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Impurity determination by thermal analysis
IMPURITY II. THE
ANALYTICA
CHIhIICA
DETERMINATION
BY
MELTING
CURVE H. 17. VAN
Inslilrrfc
for Physicd
ACTA
41
THERMAL
OF A QUICKLY \VI JK
Clmuistry (Rcceivcd
AND
W.
T .N.O., June
M.
FROZEN
SAMPLE
SMIT
UtrecJrt
z,+th,
ANALYSIS
(TJre NeilterIa~~ds)
ICJ~O)
1NTIIOL)UCTION
The heating cur\*c of a gradually frozen sample was discussed in a previous paper’. The melting curve of a sample obtained by quick freezing is now considered. When a licluicl sample is cooled rapidly to far below its freezing point, lotab ccluilibrium is cscludcd. It is hardly cvcn likely that local equilibrium (in the scnsc of part I) is present. Suclclcn freezing of a sample often results in a homogeneous solid solution*. Intcrmcdiatc stages between a builcl-up of the solid accorcling to the local ccluilibrium theory (part I) and a homogcncous solid solution may occur, depending on the rate of cooling. These intcrmccliatc stages arc likely to 1x2b~tdly reproducible and are not suitable for mathematical treatment. Thcrcforc the present discussion is restricted to the melting curve of a homogeneous solid solution.
The diffusion of the solid is assumed to be virtually restricted to a surface layer at the interface of solid and liquid. The liquicl is assumed to bc llornogcneous. Little is known about the amount of substance cont;lined in the surface laycr. It can only be stated that it usually is masimal when melting starts and approaches zero when melting bccomcs complete. The concentration within the surface layer is a function of the distance from the interface. The surface layer is assumccl to act in the same way as a solid fraction (/) of the sample which is in equilibrium with the liquid. l’hc problem will be treated as if such a fraction /, having a concentration XS is really present. Obviously f is II function of y. Suppose a fraction y of the sample has mcltcd. Since the surfilcc layer is assumed to be in equilibrium with the liquid, the total fraction of the sample participating in the equilibrium may bc represented by (~1-j- f). A ccorcling to the law of conservation of matter: (Y t- /)X
-
yxr.
t- 1s.9
(I)
where S is the concentration of the total sample and XL the concentration in the liquid. When XS and XL are small, XS/XL = k (constant) is valid. Thus -_.* Indications have been obtained that systems normally forming no solid solutions may form a supcrsaturatcd
solid solution
ou quick
frceziny.
Anal.
CJrim.
Ada,
24
(rgG1)
41-45
42
H. I’. VAN
WIJK,
W.
bf.
SMIT
(2) Morcovcr following
valid. Conscqucntly the tvhcn XL is small : XI, = Az,A-I’ is approsimately rclution between AT and y may bc consiclcrccl as a fair approximation: (3N
or since X/kl, ccluals the depression
of the find
melting
point (A-f,)
of the sample: (3lJ)
When no miscibility occurs in tlie solid, the whole solicl has permanently the same concentration and may be consiclcrccl to be in equilibrium with the liquid. Then / Since in that cast k = 0, ccln. (3:~) reduces to tlic well known White c!quals (I -y). ccluation. In order to avoid possible misunclcrstanclings it should be noted that the following discussion is restricted to solid solutions. It has dt-cncly been mentioned, that f is a function of y, being zero at y = 1. Little more is known about /. It can only be stntccl that f also clepcnds on the shape and size of the crystals. When tiny crystals arc formed the area of the interface of solicl and licluicl and thus /, may be large. The ultimate value of f equals (1 - y). Then ccln, (3;~) becomes ccln. (9) of part I, which is valid for total ecluilibriuml. lGq~ccia!!y when tiny or impcrfcct crystals arc present f is ids0 n function of time. ‘For, x5 may bc known, perfect and large crystals have a tcnclcncy to grow at the cost of small or irnpcrfcct crystds. When /is relatively small the tempcrnturc of the sample approaches its final melting point soon after melting has startccl (see ccln. 3b). Thus superhcnting of the bulk of the solid must occur, unless an additional interface is formccl. Since the formation of new intcrfacc occurs by passing an cncrgy barrier, the tcmpcrature of the sample may dccreasc suclclcnly when the new interface is formed. The formation of the new interface in the solid mny be comparccl with the formation of a vapour bubble in a liquid. As is the case with superheating of liquids, the cstcnt of superheating of solid solutions is unprcdictablc. So / may change discontinuously to an unknown estcnt. It is thus clear that a cluantitntivc cspcrimcntal proof of cqns. (3a) and (3b) is impossible. However, the foregoing permits a number of qualitative conclusions which can be clicclx~l by cxpcrimcnt : The melting curve of a sample of solid solutions obtained by cluick freezing may show the following fcaturcs, viz. (I) l’lic temperatures observed will bc higher than the corresponding tcmpcraturcs of a curve calculatccl on the basis of total ccluilibrium (see rcf.1, ccln. 9) or at lcast bc cc1ual to these. (2) ‘I’hc difference bctwecn the temperatures observccl and those obtaining to a “total ccluililxium curve” will be larger when the sample has been stored for a long .time (preferably at elcvatccl tcmpcraturcs). Anal. C/dim. Ada,
24
(rgG1)
4x-45
IMPURITY
DETERMINATION
BY
THERMAL
ANALYSIS.
