[Illustration: FIG. 76.]
§ 61. _External Work during the Expansion of Air._--If air expands
without doing any external work, its temperature remains constant.
This result was first experimentally demonstrated by J. P. Joule. It
leads to the conclusion that, however air changes its state, the
internal work done is proportional to the change of temperature. When,
in expanding, air does work against an external resistance, either
heat must be supplied or the temperature falls.
To fix the conditions, suppose 1 lb. of air confined behind a piston
of 1 sq. ft. area (fig. 76). Let the initial pressure be p1 and the
volume of the air v1, and suppose this to expand to the pressure p2
and volume v2. If p and v are the corresponding pressure and volume at
any intermediate point in the expansion, the work done on the piston
during the expansion from v to v + dv is pdv, and the whole work
during the expansion from v1 to v2, represented by the area abcd, is
_
/v2
| p dv.
_/v1
Amongst possible cases two may be selected.
_Case 1._--So much heat is supplied to the air during expansion that
the temperature remains constant. Hyperbolic expansion.