal in the knife. If left in relative isolation, the salt remains undisturbed, packed in a crystal lattice. A gas, however, which has weak forces of attraction between individual particles, will expand to fill a container without any work being done to it (at least as far as atmospheric pressure allows). A liquid will experience capillary forces, either climbing the walls of a narrow container, resulting in a concavity at its surface, or pooling away from the walls of the container, resulting in a convexity at its surface (which of the two happens depends mainly on the comparative strengths of the forces of attraction between the molecules of the liquid and those of the substance comprising the container).
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