240
Appendix D
AppendixD
241
9 CHANGE OF PHASE
9.1 A small amount of pure liquid is containedin a glasstube from
which all the air has beenremoved(Fig. D.8(a)).The volume of the
tubeis significantlygreaterthan thecriticalvolume Z. of the enclosed
liquid. (a) Describewhat happensas the temperatureis raised so
that the substancetraversesthe path XY on the P-V projectionin
Fig D.8(b). (b) If the volume of the tube were equal to V,, what
would you observeas the temperatureis raised?
9.2 Gasis containedin a glassbulb of volume250cm3(seeFig. D.9).
A capillary of length 10cm and of diameter 1mm is connected
to the bulb. Mercury is forced into the bulb compressingthe gas
and forcing it into the capillaryso that it occupiesa length of 1cm.
This processoccursisothermallyat 20"C.The initial pressureof the
gasis 10- 3 torr. (a)What is thefinal pressure
of thegasin the capillary
if it is nitrogen?(b)What is the final pressureof the gasin the capillary
Justify any
if it is water vapour?(c) How much water condenses?
(The vapour pressureof water at 20"C is 17.5torr.)
assumptions.
Initially
Finally
Figure D.9
9.3 Considerthe isothermat T on the P-V projection(Fig. D.l0).
At the point K the substanceis a mixture of liguid and vapour.
Let the massesof liquid and vapour be m, and m, and the total
massof the substance
bem.Thenthe volumeoccupiedby the mixture
at K is mrq* m"u"whereu, and nv are the specificvolumesof the
FigureD.10
liquid and vapour. Let the specificvolume of the mixture at K be
u. Showthat:
x
l i q ui d
mlu - u1): m'(u"- u)
This result,which givesthe ratio ftr"lflrt,is known as the'lever rule',
for obviousreasons.m,fm, is called the quality of the mixture.
9.4 At the criticalpoint, (APrcV)r:0 and @2PIAV2)r:0. Show
that, for a van der Waals gas,the critical point is at:
(a)
Figure D.8
,,:#,
v,:3nb,
,":h
242
AppendixD
9.5 The vapourpressureof camphoris as follows:
Temperature('C)
Pressure(torr)
30.8
1.04
55.0
3.12
243
AppendixD
62.0
4.22
78.0
7.8
By plottingln P against1/[ estimatethelatentheatof vaporization.
9.6 The equationsof the sublimationand the vaporizationcurves
of a particularmaterialare givenby:
lnP : 0.04- 6lT sublimation
lnP : 0.03- 4lT vaporization
(a) Find the temperatureand pressureof
whereP is in atmospheres.
the triplepoint.(b) Showthat the molar latentheatsof vaporization
and sublimationare 4R and 6R. You may assumethat the specihc
volumein the vapourphaseis much largerthan thosein the liquid
(c) Find the latentheat of fusion.(Hint: Consider
and solid phases.
a loop round the triple point in the P-T projection.As S is a state
function,
Uru/7r")- UsLlTrr)- {/rulTrr} :0.)
9.7 The phasediagram for 3He is as in Fig. D.11. Discussthe
variationof G along the sectionXY. What doesthis tell you about
theentropyof thesolidphasecomparedwith theentropyof theliquid
phaseat the lowesttemperature?
The resultshouldsurpriseyou.
9.8 Tin can exist in two forms,grey tin and white tin. Grey tin is
the stableform at low temperatures
and white tin the stableform
at high temperatures.
Thereis a first-ordertransitionbetweenthe
two phaseswith a transition temperature of 291K at a pressureof
l atm. What is the change in this transition temperature if the
pressureis increasedto 100atm. (The latent heat for the transition
i s 2. 20x 103Jm ol- 1; t he densit iesof gr ey and whit e t in ar e
5 . 7 5x 1 0 3 k g m - 3 a n d 7 . 3 0x 1 0 - 3 k g m - ' ; a n d t h e a t o m i c w e i g h t
of tin is 119.)
9.9 The Curie temperature for nickel for the phasechange from the
ferromagneticphaseto the paramagneticphaseis 630 K, the pressure
being I atmosphere.If the pressureis increasedby 100 atmospheres,
calculatethe shift in the Curie temperature.(In this phasetransition,
c" changeb
s y 6 . 7 J K - 1 m o l - 1 w h i l e p c h a n g e sb y 5 . 5x 1 0 - 6 K - t .
The densit yof nickel is 8. 9 x 103kgm - 3 and t he at om ic weight is
s8. 7. )
IO OPEN SYSTEMS AND THE CHEMICAL
POTENTIAL
10.1 Der ive equat ion ( 10. 3) .( Hint : Expr essU : U( S, V,N t , lf r . . . )
andsodU : ...)
10.2 Consider two systems,A and B, each composed of the same
single particle type. The two systems are contained in a chamber
surrounded by rigid adiabatic walls and they are separated from
each other within the chamber by a rigid diathermal wall which is
also permeable to the particles (see Fig. D.12). Show, using an
argument similar to the one used in section 10.2,that the condition
for equilibrium against particle exchange is the equality of the
chemical potentials.
10.3 Consider the system of question 10.2.Suppose that the two
systems,composed of the same single type of particle, are both in
the samephase,e.g. a gas on each side of the separating wall. Show
that the pressuresare equal. Would the pressuresbe equal if different
Rigid adiabaticwall
Figure D.ll
FigureD.12