686
Chapter 22
Using Equation 16.41,
ot
n'
2n
B2v(T):'+
[1
- (r'-
t)(e'zk"r
dBru _2ro3No (rr _ t) n,t*"r
3
dr
Substitute into the equation fbr
Lc
AZ
- t)l
(.+)
\RBI-/
to find
: -'"oir ^ ,r, - tr#ntkur (Pz- P,) +
For Nr(g) under the conditions specified'
I rr
2tto
LU : ---J-(n
^
No." r _
ly.€_r, ,urlp,
Rrr
2r(32'/.7 x
10-12 m1316'022
x
- ', (H)
aos
I roo(lo o t.oor
-
I-
-5.13 x 10-a bar'm3'mol-r
t
-5i.3 J'mol
of state P lV - b\ : RT
22-8. Determine Z" - Zu for a gas that obeys the cquation
We can write, from the equation of state'
P(V
-
b)
RT
/aP\
D
t;l
v
/ay\
-R
[*l
\0r /
p
-b
v
Now we substitute into Equation 22'23:
e
RT
,_e ,: (#)"(#)"
r
v-b (;)
:^
of water at25'C is 2.512 x 10-a
22-9" The coefficient of thermal expanslon
i
5
thc value of C ,, - C,, for
compressibility is 4.525 x 10 bar Calculate
25'C. The densrty of water aL25'C is 0'99705
g'mL ''
The molar volume of watcr ts
\
1
_
v:(o*6;;;)(
We can now substitute into
\/l8.ol5g\/
ldm:,\:o.ol8o68dml
\rooo'L)-"
i
'*
Equation 22'27 to find C"
C,-C,,:.K
-
-
Cu' For one mole'
atT v
(2"572
x
10 * r-')'(z9g'ts r)(0'otg0og amt)
4.525
1.815 x
x
10-s bar
dm3.bar-r
.K-l
r
and its thet'mi
mole of water
-rLZI
>I
00r
/J
009 00s
00t
6-
'edo1s e,trltsod e suq eurl prnbrl-pllos eql 'prlos
eqt;o flrsuep oqt upql ;eteer8 sr prnbrl eqt yo
,{ltsuep oql esnEJeg 'eut1 su3-ptnbrl eqt lcnrrsuoo ol lurod
,1urod
IucrlrrJ puu '1urod Burlroq lerurou
eldrr1 eqt puu eull prlos-prnbrl oq] tJnrlsuoJ ot turod 3ur11eu
luurrou pue lurod eldul eqt esn e \
'prlos
Jo ,(lrsuep < plnbrl go flrsuap
,1urod
'turod
Surlroq
:C.lt
I
Burlleru Ierurou lutle 91 I puu J.ZtE .1urod
I€rurou
t
lucrlrrc
'.uN Zt'0 pue J.EII 'turod e1du1 :e1ep 8ur,tro11og eql ue.rr8 z1 ro; urer8erp eseqd eqt qole{S .Z-€i
pue i3"yg
'ernleredrual lurod e1du1 eql ueql req8rq
sr e.tnluredruel
lurod Suqleur Isturou sll esn€Joq 'ernsserd perlddu uu repun lJetu lou seop ue8,{xo
l€ql ees uer el\
0rI jzt
>t tJ
00t
08
yo
PINbII
I
ruut8erp aql arnpord a16 .lurod
JeorlrJ3 eql le sdols eul
se8-plnbrl eq,I 'ouII su8-prnbrl oqt tcnrlsuoc 01 rurod
IeJrtrrc puu 'lurod Burlroq pugou ,ru,oa'"ta,r_
oql pue eull pllos-plnbtl eqt lJnJlsuoJ o1 lutod Suqleur
JEuJou oql pue lurod a1drr1 eqt esn u'c o.\1
isoop
'1urod Sutlroq
:'uot Vy I pue
Jele,^A se
SNOIl-NIOS CNV SIA/]IBOUd
e!rqlllnbl
t(,
ernsserd perldde up topun l1aur ue8,(xo seoq .Jo6.Zg _
l
pt'rou puu i3"7'g I7- 'tutod Sutrleur lp.urou luol g73
Lt puu x 9'tsl .rurod lecrru:
)I g'?E 'lurod e1drr1 :u1up Surmolyo; eql Sursn ua3,,{xo rog ruer8erp esuqd eql qJle{S . t-
aseLld
drrdvHf
724
Chapter 23
23-6. The slope of the melting
curve of methane is given by
dP
dT
(0.08446our'a-r
-
es)To8s
and pressurc
Using the fact that the temperature
from the triple point to arbitrary temperatures'
',:4bat,calculate
the melting pressure of methane at 300 K
the triple point are 90.68 K and 0.1 :
K gives
Integrating from the triple point to 300
P,
[ ' d, :
Jo.ttTq b",
^
1":t00
K
l^^.
