To be more specific, what I mean is to measure it in a experiment. And if the answer is no, I want to know if it is principally impossible, or just impossible due to the technic limitation of our time.

I am sorry I wan't clear enough at first. For most of the times, in the progress of experiment, the system does not have enough time to get an equilibrium state. unless you are aiming to do a thermal dynamics experiment. (in which the entropy of the system could be measured as $$\Delta S = \int_{T_1}^{T_2} \frac{C(T)}{T} \, \mathrm{d}T$$) What I want to ask is, could we develop a normal way to measure the entropy in all kinds of experiments?

Thank you for your time:-P

  • $\begingroup$ Entropy is often only proportion of order in some system. If you measure the difference between the average temperature of a cup of coffee and the highest (and lowest) temperature of a cup of coffee, you measure entropy. $\endgroup$ – foggy Apr 27 '14 at 12:22

In theory, we can compute the entropy of a system, which is given by,

$$S=k_B \frac{\partial}{\partial T}(T\log Z)$$

where $Z$ is the partition function of the system, computed by an integral over the phase space,

$$Z=\frac{1}{(2\pi \hbar)^d}\int \mathrm{d}^dq \, \mathrm{d}^d p \, e^{-\beta H(p,q)}$$

where $H(q,p)$ is the Hamiltonian, and $\beta=1/k_BT$. (The formula for entropy from the partition function not only holds in statistical physics, but may also be used in other fields, such as general relativity.) In principle, one could then compute a change in entropy,

$$\Delta S = S'-S$$

providing one knows the change in the parameters which the entropy depends on. For example, for an ideal gas of $N$ indistinguishable particles,

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

Assuming the particle number is constant, as well as the volume of the system and mass of the constituent particles, we would only need to know the change in temperature to determine the change in entropy, as the thermal de Broglie wavelength $\lambda \propto T^{-1}$. (Note: technically the above formula is not exact, as the derivation employs Stirling's approximation of the gamma function.)

Alternatively, providing we know the heat capacity as a function of temperature, i.e. $C(T)$, we may compute a change in entropy due to a change in temperature, as $$\Delta S = \int_{T_1}^{T_2} \frac{C(T)}{T} \, \mathrm{d}T$$

For certain systems, we may assume $C(T)=c_0$, where $c_0$ is constant, in which case the change in entropy is simply $\Delta S =c_0 \log(T_2/T_1)$.

Caveat: the partition function depends on which ensemble we select, e.g. the grand canonical ensemble, micro-canonical ensemble, etc.

  • $\begingroup$ There is a "now" that should be a "know" in the first line of the second part where you're addressing the specific heat. $\endgroup$ – Ignacio Vergara Kausel Apr 27 '14 at 12:58
  • $\begingroup$ @IgnacioVergaraKausel: Thanks for catching the typo. $\endgroup$ – JamalS Apr 27 '14 at 13:00

Entropy is an extensive property, like mass and volume. So to directly and experimentally measure the change of entropy of "all kinds" of systems, one would have to measure initial and final values of some extensive property that is related to entropy. The only observable extensive property that works is the number of possible states of the system. Which means you would have to be able to enumerate every possible state. For any reasonable system, that number is too large to count.


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