A sample consists of an amount n in moles of a monatomic...

A sample consists of an amount $n$ in moles of a monatomic ideal gas. The gas expands adiabatically, with work $W$ done on it. (Work $W$ is a negative number.) The initial temperature and pressure of the gas are $T_{i}$ and $P_{r}$ Calculate (a) the final temperature and (b) the final pressure.

Key Concepts

Specific Heat Capacities and Adiabatic Index

For a monatomic ideal gas, the specific heat at constant volume (Cv) and the specific heat at constant pressure (Cp) have particular values, and their ratio (? = Cp/Cv) is known as the adiabatic index. This index determines the relationship between pressure, volume, and temperature in an adiabatic process, typically expressed by P V^? = constant and T V^(?-1) = constant. These relations are fundamental in calculating final state variables after an adiabatic change.

Ideal Gas Law

The ideal gas law, expressed as PV = nRT, relates the pressure, volume, and temperature of an ideal gas, where n is the number of moles and R is the universal gas constant. This law provides a fundamental relationship that can be used in conjunction with other thermodynamic principles to determine the state of a gas under various conditions, including adiabatic processes.

Adiabatic Process

An adiabatic process is one in which no heat is transferred into or out of the system. All changes in the internal energy of the system are solely due to work done on or by it. This concept is crucial when analyzing processes where the insulation or rapidity of the process prevents significant heat exchange, leading to specific relationships between pressure, volume, and temperature.

First Law of Thermodynamics

The first law of thermodynamics states that the change in internal energy of a system is equal to the heat added to the system plus the work done on the system. In the case of an adiabatic process, since no heat is exchanged, the first law simplifies to a balance between changes in internal energy and work, making it a key principle for calculating changes in a system's state during adiabatic conditions.

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