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Definition of heat, in contrast to the definition of work, is movement of particles with no particular direction.

Definition of static pressure, in contrast to dynamic pressure, is again movement of particles with no particular direction.

My confusion is why these different notions have similar definitions at the microscopic level?

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    $\begingroup$ Have you looked at the units of heat and pressure? They are not at all the same, so your understanding of them at a microscopic level is lacking. $\endgroup$ – tpg2114 Feb 13 '17 at 10:47
  • $\begingroup$ They're not the same, but proportional. $\endgroup$ – Tobi Feb 13 '17 at 11:03
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    $\begingroup$ @tpg2114 Let's keep a constructive tone, shall we? Naturally, some understanding is lacking; otherwise no need for a question. +1 one from me to even out this uncalled for aggresiveness on an interesting question about macroscopically different properties that seem microscopically similar at first. $\endgroup$ – Steeven Feb 13 '17 at 11:23
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    $\begingroup$ To be fair, pressure does have the same units as energy density, and there is a nontrivial connection there. $\endgroup$ – rob Feb 13 '17 at 13:41
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    $\begingroup$ A simple answer: Heat (or better internal energy) is related to all degrees of freedom of the system. So it includes kinetic energy but vibrations and rotations as well. Static pressure on the other hand only depends on the momentum distribution of the molecules and is therefore only related to the kinetic energy. $\endgroup$ – Jannick Feb 13 '17 at 14:05
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If you consider an ideal gas without any mean motion, then at the molecular level all you have is molecular motion (there is no potential between them, except that associated with hard spheres), and therefore every property definable for an ideal gas must have its origin in that motion. But then the procedure for calculating any particular property (say, internal energy) is different from that for another property (say, pressure; if you are thinking of a flow, then static pressure).

As @Jannick said in his comment, internal energy accounts for other kinds of energy apart from kinetic energy of molecules. However even if the ideal gas molecule is assumed to have no further structure so that its internal energy is completely accounted for by kinetic energy of its molecules, it is still different from pressure.

Pressure at a point is defined as normal momentum flux per unit area of an infinitesimal surface located at that point (see this). Unlike what you said, a direction is essential in defining this particular momentum flux, which direction is the normal to the area element under consideration. Only under the assumption of isotropy does this particular momentum flux become independent of any direction. Internal energy on the other hand is defined as amount of kinetic energy of molecules contained in a volume. There is no flux across some area element to be considered in this definition. As you can see, even though both properties refer ultimately to molecular motion, they, by virtue of their particular definitions, are measuring different aspects of molecular motion. In fact you can vary the two properties independently of each other.

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