# Why can't we directly measure internal energy and entropy?

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

I found that entropy is related to 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 $$H$$ 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.

So, why we can’t directly measure internal energy or entropy. And is it related to the uncertainty principle?

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

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

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 $$k_\text{B}$$ the Boltzmann constant. The formula for the total entropy is a bit more complicated and only for an one-atomic ideal gas:

$$S = Nk_\text{B} \left( \ln \left(\frac{V}{N\lambda^3}\right) + \frac{5}{2} \right)$$

where $$\lambda = \frac{h}{\sqrt{2\pi m k_\text{B} T}}$$. (Here, we indeed get a quantum-mechanical constant. This related to the statistical meaning of entropy being proportional with the logarithm of the number of micro-states possible for a single macroscopic state. Through counting the micro-states, quantum mechanics comes in.)

The question on the uncertainty of thermodynamic potentials is not related with Heisenberg’s uncertainty principle. It is related with thermodynamic 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 thermodynamic states.