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In some of the descriptions I have found about surface tension, while they explain the phenomenon for liquid in an open container they use the following arguments:

1) A molecule at the surface feels a net downward force

2) A molecule which is not at the surface feels zero net force

As far as I know, even in static fluid, the molecules are always in motion and it is not logical to think any of them with zero acceleration and with zero net force.

Are those explanations correct or not?

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The molecules at the liquid to air interface feel this downward pressure due to cohesion with other liquid molecules. The molecules below the surface feel no net force (unless external force is applied) because cohesion with other molecules is pulling them equally in all directions. They are in motion, as you stated, but they are not accelerating. decelerating, or changing vectors (except for normal collisions with each other, or container walls). You might read https://en.wikipedia.org/wiki/Surface_tension and https://en.wikipedia.org/wiki/Surface_tension#Causes

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