10 litres of gas at atmospheric pressure is compressed...

10 litres of gas at atmospheric pressure is compressed isothermally to a volume of 1 litre and then allowed to expand adiabatically to 10 litres. (a) Sketch the processes on a $p V$ diagram for a monatomic gas. (b) Make a similar sketch for a diatomic gas. (c) Is a net work done on or by the system? (d) Is it greater or less for the diatomic gas?

10 litres of gas at atmospheric pressure is compressed isothermally to a volume of 1 litre and then allowed to expand adiabatically to 10 litres.
(a) Sketch the processes on a $p V$ diagram for a monatomic gas.
(b) Make a similar sketch for a diatomic gas.
(c) Is a net work done on or by the system?
(d) Is it greater or less for the diatomic gas?

Key Concepts

Isothermal Process

An isothermal process is one in which the temperature of the system remains constant throughout. In thermodynamics, when a gas undergoes an isothermal change, its pressure and volume are related by Boyle's law (pV = constant). Work done during such a process is given by the logarithmic relation between the initial and final volumes, and there is heat exchange with the surroundings to maintain constant temperature.

Adiabatic Process

An adiabatic process is one in which no heat is exchanged between the system and its surroundings. Consequently, any work done on or by the system results in a change in the internal energy and temperature of the system. For an ideal gas undergoing an adiabatic process, the pressure and volume are related by pV^? = constant, where ? (gamma) is the heat capacity ratio.

Pressure-Volume Diagram

A pV diagram is a graphical representation of the relationship between pressure and volume for a thermodynamic system. It is used to illustrate various thermodynamic processes and cycles, such as isothermal and adiabatic processes. The area under the process curve on a pV diagram corresponds to the work done during the process.

Specific Heat Ratio (?)

The specific heat ratio, denoted by ?, is the ratio of the specific heat at constant pressure (Cp) to the specific heat at constant volume (Cv). It plays a crucial role in adiabatic processes, influencing the steepness of the pV curve. Monatomic gases typically have a ? of about 5/3, while diatomic gases have a lower ? (around 7/5) due to the additional degrees of freedom from rotational modes, which affects the work and temperature change during adiabatic processes.

Net Work in Thermodynamic Processes

In a cyclic process involving both isothermal compression and adiabatic expansion, net work refers to the total work performed over the complete cycle. The sign of the net work (work done by the system versus work done on the system) depends on the nature of each individual process. The differences in specific heat ratios between monatomic and diatomic gases mean that even if the same volume limits are used, the net work output can vary, typically resulting in different work magnitudes due to the differing slopes of the adiabatic curves.

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