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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
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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.
