Work Done by an Expanding Gas
To derive the expression for the work done by an expanding gas,
to understand how it follows from the expression
for mechanical work.
Especially from the historically important perspective of making engines to convert heat energy into
work, the work in thermodynamics is defined as the work done
the exterior world,
and not vice versa as is done in the rest of classical mechanics. In classical mechanics, one always
considers the work done
the outside world. Rarely does one think about the work done
by the system. Suppose you push a large block with a certain force of magnitude
over some distance.
You have done work on the block; hence the energy of the block should increase. According to
Newton's 3rd law, the block exerts the same magnitude of force
, but in the opposite direction (i.e.,
directed back at you). Hence, the work done by the block (on you) is negative, since the direction of
motion opposes the direction of the force. In summary, you have to be careful about the sign of the
work: the same situation gives opposite signs of the work depending on whether our perspective is
classical mechanics or thermodynamics.
In thermodynamics, one often deals with liquids and gases that exert forces on their containers (i.e., the
fluids exert pressure over an area). If the container changes volume, then this force acts through a
distance and hence does work.
For a steam engine, the example pictured
here, the "container" is a cylinder whose
volume changes as the piston slides in or
out. Suppose a gas is confined within the
cylinder. The pressure of the gas is
the area of the cylinder is
. Consider the
work done as the gas expands, pushing the
piston to the right. Call the infinitesimal
distance the piston moves
does the gas exert
on the piston? (Note that the
axis is to the right in the
Express the force in terms of
, and any constants,
If the piston moves a distance
, what is
, the work done
Express the work done by the gas in terms of given quantities.
, the increase in volume of the gas?
Express the differential increase in terms of
and other given quantities.
Now find the work done by the gas in terms of the thermodynamic variables.