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My physics textbook says, "The amount of force by which molecules attract each other when they are moved away from each other by a certain distance, is lesser than the amount of force by which molecules repel each other when they are moved closer to each other by the same distance." Why?

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I find that sentence very badly written, so I hope that your book isn't always like that. In any case, the potential describing the interaction between two molecules is probably of the form: potential

From this potential you can deduce the force: $$F=-\frac{dV}{dr}$$

Now look at the potential:

  • When the molecules are closer, $r$ is shorter so $V$ has a steeper slope, so the force is stronger (also, since the slope is negative, the force is repulsive).
  • When the molecules are further apart, $r$ is longer, $V$ is less steep so the force is less strong (and the force is attractive).
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To add to the already correct answer by @Miayse:
Think of a two pieces of magnet or two balls of different electric charge: they are attracted to each other, but, once they touch, cannot get closer anymore - this would require them physically penetrating each other.

A caveat here is that we are talking about molecules, not about atoms and certainly not about two particles of the opposite charge. In the latter case, we have simple Coulomb attraction, and any repulsive force is unfathomable. Atoms and molecules are however made of many charges of different sign. So they may attract each other even when each molecule is overall neutral - although this attraction is weaker than the Coulomb law. Yet, when the two molecules are brought very close to each other, they would repel - accommodating all these charges in small space would require changing the structure of molecules, which is energetically unfavorable. Lennard-Jones potential is a simplified model of such intermolecular interactions.

To some extent this also applies to metals: electrons in metals repel each other, and the fact that there are positive ions present does not alone explain the stability of a metal (the Earnshow theorem tells us that classically such a system could not be stable). There is however a quantum contribution due to the indistinguishability of electrons (exchange energy) which keeps electrons from running too far away:

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$U=kQq/R$ is Potential energy between two charges $q$ and $Q$ at distance $R$ so when $R$ tends to zero then $U$ tends to infinity and molecules always wants to be have high stability which comes from low potential energy so they repel strongly when they comes closer that's why "The amount of force by which molecules attract each other when they are moved away from each other by a certain distance, is lesser than the amount of force by which molecules repel each other when they are moved closer to each other by the same distance.

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