Note 4 - Note 4 Entropy II Molecular basis of Entropy 4.1...

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1 Note 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 2 ln 1 coin What if we had 2 coins? How many arrangements would be possible then? Each coin would have 2 possibilities, so there would be 2 2 ways (both heads; 1st head, 2nd tails; 1st tails, 2nd heads; both tails). In general, if we have N coins, then there are 2 N different rearrangements. This means that 2 ln 2 ln N N coin N On thing we immediately notice is that coin coin N N 1 Thus, is extensive! Next, let’s consider something closer to the thermodynamic systems we have been discussing. How many ways are there to rearrange the N particles 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 V can hold V / b particles. What this means is that there are V / b different places we can put a single particle. Thus, the for a single gas particle is ) ln( 1 b V particle Now, what if we had N ideal gas particles? Each particle could be in any of the V / b locations (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 N b V different ways to arrange these particles. This leads to
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