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 is the product of the pressure and the area: F=pi(r^2)P. Show that the work done by the gas when the volume expands from volume V1 to volume V2 is W= integral (V1 to V2) PdV. Medal to anyone who can help me show that this integral is correct!

Question

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 is the product of the pressure and the area: F=pi(r^2)P.  Show that the work done by the gas when the volume expands from volume V1 to volume V2 is W= integral (V1 to V2) PdV.  Medal to  anyone who can help me show that this integral is correct!

anonymous · Answer

The definition of work is 
$$ W = \int \vec{F} \cdot d\vec{r} $$
or, in your case, since everything is in one line,
$$ W = \int F dz $$
Where F is the pressure times the area and dz is the change in height of the piston.  You can rewrite this in terms of pressure and volume.

anonymous · Answer

Because pressure is the force being exerted right?

anonymous · Answer

No, but it's related to the force... re-read what you wrote in the beginning.

anonymous · Answer

OH I totally understand now!! Thank you so much!!!