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When a substance is heated but not changing its phase, is the potential energy between the particles constituting the substance also increasing, or is it only the random kinetic energy of particles that increases?

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The potential energy between the particles constituting the substance basically depends on the average distance between particles and the nature of the interaction (ionic, dipole-dipole, etc.). For many systems, this variation is orders of magnitude smaller than the increase of the kinetic energy of particles for many systems.

However, indeed, thermal dilatation might change the average distance between particles. Temperature increase might favor disorientation of electric dipoles, contributing to the increase of the potential energy of the arrangement. In this sense, you are right.

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