Jq0.o8 ,.0'08446
K
bar'K-rssrossdr
bar'K-r'8s
: 0.084461.85
Pr: l556bar
Pz- 0 l714bar
[{:oo
r)"t -
(90.68
K)"']
K'
This is the melting pressure of methane at 300
the entire liquid-vapor coexistence curve can
23-7 . Thevapor pressure of methanol along
be express- -
very accurately by the empirical equation
ln(P/bar)
;
=
Hr
h
''F
t
;r
where x -- T lr,' and
methanol is 337.67 K.
-
{ :
-10'7?849 +
16.158201
-
* 4.373232x , -
2.38t3jj
x3
512'60
3.603425x
-
+ 4.5l.2199(1
x)'io
K' Use this formula to show that the normal boiling point
:
bar. If the normal boiling point of methanol i'
At the normal boiling point, P - 1 atm 1.01325
should hold when x :337 '61 1512'60:
331 .67 K, then the equality below
r6Js8201 - 3'603425x
:, _ l0'"''15.849
:" +
+ 4.313232xt - Z.3gtzll x3 + +.512t99(1 - x)1
0.013163 : -16'323364+|6158201-2.313.|19+1.891112-0.6801220+0735141
1n(1.01325)
70
0.013163
=
0.0132546
is the temperature at which the vapor pressure is exactll
the standard boiling
formula given in the previous problem to show that
one bar. Use the
".pi.I"l
point of methanol is 337'33 K'
23-8. The standard boiling point of a liquid
we do this in the same way
ln(1)
"
-10'"752849
x
*
?
0
as the previous problem' but substitute
-
+ rc158201 -
4.373232x2
t6.33g820
0^, -0.000143
+
-
:
331 '33 l512'60 into
3'603425x
- z.zgt3ll *' +
16.1 58207
x
4'572t99(1
2.31 1329
+
1
-
.893892
x)r
-
70
0.678668
+
0'1 31 5'/ 2
r_loru.[{
OOt
ut
091
z'lt
x9z
() 9l'zEt)() 9l'Lzil(,->r'r lotu'f Etlt'8)
tJ
-'J :
'd Ul ---;-'rl "t'JU
Ho"tV
("'t \ y :3'd
I
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'e8uer ernlereduel srql :e'to
t l'f,7 uotl€nbg esn u€f, e',rr
ul
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o"ny
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ernsse:d ;odun eql 'BZ-g'Z
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lrol (.)LOl : ull€ 80t'I
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, tn.'t't sltE g : tll ur
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:
ut
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\-t=) -j,'':^,,: z,
/ .gJ-\
:g1'97 uorlenbg asn u€c e,Lr 'e8ue-t
qll'44 luBlsuot suI€uIeJ Hoonv Surunssy
Ol
ernleredruel eerSep-ual stql le^o ernlereduel lCedse-r
eql 'J.0 I I
'rJol ELOl sI enlE^ Ieluetulreoxe
o1 uottenbe uol'(ede13-snlsn€lJ eql esn
le rela,{\ 3o e:nsserd rocle.t eql elEInJlEt
'1urod Surltoq l?tuJou
.Lt-EzurolqoJd ur
)I
.prle^ aq ot uortenbe
strtE r-lotu't)t99'0?sllelB'^ 3ouotleztrodu't1o'{dltqtuer€lotueq;-'Lz-tz
eql l€q] sreedde
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eql
eull
