I was studying about Surface Energy in fluids, and here's what I was reading:
(a)
A liquid stays together because of attraction between molecules. Consider a molecule well inside a liquid. The intermolecular distances are such that it is attracted to all the surrounding molecules. This attraction results in a negative potential energy for the molecule…
How do I know that a negative potential energy has been produced?
Also, I saw this thing right after the above statement:
(b)
The average potential energy of all the molecules is the same. This is supported by the fact that to take a collection of such molecules and to disperse them far away from each other in order to evaporate or vaporise, the heat of evaporation required is quite large.
How does the largeness of heat of evaporation in this example show that the average potential energy of all the molecules is the same?
Best Answer
There are many factors contributing to intermolecular forces, but to a first approximation the total potential energy is a sum of interactions between pairs of molecules. Also, generally, the interaction between each pair is attractive when they are far apart, and repulsive when they get so close together that the electron shells start to overlap. A very simple model is the Lennard-Jones potential. An attractive force means that the potential energy decreases as the molecules get closer together. Typically, in a liquid, the nearest neighbours of a given molecule will be at a distance close to the minimum of the pair potential: this means, a negative potential energy (taking the zero of energy to be at infinite separation). Interactions with molecules that are further away are weaker, but still correspond to negative potential energies. So, for a liquid, the total potential energy for the interaction of a given molecule with its neighbours will be negative.
The second quote is just saying that on average, all the molecules in a liquid are equivalent to each other. At any instant, their interaction energies with their neighbours will be different, but if we take an average as they jostle around, the answer will come out the same for each one. (It is implicitly assumed that we are just discussing a liquid composed of a single species; water, for example).
The implication is that, if there are $N$ molecules, and each has a total average interaction potential energy with its neighbours $-E$, then it would take an overall energy $NE$ to separate all the molecules, and this should correspond to the heat of evaporation. A bit more thought will show that this calculation overestimates the total energy by a factor of $2$ (since we would count the average interaction of molecule $i$ with neighbour $j$, and that of $j$ with neighbour $i$, separately, but really we should only count it once), so the estimate of the total evaporation energy is $NE/2$.
All of this only considers the molecules within the bulk of the liquid. Your source then goes on to consider those molecules near the surface. For those molecules, the neighbours are mostly on the liquid side, there are far fewer neighbours on the gas side, so the total potential energy per molecule is less negative than for those in the bulk of the liquid.