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Second law of thermodynamics, entropy

Problem 2.154

NN atoms of gaseous helium are enclosed in a cubic vessel of volume 1.0 cm31.0 \mathrm{~cm}^{3} at room temperature. Find: (a) the probability of atoms gathering in one half of the vessel; (b) the approximate numerical value of NN ensuring the occurrence of this event within the time interval t≈1010t \approx 10^{10} years (the age of the Universe).

Reveal Answer
2.154. (a) P=1/2NP=1 / 2^{N} (b) N=log⁡(t/τ)log⁡2≈80,N=\frac{\log (t / \tau)}{\log 2} \approx 80, where τ≈10−5 s\tau \approx 10^{-5} \mathrm{~s} is the mean time which takes a helium atom to cover distances of the order of the vessel's dimensions.
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