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,-\ I J/L
".tJ
'J
lI
's^
7
u1
eql esq
lold ot L-EZ urelqold ut uant8 d ul roJ €lnturoJ
IJ
iplle^ eq 11r,r,r uorle nbe uo'r'(ede1 3-s
Sursn
rno'(
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nIS nE
eqt )urqt no,{ oP e8uer e;nlureduel 1€q,{\
ut ue,tr8 loueqletu JoJ €]€p e:nsserd rode't eq1
rsura8e 7 u1 told oi L-EZ LuelqoJ4
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r:uqrlrnb3 asrq.l
se
eso '92-tz
o <- sounv leql eclloN
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elrqder8puupuotu€Ip
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puoluulp qslq' l le
pue
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3o,&rsuep eql e{€I
:
leql ue'ttg
elelncn,)'JE9t't+ 9681
pun
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";7'y
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(puoururp)3
*
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e8ueqc esuqd eql rePISuoJ
tlJ
09t 009
00t
'98-tZ
092
xw
tt
lt) salnssa'rd pue se.rnleladLuel qBrq le
'1utod Suqleur I?uJou eql
tou st euII se3-prnbrl eql JoJ uotlenbe eqt leql elo51
.(turod
elgrnrce
,(1e1e1duroc
I
[uJIlllJ eql ]JeSJalul ]ou
Seop
euII eql 'se8 puu pllos eql uee'4d'leq
puu 'turod e1dr.q eql'1utod pctluc eql 'su8 pue rode'r oql uee'^Aleq
u'r ?1y\ os '>l zz7' sr ;o lurod
aurl eql 3ur11o1d rq eprpo, u"dorpiq ;o urerSerp eseqdot,e ecnpo'd
''
e^los ol 3ur1nlqsqn5
tnql pug e^\ 'o'd
3ur11eru l€tuJou eq1 l€ql aou{ osl€ olv.JJol g'gtz:
'oJ
:
) g'01€ :
>r
Ev'vlz
d'J
o'J8vv'r
dr
dr
r.
>r L'9692
-LL>LI_
J
>rz'9062
-
020'61
luroo
e1d'l
pelBsol sr lurod e1du1
eqt le os .eur€s eqt eJB sernsserd rodu.t prnbrl pu€ pllos eql eJeq^A
eql
'eplpol ueSorp,{q;o urur8erp eseqd eql I{rleIS ol
pul 5 VZV'>IZZZ are splpol uaSorp'{q
urelqo-rd snora.erd oql ur utep eqt esn ',(1e,tt1cedser 'ru]€ 0'28
'St-tZ
eql pug 'ernlered1uel IBoIIIJc eql 'lurod Suqleur IeIuJou oql luql ue^IC
elnsserd
1o
letltlrc
>r L'9692
rrlr i :
>17'9062
eql lE sedols eql Jo
OIIEJ
-: lE
eql '1urod eldrrt eql
,f,
'* l,,l- :
I/,t\"lr
\xlsoszl
-dP
Pue
--
Jpldp
Jpl"dp
st lurod e1drr1
'JJol Jo sllun uI eJe seJnsserd ereq,Lr
sd ecurs
/,J\rp
/"
\)
t-tso:z'9062
,d P
se se,\Jnc se8-prnbrl puu se8-pr1os eql Jo sedols eqt elIJ'^ uec ed\
.1urod e1du1 eql le a^JnJ su8-prnbrl eqt pu€ e,trnc se8-pt1os eql Jo sedols eql Jo oIl€J eql
LVt
euqrpnb3 aseqS
elslnopJ
742
Chapter 23
To find the standard molar Gibbs entropy, we use the Maxwell relation (Equation 22.46)
/^
: _3.:03 J.mor
' : eA
ar
\
/,
_
A. s"
I
A,Go: A,E _ TL,s'(Equation 22.13),wefindthatA.H':
Substitutinginto
1g95J.mol-r.