II.
43
(3) After prolonged storage of the sample, its melting curve may resemble the curve of an extremely pure substance. (4) Depending on the rate of formation of new interfaces, the melting curve of such a sample may show a maximum.
DEPHESSIO.NS
FOUND
AND
CALCULATED
CONTAISISG
ON 0.181
A
QUICiiLY
DICd”/,
FROZEN
SAMPLE
LIT
AT fmtnll 1:roct
iOIl
--
liquid
.-
&l/tout rrtttreuli~lg .-__ _I_
__-_.___.L____-_
--_
OF
. ..-_ --------_
--
uflrr 3 II attttrulinl: ---..~_--_
cljfrr 1” days utsnrulitrg
_
cnlcrtlntrd
ucrorJitlEthe told equilibrium lACOry
0.1
0.13
0.1
0.10
0.22
0.2
0.00
0.08
0.08
0.18
0.3 0.4
0.08 0.08
0.08 0.08
0.0s
0.1
0.08
0.12
I
l’AI3l.IS
DIPHENYL
PR&NANTHRISNlS
.I
I I
I
DEPI~lSSSIONS
FOUND
AND
CALCULATED
0.762
CONTAI,NING -_.- __^..._.. -.---_~.
.^_._ .._
ON
_.
A
1110l’~”
QUICKLY OF
_-__..__
__ ._
..l’F /r,ut:d -.. ___ ..___-- .-.-------_ .---.. ruifhltl u/h _j II cortrdulirr~ aunrrrling ._.__ ----._. _ _-.__..... --..--_-. ._-----_._
f~vrnr’lio~i liyctid __--.--
VHOZI’.S
SAMPLIS
OF
_...
,__.. ---
._.. -___
_-___-. _.-rlfkr IO Jags cctrtrlW/lIl~
__-. _-.-._- -... .-
,l7- cdrrtlatevl acrortliug the totnl cquiltbrirrm thWy
.__.-.._.
0. I
0.79
0.G7
0.40
0.2
0*3
0.05
0.5’3
D*.59
0.38
0.4
0.55
0.52
a.37
0.55
0.5
0.51 0.43 0.40 0.30
0.49 0.46 0.40 0.30
0.37 0.37 0.30 0.30
0.49 0.43 0.39 0.30
0-G
0.7 I .o
I3lSPRI~SSIONS
FOUND
ASD
0.55
CALCULATII1>
COSTAISING _-._.-__-_l__-
___^___.._
ON
0.300
-__
_._-.._------Fetrliou Iiqlrid without cc~tnrulitr~ ______I__.___--_-___------
0.95 a*77
0.37
A
nlO1°/o
QUICKLY OF
0-Q
FROZRN
SAMI’LIS
TIN
._~__ __.___.-_--.-._._ d’l’
41’1’foulrJ
cdcv/ntd
._-__ ..__,_-- _.__ __.._._ uccorrlitt~the fold rqttiltbriuttt cr/lrr3 h uftrr 20 days lhrovy rrt8trcnlirrg atlrtrulirlg _ ___.____--_--._.-_ _.I_ _I-
0.1
I .Yo
I.51
I.10
I.39
I.34
0.98
I
0.3 044
1.20
1.20
0.88
I.22
I .07
1.05
0.80
I .08
0.5
0.97
0.98
0.90
0.95 0.87 o.G5
0.75
0.0
0.Gg
0.90
065
0.64
0.65
01’
HIShlUfiI
0.2
1.0
DIPHENYL
I’rIISSANTI1I~ENl1
Arml.
