# Is it possible to determine the absolute value of entropy of a system?

I came across a question (in some assignments I had received) where it was asked:

Can the absolute value of entropy of a system can be measured?

and to my surprise, the answer said yes. Till now, I knew that entropy is a state function, just like enthalpy and internal energy, and that only change in entropy can be measured.

Am I missing something here?

## 2 Answers

Well, in practical I would say it is not possible. But there is a way to theoretically approximatly calculate an absolute value for entropy.

According to the third law of thermodynamics the entropy change can be calculated as

$$\mathrm d S = \frac{\delta q}{T}\, .$$

Per definition, a perfect crystal at $\pu{0 K}$ would have an entropy $S=0$.

So when you now heat that crystal and measure the change in entropy, since your initial $S$ was zero, the change is equal to the absolute value.

One problem is though, that the above equation only holds for constant temperatures. So the steps at which you measure the entropy changes have to happen at very (infinitesimal) small temperature changes.

• "very (infinitesimal) small temperature changes." that means our crystal would always stay at almost $\pu{0K}$, won't it? – Gaurang Tandon Feb 18 '18 at 13:52
• @GaurangTandon depends on how many measurements you would make :) – Fl.pf. Feb 18 '18 at 14:05
• That's reasonable :P – Gaurang Tandon Feb 18 '18 at 14:08
• Still, I wouldn't call it absolute by any means. It's just that it can't be reduced by cooling. If someone took into account freedom of movement of all electrons, I have a feeling that neutronium at o K would have much lower entropy then ordinary crystal. – Mithoron Feb 19 '18 at 0:54

Use the third law, 'the entropy of a perfect crystal is zero at the absolute zero of temperature' and then measure the heat capacity $C_P$ vs temperature and use $\displaystyle S_T-S_0=\int_0^T \frac{C_P}{T}dT$. Many textbooks show plots of $C_P$ vs T. Additionally the entropy can be obtained ( for molecules in the vapour phase) by spectroscopic means for the internal energy and using the Sakur-Tetrode equation for the translational energy. Partition functions are calculated accurately from spectroscopic data for the vibrational and rotational energy. If $Z$ is the partition function $\displaystyle S=R\ln(Z)+RT\left(\frac{\partial \ln(Z)}{\partial T} \right)_V$ where the partition function is $Z=\displaystyle \sum_{levels}e^{-E_i/kBT}$ where $E_i$ is the energy of the $i^{th}$ electronic, vibrational or rotational energy level.