I learned about entropy in chemistry, I saw that we can measure $\Delta H$, but can't directly measure the $H$. So I searched about 'why we can't directly measure the '$H$'?', but I couldn't find the exact reason.

I found that the entropy is related to the internal energy. So I searched about internal energy, and I could find a simple reason for 'why we can't'. We can't directly measure the H or internal energy because when we try to measure the internal energy, the internal energy changes. When I found this, I thought that it may be related to the 'uncertainty principle'.

What I am curious about is the exact explanation about 'why can't we directly measure internal energy or entropy'. And is it related to the uncertainty principle?

  • $\begingroup$ Thank you to people that edit my question:) $\endgroup$ – 정우남 Apr 28 at 9:05

In general it is difficult to quantify the absolute values of diverse thermodynamical potentials of matter due to their complicated structure, however, for simple-structured matter it is possible. The best example for this is the classical (not quantum!) ideal gas where the internal energy $U$ and enthalpy $H$ are given by

$$ U = c_V N k_B T \quad \text{and} \quad H = c_p N k_B T$$

where $c_V$ and $c_p$ are the dimensionless specific heat capacity at constant volume respectively at constant pressure (BTW $c_p = c_V + 1$ for an ideal gas). Furthermore $N$ is the number of particles $k_B$ the Boltzmann constant. The formula for the total entropy is a bit more complicated and only for an one-atomic ideal gas:

$$ S = Nk_B [ \ln (\frac{V}{N\lambda^3}) + \frac{5}{2}] $$

where $\lambda = \frac{h}{\sqrt{2\pi m k_B T}}$ (Here indeed comes a quantum-mechanical constant in. This related with the statistical meaning of entropy being proportional with the logarithm of the number of micro-states possible for a single macroscopic state. Upon counting the micro-states quantum mechanics comes in).
The question on the uncertainty of thermodynamical potentials is not related with Heisenberg's uncertainty principle. It is related with thermodynamical fluctuations. As any kind of matter an ideal gas is in permanent energy exchange with the outer world. So the formulas given above are actually mean values as the actual values fluctuate. However, due to the large number of particles contained in a macroscopic amount of matter, the relative values of these fluctuations with respect to the corresponding total macroscopic values are very small. So if a limited number of gas particles (for instance those close to the wall of the container of the ideal gas) exchange their energy with the outer world -- positive or negative -- does not substantially change the total amount of internal energy, so the mean value (taken over a time scale typical for macroscopic physics) remains constant. Last note: All given formulas are to be understood for equilibrium thermodynamical states.

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  • $\begingroup$ Thank you for your wonderful answer:) $\endgroup$ – 정우남 Apr 28 at 10:48
  • $\begingroup$ 환영합니다/you are welcome $\endgroup$ – Frederic Thomas Apr 28 at 13:32
  • $\begingroup$ Oh!!!Thank you so much:) I am really really happy to your Korean answer:) You are so smart and nice person! $\endgroup$ – 정우남 Apr 28 at 16:16

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