#exact differential

[engineeringmathgeek]

How to integrate exact differentials. Here I show how to integrate exact differentials using several examples that you'll find easy to follow.

[morkuzhambu-blog]

Inexact to exact..

3 Uf!! Are you alive? With your previous post posted ages ago, I thought you were long gone for good!!

Sorry about that. But I think its time for me to trouble all of you with my babbles.

Am happy that you admit that those are babbles!

Anyways, this post will talk about inexact differentials and integrating factors. Eventually we ll get to see what entropy is (not what it means!). A small correction. We will see what Clausian entropy is.

I had sent your previous post for carbon dating. It says that the post was 9 months ago. So give me some time to look at the post and come back to this one.

Take all your time. But just to summarize, exact differential is one that is a complete/ total differential of a function. The others fall under the category of inexact differentials.

Inexact differentials can be converted to exact differentials if you can find an integrating factor. (Does it ring a bell on some Ordinary Differential Equation Solving technique that you had studied way back in your high school?. Its the same. Even there we used integrating factor to convert the inexact differential to an exact one.)

As the name suggests its a factor. Meaning it should be multiplied with the inexact differential.

If you want to know more on integrating factors, please refer to any standard Calculus text book.

Ah! Accept that you don't know!! Why do you redirect us to some other thing? People close the tab! This blog is worthless!!!

Now let us see an example of how integrating factor can convert an inexact differential to an exact one. We know that Energy is an inexact differential. From the first law of thermodynamics,

Let us consider for the moment that we are dealing with an ideal gas. 

The internal energy can be rewritten as,

Hence,

Divide both sides by T in above equation and substituting ideal gas equation.

Hence, is an  exact differential. We know that  . Hence entropy is an exact differential. Hence, is the integrating factor.

Now that we appreciate the math behind it, it is time for us to understand the physical meaning of entropy.

Next post will give you a complete idea of how Clausius defined Entropy and also on the generic form for entropy devised by Boltzmann.