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Delve into entropy, the First Law, and the Euler Criterion in physics, with examples and explanations, to grasp these concepts deeply. Explore ideal gas equations and how Claussius made pivotal contributions.
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Entropy – first look Let’s write the First Law in this form: And let’s examine the Euler Criterion: -- obviously, not !!! Why “obviously”? The above means that Can that be true? Let’s use the ordinary chain rule:
In order to prove that something is not true in math, it’s enough to find JUST ONE contradicting example. Take the ideal gas equation: pV = kNT And CV is not zero, either – we can finally conclude that cannot be an exact differential. What did Claussius do that secured him a solid site in the history of physics? Well, he wanted to “reconstruct” in such a way that would make it exact. Let’s again take
and divide the whole equation by T: Let’s try the Euler Criterion now: For ideal gas U=(3/2)kNT so the latter derivative and thus the whole thing is zero.
Now, let’s check the other derivative: Now, we will do some juggling with the ideal gas equation:
We use the gas equation one more time to get: And when we apply this result to the expression on the preceding page, we obtain: So, we have shown that IS AN EXACT DIFFERENTIAL!!! We call S the entropy.