I&#
Chini.
.40
Aciu,
24 (196x)
41-45
H. P. VAN
44
WIJK,
W.
M. SMIT
‘The samples used in the quick frczzing csperiments were the same samples as usecl during the cxperimcnts on gradual freezing (see part I) _ The heating curves were cletermind With the thin-film apparatus 213.The samples were melted in the measuring vessel and then frozen within IO-zs sec. After the sample had been storccl for a definite time, its melting curve was clctcrmincd. The csperimentnl results, which arc summarisecl in Tables I, i1 and III, illustrate the first three conclusions; set p. 42). Fig. T may be consiclcrcd as an illustration of conclusion 4. The curve prescntccl was obtainccl with a sample of tin containing o..Tor> mol U/0of bismuth. The sample was quickly frozen and then stored for three hours at CLtcmpcraturc of z&O. Analogous curves were ohtainctl on previous occasions with organic substnnccs~. t.mp.rotur. lC
l;iK. I. Melting curve of :I quickly frown solution of 0.300 mol o/o bismuth of annealing ilt 2~8~.
in tin after three hours
This discussion is rcstrictcd to solid solutions obtained by quick freezing. Since so little is known nbout the function f, which is d&x-mined by the behaviour of the diffusion layer, it must lx concludccl that an impurity &termination based on only one curve of CLquickly frozen sample is of dubious value. As already mentionccl it mny ll:~l>p~n thnt f eclui~ls (r-3’). This may occur wlicn il complex of conditions is fulfillccl, v,iz. ((1) the rate of diffusion in the solicl is relatively high; the sample consists of a li\rgC! number of tiny and impcrfcct crystals with only 8 small tendency to transform into large and pcrfcct crystals; (0) tllc melting curve is cletermincd according to il qUiXSistatic metl~od, the sample being kept ullcler acliabatic conditions only to the extent recluirccl to obtain a constant tcmpcraturc after addition of n certain amount of heat. These conditions seem to have been met previously~ during the cletcrmination of the melting curve of n sample of cliphcnyl contiuninatecl by phenanthrcnc. At least it aplxarccl that the equation for total equilibrium was upplicablc in that case. However, in scvcral other cases the conditions for attainment of total ccluilibriuni seem to lx iLbSent. ~AuLr:Ya, for instance, investigated a number of quickly frozen samples of solid solutions and found his impurity determinations invariably low. In some cases only 6% of the amount of impurity ilcldcd coulcl bc trncccl. rlriul. Chim. Ada, 24 (xgGr) 4x-45
IMPURITY
DETERMINATION
BY
THERMAL
ANALYSIS.
II.
45
It may be useful to note here that the comparative method is applicable to quickly frozen samples, provided that the function f of the two samples to be compared is the same. Then, as can be shown easily, eqn. (12) of part I is validl. Since it appears that the same equation of the comparative method is applicable to samples showing total equilibrium as well as to both gradually or quickly frozen samples, it is comprehensible that the comparative method yields acceptable results in so many casts. The comparative method only demands the same pretreatment of both samples to be compared. Any special treatment is not required. Nevertheless it should be stressed that cluick freezing is not recommended, even when the comparative method is applied. In addition to the risk of unforeseen variations in the behaviour of the quickly frozen sample, it should be remembered that quickly frozen mixtures show smaller depressions than any other mixture does. ACKNOWLEDCEMISNT
We wish to thank Mr. A.
VAN
KUYK
for his accurate mcasuremcnts.
A temperature-heat content relation is dcrivcd for a homogeneous solid solution of a binary system showing a very rcstrictcd rate of diffusion in thu solid. It is shown cxpcrinicntally that quickly frozen samples behave like homogeneous solid solutions. According to the new theory the dcprcssions dctcrmincd on pztrtly molten satnplcs arc usually small and tlcpcnd on the conditions; thus purity clctcrlllirl:ttir)rls Lxxx1 on hcatinycurvcs of quickly frozen samples arc of dubious value. The theory shows tll:rt supcrhcating of the solid may occur when a solid solution is melted; this is confirmed cxperinlt~nlully.
