Hello everyone,
In heat conductivity molecules vibrates, right? Like one molecule vibrates it vibrates the other and it goes on and on and on... So amplitude increases. So if molecules vibrate then why can't we see the object vibrating or at least any kind of movement of the object as objects are made up of molecules?
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Two reasons: (1) the molecules don't all vibrate in phase, but rather vibrate randomly, and (2) the amplitude of motion is very, very small.

I believe the movement is expressed by the expansion of the material.

Let me give you the example: The big wall and bricks in it. So if you move 1 brick then it will move the nearby one and on and on. (Heat Conductivity is the transfer of heat from 1 molecule to another. Right?)
So ultimately the wall will fall. Same is in the case of Heat Conductivity, heat transfers from molecule to molecule and as the object is made up of molecules so it should be obvious that the object will move as components in it will move.

And yeah expansion is the result of it but can anyone give me example of the expansion of some object which expands due to heat conductivity.
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Most materials expand when heated. Metals are a good example.

As for your bricks-in-a-wall analogy - when one object pushes on another the movement of the second object is not instantaneous. From F=ma it's clear that given some pushing force from the fisrt atom the acceleration of the neighboring atom is finite, which means it doesn't move precisely in sequence with the first. Further, that second atom has its own vibration going on as it too has been heated, and there is no reason to think that its vibrations will be in synch with the first. Hence as you consider the vast numbers of atoms in the material the atoms vibrate essntially randomly, and the net movement of material is zero.