1Note 4. Entropy II: Molecular basis of Entropy 4.1 The molecular basis of Entropy We have discussed some mathematical aspects of entropy, but without a microscopic interpretation, entropy is often a vague and abstract concept. I will now give a microscopic understanding of entropy. This will also (hopefully) help to shed light on the nature of heat as well. Let’s start by defining a new quantity . We define as the ln (natural log)of how many different arrangement a system could have. For example, consider flipping a coin. When it lands, it can be either heads or tails. Thus, there are 2 different arrangements and 2ln1coinWhat if we had 2 coins? How many arrangements would be possible then? Each coin would have 2 possibilities, so there would be 22ways (both heads; 1st head, 2nd tails; 1st tails, 2nd heads; both tails). In general, if we have Ncoins, then there are 2Ndifferent rearrangements. This means that 2ln2lnNNcoinNOn thing we immediately notice is that coincoinNN1Thus, is extensive! Next, let’s consider something closer to the thermodynamic systems we have been discussing. How many ways are there to rearrange the Nparticles of an ideal gas in a volume V? Let’s say that each particle has a volume b. For simplicity, let’s imagine that the box is cubical and that are gas particles are little cubes as well. Then, we ask, how many ways can we arrange a cube of volume b in a cubical box of volume V ? One way to answer this question is to fill the large box with particles. The box of volume Vcan hold V/bparticles. What this means is that there are V/bdifferent places we can put a single particle. Thus, the for a single gas particle is )ln(1bVparticleNow, what if we had Nideal gas particles? Each particle could be in any of the V/blocations (note that since these particles are ideal, they can be in the same place possibly. Remember that molecules in ideal gas do not see each other!). Thus, there are NbVdifferent ways to arrange these particles. This leads to
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