86heated wall, and cooler atother end(seeProblem4.5). Expanding the
temperatureTalong the length of the tanker in aTaylor series and keeping
the first two terms(since thetemperature differencebetween the walls is
small compared to we have
We may write theideal gas law as afunction ofposition in the tanker:
where is the gas concentration.Rearranging, wehave
The total numberNof molecules in the cylinder isgiven by
whereA is the cross-sectional area of thetanker. Alternatively, we can
integrate (S.4.4.3)exactly andexpand the resultinglogarithm,whichyields
the sameresult. The total number ofmoleculesoriginally in thetank is
Since thetotalnumber ofmolecules in the gasbefore and after heating is
the same, (no phasetransitions), wemay equate(S.4.4.4) and
(S.4.4.5), yielding
SOLUTIONS