82 ProMema d Solutions on Thermdynamics d Statiatieal Mechanic8
1084
Consider a gas which undergoes an adiabatic expansion (throttling
process) from a region of constant pressure p; and initial volume Vi to a
region with constant pressure pf and final volume Vf (initial volume 0).
Fig. 1.27.(a) By considering the work done by the gas in the process, show that
the initial and final enthalpies of the gas are equal.
(b) What can be said about the intermediate states of the system?(c) Show for small pressure differences Ap = pf - pi that the temper-
ature difference between the two regions is given by AT = -(Ta - l)Ap,V
CV
where a = - ( av) and cp = (g)
v dTp P
(d) Using the above result, discuss the possibility of using the process
to cool either an ideal gas, or a more realistic gas for which p = RT/(V- b).
Explain your result.
(SUNY, Buffalo)Solution:
which is equal to a reduction of the internal energy:(a) The work done by the gas in the throttling process is pfVf - pix,
ui - Uf = pfvf - piVi.
Thus Ui + pi& = Uf + pfVf, i.e., Hi = Hf.
(b) Because the process is quasi-static, the final and initial states can
be any two intermediate states. Thus the conclusion is still valid for inter-
mediate states.(c) From dH = TdS + Vdp = 0 and
dS= (g)pdT+($)Tdp=$dT-(g) C dp,
P