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Internal energy at a specific state can't be calculated, but using kinetic theory of gases and the law of equipartition of energy, average kinetic energy is directly proportional to temperature. For an ideal gas, internal energy is due to kinetic energy, as we neglect potential energy. This means internal energy is a function of temperature only. This gives for a monoatomic gas, the equation: $$\langle E\rangle=\frac{3}{2}RT,$$

from $U=\frac{f}{2}RT$ from law of equipartition theorem, where $f$ is number of degrees of freedom.

So why can't we find internal energy of a particular state?

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    $\begingroup$ We can, in the case you mentioned it is function of temperature and number of particles only. However, in general, internal energy is not a function of those two variables only, but may depend on density and possibly other variables. $\endgroup$
    – Ján Lalinský
    Commented Aug 26, 2017 at 22:51
  • $\begingroup$ but here we are discussing about ideal gas. for an ideal gas it depends only on temperature $\endgroup$
    – L0705
    Commented Aug 4, 2018 at 16:01

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As internal energy is a state function so its change is defined. To find its value in a given state we have to consider a reference state of the system in which it is taken zero arbitrarily. In your result of kinetic theory zero kelvin is the reference temp or state.

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  • $\begingroup$ But in kinetic theory we took absolute terms like temperature, pressure we did not take any reference state $\endgroup$
    – L0705
    Commented Aug 28, 2017 at 1:53
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You seem to be taking an almost universal rule and understanding it as absolutely universal.

The limitation on knowing internal energies is (a) if we only know the macrostate and (b) applies to almost, but not quite, all systems.

The 'almost' is important here because for a few—generally very simple—models and a few physical systems that are very well represented by those models we can compute the internal energy knowing only the macroscopic variables.

And the ideal gas is the premier example of such a system.

So, the short answer to your question us that the ideal gas is one of a small number of exception to the general rule.

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The internal energy can be calculated only for some systems, which includes the ideal gas example you used.

For real gases, internal energy depends upon other things as well, for which we cannot derive a relation.

Hope this helps. Ask anything if not clear. Have a wonderful day :)

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Even monoatomic gases are made of smaller parts, nuclei, electrons, and other sub-atomic particles, all of which are capable of storing energy. When we write $$\langle E \rangle = \frac{3}{2}RT$$ we are making the implicit assumption that under the conditions of our system these other forms of energy do not change appreciably. But this doesn't change the fact that there are contributions to internal energy that we are not including in this formula.

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Internal energy refers to the energy of matter which is due to its internal state.... it is the energy possesed due to kinetic and potential energy of the system...

it consists of all the following:

  1. Electrostatic potential energy of subatomic particles..
  2. Thermal energy of those particles...( kinetic energy and potential energy) this one is sub included in previous one as potential energy is just electrostatic repulsion
  3. Gravitational potential energy

we can summarise all these in just thermal energy of The system.. of its subatomic particles....

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