morkuzhambu

About the Blog! • What did you assume you were going to post in this blog? Some interesting aspects of science and related trivia. • whew! Yet another lousy blog! Why waste a blog? Its a selfish blog to try and communicate effectively about the very few stuffs I understand • Whatever! Now you will tell me who all are behind this!! Name all of them who poisoned your mind towards starting this! I would like to thank Bharath Srinivasan of JNCASR first of all for he was my first inspiration and even now continues to inspire me. And Dr. Sudhir Jain!!!! • Alright!! Thats it!! Show is over. Let me see what you are upto!!!

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And so you can describe many non-trivial functions as a polynomial. And you can easily derive polynomials, you can integrate them and so forth. Polynomials are really nice! I can't help imagining smiling polynomials! -Prof. Gilbert Strang

Photo courtesy: www.mit.edu/.

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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.

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In science, there is only Physics; all the rest is Stamp collecting. - Lord Kelvin

Photo courtesy, Wikipedia.

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Exact Differential (What exactly is it?)

2  Disclaimer: This post assumes that the reader comprehends the basics of Calculus.

No one bothered to comment on your previous post! Don't you think that its a good sign to stop blogging?

In integral calculus, the common representation of an integral is 

                                           This usual notation is just a special case of the more general form,                                                                  

where c represents the curve of integration. If the curve is 1 dimensional, then its called a line integral, surface integral in case of a 2 dimensional curve and volume integral for a 3 dimensional curve.

A mathematical differential is said to be exact, as contrasted with an inexact differential, if it is of the form dQ, for some differentiable function Q. The form A(x, y, z) dx + B(x, y, z) dy + C(x, y, z) dz is called a differential form. A differential form is exact on a domain D in space if A dx + B dy + C dz = df  for some scalar function  f throughout D. This is equivalent to saying that the field is conservative.

Thanks for puking it directly from Wiki. But with due respect to the author, we don't understand a bit!

ha..!!

Let me explain. The integration that we are familiar with is actually a line integral along X-axis between limits a and b on X-axis. In a line integral, we integrate along a line. Meaning to say, we trace the area enclosed by the function f on the curve c.

An exact differential is a function which gives you the same numerical value when we integrate it along any path between the initial (a) and final limits (b).In the figure below we integrate along X-axis and X=Y line.

Think the skies are clearer!!

To understand what the above statement means, we need to understand 'integrating along a curve'. Let us explain it by a simple example.  can be evaluated along X- axis by substituting y=0.

Along X=Y, we substitute x=y and evaluate the integral.

P.S. One might wonder as to how area subtended with x=y line is higher than that with the X-Axis. It is to be noted that the integrands are different in each case and hence the areas differ.

Thus, as we are clear with the concept of 'integrating along a curve', we go on to define exact differential. A function/integrand is said to be an exact differential if the value of the defenite integral remains the same along all paths of integration between the lower and upper limit.

If the value varies with the path of integration, then it is referred to as an indefenite integral. Indefenite integrals have applications especially in mechanics. From a logical point of view, it is clear that the work done varies with the path along which it is done. Work is defined as 

                                             Graphic image for Work done along different paths.

Hence, the Force is more often an inexact differential. Since Work depends both on the state (represented mathematically by the limits of integration) and the path (represented by the c), they are exempted from a set of thermodynamic variables called 'State variables'.

For a state variable. where G(x) is the state variable.

It is interesting to note that in case of an exact differential, the area subtended is the same with any curve. It would be more enlightening to know the condition that the integrand should satisfy, in order for it to qualify as an exact differential.

To sum up,exact differentials can be integrated along any path and still the value of the defenite integral would remain the same.

Consider the total differential . Also consider the differenatial equation of the form . Comparing both, we get a condition for the functions to satisfy in order for them to be equal, i.e. . If the above condition is satisfied, then,

                                                       On integrating, .

In other words,(iff they satisfy the condition). It means that the integral gives a scalar constant and not a function and hence along whichever path we integrate, the value of the integral will not change (independent of path). Thus for a differential to be called exact, it should be a total differential of a function. 

In further posts, we will discuss on how to convert an inexact differential to an exact differential and more importantly its application on to thermodynamics (especially Claussian entropy).

whew! I need a days sleep!

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Just because something is familiar to you, it does not mean you have understood it. - Prof. L. Mahadevan, Harvard University

Photo courtesy, Harvard University.

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Venturi effect

1 Disclaimer: I know it is a trivial issue and doesn’t qualify as a ‘scientific’ enough post.  

Then Why do you blog this?!

Venturi effect states that the fluid pressure decreases when it passes through a constricted pipe. It was found by an Italian Physicist Giovanni Battista Venturi. 

Yeah! I know it! Whats so great about it?

Let us take a look at the dynamics once. The fluid passes through a pipe and then encounters a constriction. Hence, the velocity of the flow increases.  It implies that the fluid accelerates. As a result, the exit stream has a high Force (F=m*a). Hence, the pressure (F/A) has to increase.

Woila! Dumbo

But Venturi effect states that the fluid pressure decreases!

Arghh

It can be easily verified practically that the pressure infact increases when the fluid meets a constriction. When one places hand perpendicular to the flow of water through a hose pipe and constricts the mouth of it, one can feel the increase of pressure. Then, is Venturi effect wrong?

No. The common misconception that most books bring (with due due dueeee respect to all the authors) is that they don’t state clearly what they denote by the term ‘pressure’. The term is actually ‘pressure head’. Pressure head is measured by placing a small tube perpendicular to the flow of fluid. Ref Fig 2. The pressure is calculated from the height of fluid in the tube using P = h dg (h=height, d=density, g=acceleration due to gravity). Hence, Pressure head measures the pressure exerted by the fluid on the walls of its container.

Now let us relook at the Venturi effect. If the fluid meets the constriction, then the velocity increases and hence pressure increases along the direction of the flow. We have not supplied any energy to the system and hence the fluid , in order to increase the pressure along the flow, must decrease pressure perpendicular to the flow. Hence, the pressure head decreases.