Because both graphite and diamond are incompressible, we can write (as in Problem
Gi.upr'
:
ci.*n + V *urn(P
4,u'
:
do,"'
-
23-18\
Po)
and
* {,",{e -
ro)
Combining these two equations gives
A,G
- l,d
+ (%,",
-
T
r_r)(
p*
po)
When graphite and diamond are in equilibrium, A,G : 0. Substituting into the equation given in
the problem, we see that at 25'C, L,G' : 2898 J.mol-t. Then
0: A,G'. (#-;=
:2898 J.mol-r
-P:
|
t.916
15
-
- t-h;=)
(1.916
x
x l0 r 6r:--^1-r
rt,ot
10-3 dm3.mol-r)1p
-
g mol-r)1P
1
bar)
/?RqRr.-^r-r1/o'o82o6dml
(zals
l'mot- )
(
- l bar)
,ffi
!1) +ru-
000 bar
23-37. Use Equation23.36 to calculate k" - En for Kr(g) at298.15 K. The literature value
is
-42.12 kJ.mol-r.
lt" - Eo: -RZ ln
We do this in the same way we found
p' -
l({\wl
L\y/ P" )
(23.36)
Eofor A r(g) in Section 23-5. First,
qo(V.Tl (2rmk"T\3'2
v\r'/
-r-l
_l
(2n)(1.391
_|l
x
10-25
kg.mol-')(t.381 x l0-23 J.K-')(298.15
L
:7 .422 x 1032 m-3
kuT _ (1.381 x 10-r3 J.K-')(298.15
Po
1"0 x 105 Pa
(6.626
Substituting into Equation 23.34
x
K)
10-34
:
K)lt/'
J.s;2
4.716
"
|
16*26
--3
giv es
If4) g-l
' ti" - En:
u -Rr," L\Y/
P")
: -R(298.15 K) lnt(7 422 x 1032 m-,) {+.t t6 x 10-26 m-3)l
: -4.212 x 104 J.mol-l
^a, rn
^arcn
- ft- +).ur-
r ^a,$
= ^?.b
=ft-tl "$e,'P
--11,!
tJ-*={.P;15
-(4.
Ap
F=nP
S '!4,"
I * q'.r-n
ltu-n\
!*'
Y
=
lfl
-\lju
,Fb
/A
H'n(*)
r/^e) ,..
^1w\
\;alr
d-
tlrte
t +,f a/ l€ )
Ap'r- = /tp \Te)
(fo*rrttotl) o
'^$Fy{^M?
il)t,&U,o),
"-
n
1:tE {k}
3t08 Yoq0
nemt
ao',unr
wpwb,,
\Te /=nP
J_
ffi
(",
J=SF
--'ft)"-ffi=d
0
t\)rsv=
r,rr"v =
r,ortv
rffi3/-o/t-\'
-.*\t +il "'t: -L&V
-fJU,It p dY
o.=nZ-T
dV +
= -nrll
r^(#*J
tanP(# h)
$*u=
-"nl(t-;.)+nRrr,r.(ffi)
$*u=
nfrrhtr#
@ cp-cv =ry
, n=L(#),)
Jt"u: V=nBr 50 eI=
,
t
.$'N--r-.ff
\v Vt
Ylt
Ce-Cv=fJ
ft.-F)
4^nz(# -b)
K-ar,d
"
+ (ry),
*F
(#\=
=oL
flq-oitrl = wbb'tlql
(r)(# W
=r-l
w6)$+ilb)=d--! ?sset*se
\BJ =d
nd g'TggQLofro)=dy
6>t
St
'$tC
)
ft.ur. f,
\M)w
ooo,rrblrgr$.9o*tloo,1, )
=dV
)S/'Ele ')uQ ='tt W
?l5l'8?e =)"9- = lL W
(trlrt
4'hV
urtr.{=utsV -- dV
.-!- 'rf SnttV
,df
T,rJorE =dprJ
?o rlq 1
n*o.v
'' iF nW
=dP
7u/
"N
H= Tl =-dl
e
\ndilov +
,ruqwr
€ovt
ffi=4
t =:'ttul
'u/ [ 'L\
*n"uttu't]
J= -llLo (
@ VP*r,a.= a,b4 To-ut
-n6.f,"C
T= -luD'L
1= -tA\
T=
VPario. = 0,3,63T0-w
VPrr^.
