## Granular EntropyMarch 14, 2016 For my dissertation, many decades ago, I measured the high temperature reaction of metals forming alloys. To do this, I teamed with a graduate student colleague to construct a high temperature calorimeter. Fortunately, he was good at doing mechanical things, like wrestling with vacuum systems, while I had a talent for electronics, so the system performed its intended task. Eventually, I was able to build a second system to increase throughput (see figure).
ΔG of the transformation is zero,
ΔH is the enthalpy of vaporization, and ΔS is the vaporization entropy. We can see that entropy is just as important as enthalpy, at least in this example of boiling, since the terms are equal.
Entropy is a strange property of matter that arises from the principle that objects can be arranged in different ways. Students of physics and chemistry are familiar with Boltzmann's entropy formula,
S is the entropy, K is the Boltzmann constant (1.38062 x 10_{B}^{-23} joule/kelvin), and Ω is the number of states accessible to a system.
p; viz.,
_{i}^{250}. This number greatly exceeds the total number of atoms in the universe, which is estimated to be 10^{80}. The packing is reminiscent of the random packing of oblate spheroids I discussed in an earlier article (Packing, November 30, 2010).
"Because our indirect approach relies on the observation of a small sample of all possible configurations, the answers it finds are only ever approximate, but the estimate is a very good one... By answering the problem we are opening up uncharted territory. This methodology could be used anywhere that people are trying to work out how many possible solutions to a problem you can find."[2] ## References:- Stefano Martiniani, K. Julian Schrenk, Jacob D. Stevenson, David J. Wales, and Daan Frenkel, "Turning intractable counting into sampling: Computing the configurational entropy of three-dimensional jammed packings," Phys. Rev. E, vol. 93, no. 1 (January, 2016), Document no. 012906, DOI:http://dx.doi.org/10.1103/PhysRevE.93.012906.
- How many ways can you arrange 128 tennis balls? Researchers solve an apparently impossible problem, St. John's College, Cambridge, Press Release, January 27, 2016.
- Ning Xu, Daan Frenkel, and Andrea J. Liu, "Direct Determination of the Size of Basins of Attraction of Jammed Solids," Phys. Rev. Lett., vol. 106, no. 24 (June 17, 2011), Document no. 245502.
- Daniel Asenjo, Fabien Paillusson, and Daan Frenkel, "Numerical Calculation of Granular Entropy," Phys. Rev. Lett., vol. 112, no. 9 (March 7, 2014), Document no. 098002.
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