A Comprehensive Study of Euler's Reciprocal Rule in Thermodynamics ...

Greetings to everyone! In chemical thermodynamics, the concept of exact differentials plays a crucial role, particularly when analyzing the behavior of ideal gases. An exact differential is a mathematical concept that relates to a function whose differential is independent of the path taken between two points in a system. In the context of chemical thermodynamics, one commonly encountered equation where the concept of exact differentials is applicable is the ideal gas equation.

The ideal gas equation is given by:

PV=nRT

where:

P is the pressure of the gas,
V is the volume of the gas,
n is the number of moles of the gas,
R is the ideal gas constant, and
T is the temperature of the gas in Kelvin.

To examine the exact differentials associated with the ideal gas equation, let's consider the differential form of the ideal gas equation:

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If Z is a function of the variable x and y,
then we can write as Z = f(x,y)
math-20240105 (4).png then Z will be termed as an exact differential entity.
Ok! So we understood how to know if an entity is an exact differential or not. If both the sides are equal, then it is an exact differential. Let's now take an actual example to verify the validity of the theory.

example 1) if Z = 8x²y², Find out whther Z is an exact differential or not?

math-20240105 (5).png
Since, both RHS and LHS are equal. Hence, Z is an exact differential.

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Great! We have learnt how exactly we can apply the rule to verify if the entity is an exact differential or not!

Now, Let's apply the rule to thermodynamics.

Let's see how this rule can be applied to the ideal gas equation to verify the exact diffeential property of the state variables or state functions i.e., P, V and T.

Problem) Show that molar volume, V is an exact differential using the ideal gas equation for 1 mole of gas.

math-20240105 (7).png
We know that R is universal gas constant, but temperature, T and pressure, P can vary.

So, we can write, molar volume V as a function of P and T.

math-20240105 (8).png
Since, both RHS and LHS are equal, hence we can conclude that molar volume, V is an exact differential.

Van der Waals Gas Equation

Now, let's move onto the Van der Waals gas equation. The Van der Waals equation is a modification of the ideal gas equation that takes into account the finite size of gas molecules and the attractive forces between them. The Van der Waals equation is given by:

math-20240105 (9).png

P is the pressure,
V is the volume,
n is the number of moles,
R is the ideal gas constant,
T is the temperature,
a is the Van der Waals constant related to the attractive forces between molecules, and
b is the Van der Waals constant related to the volume occupied by one mole of the gas molecules.

Now, let us calculate the partial differential of P with respect to T at constant volume, V.

math-20240105 (10).png

What we learnt?

Understanding exact differentials is significant because they allow for the integration of thermodynamic quantities along a specific path, making it easier to analyze and predict the behavior of the system. In the case of the ideal gas equation, the concept of exact differentials aids in the study of how pressure, volume, and temperature are related in a consistent and well-defined manner for ideal gases.
In contrast to the ideal gas equation, the Van der Waals equation introduces additional complexities due to the correction terms (a and b), making the differentials inexact. This complexity reflects the non-ideal behavior of real gases, where molecular interactions and finite molecular size have a significant impact on the gas properties.