Problem NO. 1 : Law of the Adiabatic Process
Reading R. Resnick – D. Haliday thermodynamics, I’ve found something well known me although it’s making me wonder every single time when I think about it. It’s the derivation of the thermodynamics law that goes like this:
From first law of thermodynamics , but in the adiabatic process we have and instead of we put
And this is the moment that I do not understand! Why, to hell do we put instead of if both of the variables DO CHANGE in time?! Even if it’s in diffrential, it should be that second one.
If someone knows the reasons why it goes like that, please comment this post and explain it to me.
Futher we get because , and from it
On the other hand from the perfect gas equatation brought to the diffrental form we get After comparing those two expressions for , remembering that and and some simple algebraic transformation, we recieve
Integrating gives us finnaly , hence
By the way I’ve noticed that there exist a familiar strange derivation in mathematics, for derivative from product of two functions, maybe because it’s just a simplified derivation of something more complicated.
Now the trick we’ll use to evaluate this limit is to add and subtract to the numerator here. That is, in effect we’re adding zero and leaving it alone, but the formula will be easier to work with. In particular, we can start splitting it up using the laws of limits.
Of these four limits, the fourth is the limit of a continuous function because doesn’t depend on . The second and third are just the definitions of and . The first limit goes to because, as we showed, all differentiable functions are continuous. And so we have the rule
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Tags: adiabatic, physic, process, thermodynamic