Reason for inability to find the absolute value of internal energy

Today in my thermodynamics lecture, my teacher told me that it is not possible to find the absolute value of internal energy and so we have to calculate the change in internal energy.


So, my question is why isn't it possible to find the absolute value of internal energy? Is this statement true?




In order to answer this question, one needs to define what the absolute energy of a system is. Energy can be trapped in a system in ways not yet discovered or fully understood. Think of the energy associated with mass ($E=mc^2$), which is a result that was not known to the early founders of thermodynamics.


We need to define reference points, which we can use to compare our products with. Usually this "zero" point is defined as the energy of the individual atoms in their most energetically favorable state (phosphorus is an exception, since red phosphorus is much more available), such as $\ce{H2}$ and $\ce{He}$.


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I also heard that and it's not exactly true, you could find total value using mentioned earlier $E=mc^2$. It shouldn't be called absolute, though, as mass-energy depends on frame of reference - masses measured between [inertial](https://en.wikipedia.org/wiki/Inertial_frame_of_reference) and [non-inertial frames of reference](https://en.wikipedia.org/wiki/Non-inertial_reference_frame) are different. Also, as pointed in comments, you can't account for [zero point energy](https://en.wikipedia.org/wiki/Zero-point_energy) of [vacuum](https://en.wikipedia.org/wiki/Vacuum_state) this way, although I wouldn't call it a problem, as this energy is the same when the object of measurement is absent.


Still, if you weighed something and multiplied this mass by $c^2$ you would get its total energy. It would be a very rough estimate, though, as [one gram is equivalent to about 90 terajoules of energy](https://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalence), which is really large amount. Even if you would measure a very small mass with great precision, it would still be a rough estimate.


 8

In *classical* thermodynamics the change in internal energy is defined by the first law as
$$\Delta U = q + w$$
so that only the difference in $U$ is known; $q$ is the heat absorbed by the 'system' and $w$ the work done on the system. 


For example in a closed system (no exchange of matter with environment) we can write for a reversible change
\begin{align}
 \mathrm{d}q &= T\mathrm{d}S \\
 \mathrm{d}w &= -p\mathrm{d}V 
\end{align}
and then if the only form of work on a gas is volume change
$$ \mathrm{d}U = T\mathrm{d}S - p\mathrm{d}V$$
and this is the fundamental equation for a closed system. Thus only difference in internal energy are measurable from thermodynamics, and this follows from the first law. (Even if you integrate this equation from say state $a$ to $b$ the result will be $U(b)-U(a)=\ddot {}$ in other words $\Delta U$.)


Thermodynamics was developed before the nature of matter was known, i.e. it does not depends on matter being formed of atoms and molecules. However, if we use additional knowledge about the nature of molecules then the internal energy (and entropy) can be determined from statistical mechanics.


The internal energy ($U$ not $\Delta U$) of a perfect monoatomic gas is the ensemble average and is 
$$U=(3/2)NkT$$
or in general $U=(N/Z)\Sigma\_ j\exp(−\epsilon\_j/(kT))$ where $Z$ is the partition function, $k$ Boltzmann constant, $\epsilon\_j$ energy of level $j$, and $T$ temperature and $N$ Avogadro's number. The absolute value of the entropy $S$ (for a perfect monoatomic gas) can also be determined and is given by the Sakur-Tetrode equation.


 7

The statement that it is impossible to calculate the absolute energy at an ambient T seems to be intuitively incorrect, difficult possibly, but not impossible. Since U is a state function the value of U for a given set of conditions is invariable and can be set as an arbitrary zero and changes in U calculated. So all one must do is work in reverse to absolute zero. This is done for entropies using the 3rd Law. The general approach in thermodynamics is that it is entirely accurate to work with the changes in state functions and there is NO NEED to calculate absolute internal energies from absolute zero.


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