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One explanation I read is that soap's long non-polar end will seek the air at the interface between water and air, and that the ionic end will form ionic-dipole bonds with the water. In the process of moving to the surface, the soap molecules force apart the water molecules, and hence the water molecules no longer have hydrogen bonds with each other, and the surface tension is weakened. However, wouldn't the ionic heads of the soap molecules just form stronger ionic-dipole bonds with the water, increasing inter molecular forces, and therefore increasing surface tension? What is the true explanation for why soap reduces surface tension of water?

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Apoorv did not give the right explanation, because this would imply, that surface tension goes especially down, when micelles are formed, but the opposite is the case. Compare e.g.: first chapter of The Colloidal Domain by D.Fennell Evans and Håkan Wennerström.

First thing to note is that pure water has a higher surface tension than hydrocarbon chains which is due to the strong dipole dipole interaction.

If there is a very dilute solution of surfactant in water no micelles are formed but you have monomers "floating around". In contrast to ions in water they are not homogenously distributed, but they concentrate especially on the surface where their polar head is directed to the water and their hydrophobic chain is directed to the air. In the end you substituted water molecules with hydrocarbon chains which leads to a lower surface tension.

If you increase the concentration of surfactant, they start to form micelles in the water (and not on the surface). So after reaching this critical micelle concentration (called CMC) the surface tension stays quite constant. This constant behaviour is used experimentally to estimate the CMC.

See the figure below Equation 11: http://infohost.nmt.edu/~jaltig/SurfaceTension.pdf

Please note that Apoorv described some kind of reversed micelle. The normal way is that they stick their polar head to the water and have their hydrocarbon chain inside the micelle.

In the picture the white group is the polar head: https://en.wikipedia.org/wiki/Micelle#/media/File:Phospholipids_aqueous_solution_structures.svg

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The polar end of the surfactant is attracted to the polar water molecule. When enough surfactant molecules attach to a water molecule, it gets covered in surfactant and forms a unit. These units have a weaker force of attraction because only the non polar head of the surfactant is exposed thereby lowering the surface tension.

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