(III) Consider an ideal gas of n moles with molar specific...

(III) Consider an ideal gas of $n$ moles with molar specific heats $C_{V}$ and $C_{P}$ (a) Starting with the first law, show that when the temperature and volume of this gas are changed by a reversible process, its change in entropy is given by $$d S=n C_{V} \frac{d T}{T}+n R \frac{d V}{V}$$ (b) Show that the expression in part (a) can be written as $$d S=n C_{V} \frac{d P}{P}+n C_{P} \frac{d V}{V}$$ (c) Using the expression from part $(b),$ show that if $d S=0$ for the reversible process (that is, the process is adiabatic), then $P V^{\gamma}=$ constant, where $\gamma=C_{P} / C_{V}$

(III) Consider an ideal gas of $n$ moles with molar specific heats $C_{V}$ and $C_{P}$ (a) Starting with the first law, show that when the temperature and volume of this gas are changed by a reversible process, its change in entropy is given by
$$d S=n C_{V} \frac{d T}{T}+n R \frac{d V}{V}$$
(b) Show that the expression in part (a) can be written as
$$d S=n C_{V} \frac{d P}{P}+n C_{P} \frac{d V}{V}$$
(c) Using the expression from part $(b),$ show that if $d S=0$ for the reversible process (that is, the process is adiabatic), then $P V^{\gamma}=$ constant, where $\gamma=C_{P} / C_{V}$

Key Concepts

First Law of Thermodynamics

This principle states that energy cannot be created or destroyed in an isolated system, only transformed from one form to another. It forms the basis for deriving relations between work, heat, and internal energy in thermodynamic processes, and is essential for connecting heat transfer with changes in state variables such as temperature and volume.

Entropy

Entropy is a measure of the disorder or randomness in a system, and its change quantifies how energy disperses at a given temperature. In thermodynamics, the change in entropy helps to determine whether a process is reversible or irreversible, and is central to understanding the second law of thermodynamics.

Ideal Gas Law

The ideal gas law relates the pressure, volume, and temperature of an ideal gas through the relation PV = nRT. This relation provides a straightforward way to connect macroscopic state variables and underpins the derivations of many thermodynamic expressions for ideal gases.

Reversible Process

A reversible process is an idealized transformation that occurs infinitely slowly, ensuring that the system remains in thermal equilibrium with its surroundings at all times. This concept is crucial because it allows the use of exact differentials to integrate thermodynamic variables like entropy.

Specific Heat Capacities

Specific heat capacities (C_V and C_P) represent the amount of heat required to change the temperature of a substance per unit temperature change at constant volume and constant pressure, respectively. They are fundamental in quantifying energy changes and linking temperature variations to heat transfer in thermodynamic processes.

Adiabatic Process

An adiabatic process is one in which no heat is exchanged between the system and its surroundings. In such a process, changes in internal energy are solely due to work done by or on the system, leading to unique relationships among state variables, which are particularly important in deriving conditions like the adiabatic relation for an ideal gas.

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