Exploring Total Differentials and Cyclic Rules as Mathematical Maes...

Greetings to everyone! In my previous post, we have learnt about Euler's reciprocal rule and how to use the rule to find exact differential entities. Today, we shall move one step further and explore the possibilities of another differential term called as total differential. Since we have well understood how to apply exact differential to ideal gas equation and Van der waals equation, understanding total differential will not be a headache. Trust me! We will also learn to apply cyclic rule in thermodynamics.

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Total differential

The concept of the total differential is a fundamental mathematical tool that plays a crucial role in understanding how properties of a system change as it undergoes various processes. The total differential provides a precise and comprehensive way to analyze these changes, offering valuable insights into the behavior of thermodynamic variables and the relationships between them.

Now, let us consider an entity z and suppose z is a function of the variables x and y.

Here, dz is the total differential of z and δz/dx and δz/dy are the partial derivative of z with respect to x and y respectively.

Given z is an exact differential, then it will follow Euler's reciprocal rule.

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The total differential allows for a precise representation of how a function changes as each independent variable undergoes an infinitesimally small change. This concept is crucial for analyzing thermodynamic processes, where the values of state variables evolve continuously.

Cyclic rule

The cyclic rule, a fascinating concept in advanced calculus and mathematical physics, unveils a unique relationship among three partial derivatives. Specifically, it dictates that the product of three partial derivatives, taken in cyclic order, results in an intriguing mathematical identity: the cyclic product is equal to -1.

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Here, we can see x, y and z terms are in a cyclic order and then only we can apply the rule. This rule finds great use and applications in thermodynamics.

Learning Concepts through Problems

We have discussed exact differential, total differential and cyclic rule. Now, let us learn to apply this concepts to problems. This part is the most important because the only way to retain the concepts is to solve questions based on the theory you learnt and understand the application part through practice.

Problem) The exact differential df of a state function f(x,y) among the following is -
(a) xdy(b)dx - x/y dy(c) ydx - xdy (d)1/y dx - x/y² dy

Solution:
To find out which among the above is an exact differential for df, we need to apply the Euler's rule to all the optional values and see if it satisfies or not.The term that satisfies the Euler's rule will be the answer that we are looking for!

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Problem) From one mole of ideal gas,
(δP/δT)_V(δV/δT)_P(δV/δP)_T = ?
(a) -1(b) -R²/P²(c)+1(d)R²/P²

Solution:

I'm pretty sure, most people will apply cyclic rule to the above question and end up getting the answer as -1 i.e., option (a). Right?

Well NO sir! That's not the correct answer! Because cyclic rule will not be applied here. Why so? cause P,T and V terms are not in a cyclic form; which can be seen that there are two δV term in numerator. Hence we need to calculate each values separately.

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Problem)The internal pressure Π_T = T(δP/δT)_V - P for one mole of Van der Waals gas is?
(a)a/V²(b)a/V²(RT/V-b) (c)zero (d) RT/V-b

Solution:

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What we learnt?

The total differential is a powerful mathematical tool in the field of thermodynamics, providing a systematic and precise way to describe changes in state functions as a system undergoes various processes. Its applications extend to the analysis of thermodynamic cycles, the derivation of important relations.
The cyclic rule, with its simple yet profound mathematical result, offers a valuable tool for simplifying expressions involving three partial derivatives.