Lors dc lcur 6tudc sur la dbtcrmirurtion dcs impurctbs par analyse thcrrnique, lcs autcura ont constatd quc dcs &hantillons refroidis rapidcmcnt se comportaicnt commc cles solutions homog&ncs solidcs. ZUSAMMENFASSUNG Bci Bestimrnungcn des IZeinheitsgradcs durch thcrmische Analyst rasch abgckiihltc Proben wit homogcnc fcstc Ldsungcn vcrhaltcn.
wurdc gcfundcn,
dass sic11
REFERENCES 1 H. 2 W. 3 W. 4 W. 6 W. flJ.
1:. VAN WIJK AND W. RI. SMIT, And. Clrirn. Acta, 23 (1gGo) 545. M. SMIT, 12ec. Iruv. cltim., 75 (rg5G) I 309. M. Sr.rrr AND G. KATEMAN. Amal. Chirn. Acta. 17 (1957) 1Gr. hl. SMIT, Thesis, Free University, hmstcrdnm, 1946. M. SnlIT, J. H. RUYTER AND M. 1:.VAN WIJK, And. Claim. rlctu, l-1.BADLEY,J. Phys.Cherw., 63 (1959) Iggr.
22 (1960) 8.
A tral. Chi9n.
Acta,
24 (xgGx) 41-45
ANALYTICA
CHIhIICA
DETERMINATION
BY
MELTING
CURVE H. 17. VAN
Inslilrrfc
for Physicd
ACTA
41
THERMAL
OF A QUICKLY \VI JK
Clmuistry (Rcceivcd
AND
W.
T .N.O., June
M.
FROZEN
SAMPLE
SMIT
UtrecJrt
z,+th,
ANALYSIS
(TJre NeilterIa~~ds)
ICJ~O)
1NTIIOL)UCTION
The heating cur\*c of a gradually frozen sample was discussed in a previous paper’. The melting curve of a sample obtained by quick freezing is now considered. When a licluicl sample is cooled rapidly to far below its freezing point, lotab ccluilibrium is cscludcd. It is hardly cvcn likely that local equilibrium (in the scnsc of part I) is present. Suclclcn freezing of a sample often results in a homogeneous solid solution*. Intcrmcdiatc stages between a builcl-up of the solid accorcling to the local ccluilibrium theory (part I) and a homogcncous solid solution may occur, depending on the rate of cooling. These intcrmccliatc stages arc likely to 1x2b~tdly reproducible and are not suitable for mathematical treatment. Thcrcforc the present discussion is restricted to the melting curve of a homogeneous solid solution.
The diffusion of the solid is assumed to be virtually restricted to a surface layer at the interface of solid and liquid. The liquicl is assumed to bc llornogcneous. Little is known about the amount of substance cont;lined in the surface laycr. It can only be stated that it usually is masimal when melting starts and approaches zero when melting bccomcs complete. The concentration within the surface layer is a function of the distance from the interface. The surface layer is assumccl to act in the same way as a solid fraction (/) of the sample which is in equilibrium with the liquid. l’hc problem will be treated as if such a fraction /, having a concentration XS is really present. Obviously f is II function of y. Suppose a fraction y of the sample has mcltcd. Since the surfilcc layer is assumed to be in equilibrium with the liquid, the total fraction of the sample participating in the equilibrium may bc represented by (~1-j- f). A ccorcling to the law of conservation of matter: (Y t- /)X
-
yxr.
t- 1s.9
(I)
where S is the concentration of the total sample and XL the concentration in the liquid. When XS and XL are small, XS/XL = k (constant) is valid. Thus -_.* Indications have been obtained that systems normally forming no solid solutions may form a supcrsaturatcd
solid solution
ou quick
frceziny.
Anal.
CJrim.
Ada,
24
(rgG1)
41-45
42
H. I’. VAN
WIJK,
W.
bf.