YP{=
[1,q3
Tq,r
553bTma"
AHffr= a\\8.
A\\eus
AS*p : llaq CAr,r^
/CW*%4\:
'n(k)=
h ti
lu3
- fu
#)
=
+#?S
^[g')=
lr3 __l
//n( 0, 7b
lorrt\
e. b4 a,swl
Alwpt b\s,a 1"'
?
+)
As.rn\.. t-Y - t \
(E,sl4sr/r,n"t) \ lbl,lSK l4ln;rs )
K
Asro\'= +
- A$*\\o
A
Krlp"L
e^sl\..,
T^,1r4" pt,
* ( #"-* - #,,,)
|C}ryo1l.,o'^€,6-
+-\"tu\\"(+
tJS
Al]r*, Al-lpos #
*Aygp_H"/!_f\
L-- t T, ar)
A f\rrn s Tl)eUa^au
3
Fnot L\lsubl
%
kxlrrhrf
W".
T,WPt"" Fn *e*daa," fW
-A!-)
P
t,, T.
!/(
{ k*,p' A fuyt'
Pt!tr'e44-
:px
i^/A=
A$rbll--S
tPr/- &lT T/
-t^
(p) -,e/^(p'') =
+t( + })
A(p.)=i/ncq)-ry(+
+)
U^ {*',rL #,^ %rdry
-l,n(p,)= l^cg)
,o\L
-l)iu""
^y(+ {)
A+ i-rfi"
p'ar, p,
N-,,u
Sifs
-
pr .#^ tlG'
,q.,"tpJ-+p(+-k)-,t^Q.) jrn
[0,affi)-
brye D
'
3J;wYr
! !9' l+ L\
- J; =,bn [ss'a6) -U4
t'3i4s (" lry,tsz
r3l4-L rr l4t l'i)-
lt. ro5 +31_CU
l1
t; A13,0\l "
-.33,3?3*;q$03 r tt 0AY4
tst,ll'lr
+ryu_4il'ww
T1
r, N \?osR
Tr
@
t-S'c). Sr*, fu
AbLO"C)=C\/y'^ {
[,,Jo uJa^^l AG
'tlA*-
Ce.0b\ \.\4[,h^hl4
\ )T)p
fM=-au
\Ep
\ ['h{,
A6(r?)
at
AG
Cs'c)
qbt,tf, K
U!-
waL,ata Oo(t
Wz
- at{
T,
I a trtu/'\
I-
rnsr,,r,,v,al
bnxnru- w c\F- L'oLilcd
Ato -- \G{*r"
fum ,
All = A[\+,e.,r.
}t
bstr,,vwo l
_
-
AG ta6?,trK) = + Nt\ (o,o rt:os)
A\{ , 1la. $" Ucrp.r- ?"*
^ail,
?
-s'C = 99 * tO-3 bou
V Ps"r @ -S"C = e[- lo-t booVPun
@
,L'''falJp3)= AU{,p/
\ - \-\
tlffir/- T-tem-Fens7
[C,r rr.qsc\ L[\t"*y
= -ARluu = -
e/noL
Af\trn u AS"p
UJr* Fr,* hV,,fuwr,x^tr?
"lrnteg^tD-) 4/vr(P*r,r) = Ak+(-0, oo@68ant)
/in (A.bxtu=) --tu" (Pil,uD= All"uu (-0,
Alw
aul'+1ar+ f,t^c !if^a,
$n (09rtD{)-.0h (4l,* lD-)
-
w)obrz&)
Afl$,ro
G(a6gJsK)
=
:
p,Wq.TAB(AHuop
- AHsr)