A sample of gas expands from an initial pressure and volume of 10 Pa and 1.0 m^3 to a final vo

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A sample of gas expands from an initial pressure and volume...

A sample of gas expands from an initial pressure and volume of $10 \mathrm{~Pa}$ and $1.0 \mathrm{~m}^{3}$ to a final volume of $2.0 \mathrm{~m}^{3} .$ During the expansion, the pressure and volume are related by the equation $p=a V^{2},$ where $a=10 \mathrm{~N} / \mathrm{m}^{8} .$ Determine the work done by the gas during this expansion.

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Key Concepts

Thermodynamic Work

Thermodynamic work done by a gas during expansion is calculated as the integral of pressure with respect to volume. This represents the energy transferred by the system doing work on the surroundings and is graphically interpreted as the area under the process curve on a pressure-volume (P-V) diagram.

Definite Integration

Definite integration is the mathematical process used to sum contributions of infinitesimal work over a change in volume. When pressure is a function of volume, the work is evaluated by integrating the given pressure function between the initial and final volume limits.

Pressure-Volume Relationship

The pressure-volume relationship defines how pressure varies with volume during a thermodynamic process. Understanding the specific functional dependency, such as a power law relationship, is critical for setting up the correct integral to compute the work done by the gas.

Variable Pressure Process

In a variable pressure process, unlike an isobaric process where pressure remains constant, the pressure changes with volume. This necessitates using integral calculus to determine the net work done as the gas expands or compresses, reflecting the continuous variation of pressure during the process.

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