SMIT
(2) Morcovcr following
valid. Conscqucntly the tvhcn XL is small : XI, = Az,A-I’ is approsimately rclution between AT and y may bc consiclcrccl as a fair approximation: (3N
or since X/kl, ccluals the depression
of the find
melting
point (A-f,)
of the sample: (3lJ)
When no miscibility occurs in tlie solid, the whole solicl has permanently the same concentration and may be consiclcrccl to be in equilibrium with the liquid. Then / Since in that cast k = 0, ccln. (3:~) reduces to tlic well known White c!quals (I -y). ccluation. In order to avoid possible misunclcrstanclings it should be noted that the following discussion is restricted to solid solutions. It has dt-cncly been mentioned, that f is a function of y, being zero at y = 1. Little more is known about /. It can only be stntccl that f also clepcnds on the shape and size of the crystals. When tiny crystals arc formed the area of the interface of solicl and licluicl and thus /, may be large. The ultimate value of f equals (1 - y). Then ccln, (3;~) becomes ccln. (9) of part I, which is valid for total ecluilibriuml. lGq~ccia!!y when tiny or impcrfcct crystals arc present f is ids0 n function of time. ‘For, x5 may bc known, perfect and large crystals have a tcnclcncy to grow at the cost of small or irnpcrfcct crystds. When /is relatively small the tempcrnturc of the sample approaches its final melting point soon after melting has startccl (see ccln. 3b). Thus superhcnting of the bulk of the solid must occur, unless an additional interface is formccl. Since the formation of new intcrfacc occurs by passing an cncrgy barrier, the tcmpcrature of the sample may dccreasc suclclcnly when the new interface is formed. The formation of the new interface in the solid mny be comparccl with the formation of a vapour bubble in a liquid. As is the case with superheating of liquids, the cstcnt of superheating of solid solutions is unprcdictablc. So / may change discontinuously to an unknown estcnt. It is thus clear that a cluantitntivc cspcrimcntal proof of cqns. (3a) and (3b) is impossible. However, the foregoing permits a number of qualitative conclusions which can be clicclx~l by cxpcrimcnt : The melting curve of a sample of solid solutions obtained by cluick freezing may show the following fcaturcs, viz. (I) l’lic temperatures observed will bc higher than the corresponding tcmpcraturcs of a curve calculatccl on the basis of total ccluilibrium (see rcf.1, ccln. 9) or at lcast bc cc1ual to these. (2) ‘I’hc difference bctwecn the temperatures observccl and those obtaining to a “total ccluililxium curve” will be larger when the sample has been stored for a long .time (preferably at elcvatccl tcmpcraturcs). Anal. C/dim. Ada,
24
(rgG1)
4x-45
IMPURITY
DETERMINATION
BY
THERMAL
ANALYSIS.
II.
43
(3) After prolonged storage of the sample, its melting curve may resemble the curve of an extremely pure substance. (4) Depending on the rate of formation of new interfaces, the melting curve of such a sample may show a maximum.
DEPHESSIO.NS
FOUND
AND
CALCULATED
CONTAISISG
ON 0.181
A
QUICiiLY
DICd”/,
FROZEN
SAMPLE
LIT
AT fmtnll 1:roct
iOIl
--
liquid
.-
&l/tout rrtttreuli~lg .-__ _I_
__-_.___.L____-_
--_
OF
. ..-_ --------_
--
uflrr 3 II attttrulinl: ---..~_--_
cljfrr 1” days utsnrulitrg
_
cnlcrtlntrd
ucrorJitlEthe told equilibrium lACOry
0.1
0.13
0.1
0.10
0.22
0.2
0.00
0.08
0.08
0.18
0.3 0.4
0.08 0.08
0.08 0.08
0.0s
0.1
0.08
0.12
I
l’AI3l.IS
DIPHENYL
PR&NANTHRISNlS
.I
I I
I
DEPI~lSSSIONS
FOUND
AND
CALCULATED
0.762
CONTAI,NING -_.- __^..._.. -.---_~.
.^_._ .._
ON
_.
A
1110l’~”
QUICKLY OF
_-__..__
__ ._
..l’F /r,ut:d -.. ___ ..___-- .-.-------_ .---.. ruifhltl u/h _j II cortrdulirr~ aunrrrling ._.__ ----._. _ _-.__..... --..--_-. ._-----_._
f~vrnr’lio~i liyctid __--.--
VHOZI’.S
SAMPLIS
OF
_...
