Thermodynamics The Ideal Gas Law P V=n r T is used to...

Thermodynamics The Ideal Gas Law $P V=n r T$ is used to describe the relationship between pressure $P,$ volume $V,$ and temperature $T$ of a confined gas, where $n$ is the number of moles of the gas and $r$ is the universal gas constant. Show that $$ \frac{\partial V}{\partial T} \cdot \frac{\partial T}{\partial P} \cdot \frac{\partial P}{\partial V}=-1 $$

Thermodynamics The Ideal Gas Law $P V=n r T$ is used to describe the relationship between pressure $P,$ volume $V,$ and temperature $T$ of a confined gas, where $n$ is the number of moles of the gas and $r$ is the universal gas constant. Show that
$$
\frac{\partial V}{\partial T} \cdot \frac{\partial T}{\partial P} \cdot \frac{\partial P}{\partial V}=-1
$$

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Key Concepts

Cyclic (Triple Product) Rule

The cyclic rule for partial derivatives states that when three variables are interrelated, the product of the three partial derivatives—each taken while holding the third variable constant—equals -1. This relationship emerges from the implicit function theorem and reflects the inherent consistency in the way the variables depend on each other in systems described by an equation of state.

Partial Derivatives

Partial derivatives describe the rate at which a function changes with respect to one variable while keeping the other variables constant. This concept is essential in multivariable calculus and is widely used in thermodynamics to study how state variables like pressure, volume, and temperature change individually in response to small perturbations.

Chain Rule for Partial Derivatives

The chain rule in the context of partial derivatives is a method that connects the rates of change between interdependent variables. It allows one to express the derivative of one variable with respect to another via an intermediate variable, thereby linking different paths of differentiation. This technique is crucial when analyzing systems where variables are functionally dependent, such as thermodynamic systems.

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