(a) The gas law for a fixed mass m of an ideal gas at...

(a) The gas law for a fixed mass $m$ of an ideal gas at absolute temperature $T,$ pressure $P,$ and volume $V$ is $P V=m R T,$ where $R$ is the gas constant. Show that $$\frac{\partial P}{\partial V} \frac{\partial V}{\partial T} \frac{\partial T}{\partial P}=-1$$ (b) For the ideal gas of part (a), show that $$T \frac{\partial P}{\partial T} \frac{\partial V}{\partial T}=m R$$

(a) The gas law for a fixed mass $m$ of an ideal gas at absolute temperature $T,$ pressure $P,$ and volume $V$ is $P V=m R T,$ where $R$ is the gas constant. Show that
$$\frac{\partial P}{\partial V} \frac{\partial V}{\partial T} \frac{\partial T}{\partial P}=-1$$
(b) For the ideal gas of part (a), show that
$$T \frac{\partial P}{\partial T} \frac{\partial V}{\partial T}=m R$$

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Key Concepts

Ideal Gas Law

The ideal gas law is a fundamental relation in thermodynamics that connects the pressure, volume, and temperature of a given mass of an ideal gas with a gas constant. It encapsulates how the macroscopic properties of gases are interrelated and is frequently used as a starting point in deriving various thermodynamic identities and differentiations between state variables.

Partial Differentiation

Partial differentiation is a mathematical tool used to investigate how a function changes as one variable changes while holding the others constant. In the context of thermodynamics, it allows one to explore how a state variable such as pressure, volume, or temperature is influenced by changes in another, under constraints such as a fixed mass or in a particular thermodynamic process.

Chain Rule and Cyclic Relations in Partial Derivatives

The chain rule for partial derivatives is a method of relating the rate of change of interconnected variables. In thermodynamics, cyclic relations, which are consequences of the chain rule, enable one to link partial derivatives of different variables (such as pressure, volume, and temperature) in a cyclic manner. This relation, often represented as the product of three partial derivatives equaling -1 in systems with three variables, is pivotal in understanding interdependencies in state functions.

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