   Chapter 5.4, Problem 29E

Chapter
Section
Textbook Problem

# When gas expands in a cylinder with radius r, the pressure at any given time is a function of the volume: P = P ( V ) . The force exerted by the gas on the piston (see the figure) is the product of the pressure and the area: F = π r 2 P . Show that the work done by the gas when the volume expands from volume V 1 to volume V 2 is W = ∫ V 1 V 2 P   d V To determine

To show:

The work done by the gas when the volume expands from volume V1 to volume V2 is

W=V1V2P dV

Explanation

1) Concept:

The work is calculated by using the formula W=abf(x)dx

2) Given:

The pressure is the function of volume P=P(V) and F=πr2P

3) Calculation:

Since the pressure P=P(V) is a function of volume, and the volume V=πr2x is a function of x, therefore, the pressure can be written as a function of x

Therefore, if V1=πr2x1 and V2=πr2x2 then the work done by the gas when the volume expands from volume V1 to volume V2 is

W=x1x2F(x) dx

Given F=πr2P

W=x1x2πr2P(V(x)) dx

Substitute (x)=πr2x, therefore,dV<

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