,__.. ---
._.. -___
_-___-. _.-rlfkr IO Jags cctrtrlW/lIl~
__-. _-.-._- -... .-
,l7- cdrrtlatevl acrortliug the totnl cquiltbrirrm thWy
.__.-.._.
0. I
0.79
0.G7
0.40
0.2
0*3
0.05
0.5’3
D*.59
0.38
0.4
0.55
0.52
a.37
0.55
0.5
0.51 0.43 0.40 0.30
0.49 0.46 0.40 0.30
0.37 0.37 0.30 0.30
0.49 0.43 0.39 0.30
0-G
0.7 I .o
I3lSPRI~SSIONS
FOUND
ASD
0.55
CALCULATII1>
COSTAISING _-._.-__-_l__-
___^___.._
ON
0.300
-__
_._-.._------Fetrliou Iiqlrid without cc~tnrulitr~ ______I__.___--_-___------
0.95 a*77
0.37
A
nlO1°/o
QUICKLY OF
0-Q
FROZRN
SAMI’LIS
TIN
._~__ __.___.-_--.-._._ d’l’
41’1’foulrJ
cdcv/ntd
._-__ ..__,_-- _.__ __.._._ uccorrlitt~the fold rqttiltbriuttt cr/lrr3 h uftrr 20 days lhrovy rrt8trcnlirrg atlrtrulirlg _ ___.____--_--._.-_ _.I_ _I-
0.1
I .Yo
I.51
I.10
I.39
I.34
0.98
I
0.3 044
1.20
1.20
0.88
I.22
I .07
1.05
0.80
I .08
0.5
0.97
0.98
0.90
0.95 0.87 o.G5
0.75
0.0
0.Gg
0.90
065
0.64
0.65
01’
HIShlUfiI
0.2
1.0
DIPHENYL
I’rIISSANTI1I~ENl1
Arml.
I&#
Chini.
.40
Aciu,
24 (196x)
41-45
H. P. VAN
44
WIJK,
W.
M. SMIT
‘The samples used in the quick frczzing csperiments were the same samples as usecl during the cxperimcnts on gradual freezing (see part I) _ The heating curves were cletermind With the thin-film apparatus 213.The samples were melted in the measuring vessel and then frozen within IO-zs sec. After the sample had been storccl for a definite time, its melting curve was clctcrmincd. The csperimentnl results, which arc summarisecl in Tables I, i1 and III, illustrate the first three conclusions; set p. 42). Fig. T may be consiclcrcd as an illustration of conclusion 4. The curve prescntccl was obtainccl with a sample of tin containing o..Tor> mol U/0of bismuth. The sample was quickly frozen and then stored for three hours at CLtcmpcraturc of z&O. Analogous curves were ohtainctl on previous occasions with organic substnnccs~. t.mp.rotur. lC
l;iK. I. Melting curve of :I quickly frown solution of 0.300 mol o/o bismuth of annealing ilt 2~8~.
in tin after three hours
This discussion is rcstrictcd to solid solutions obtained by quick freezing. Since so little is known nbout the function f, which is d&x-mined by the behaviour of the diffusion layer, it must lx concludccl that an impurity &termination based on only one curve of CLquickly frozen sample is of dubious value. As already mentionccl it mny ll:~l>p~n thnt f eclui~ls (r-3’). This may occur wlicn il complex of conditions is fulfillccl, v,iz. ((1) the rate of diffusion in the solicl is relatively high; the sample consists of a li\rgC! number of tiny and impcrfcct crystals with only 8 small tendency to transform into large and pcrfcct crystals; (0) tllc melting curve is cletermincd according to il qUiXSistatic metl~od, the sample being kept ullcler acliabatic conditions only to the extent recluirccl to obtain a constant tcmpcraturc after addition of n certain amount of heat. These conditions seem to have been met previously~ during the cletcrmination of the melting curve of n sample of cliphcnyl contiuninatecl by phenanthrcnc. At least it aplxarccl that the equation for total equilibrium was upplicablc in that case. However, in scvcral other cases the conditions for attainment of total ccluilibriuni seem to lx iLbSent. ~AuLr:Ya, for instance, investigated a number of quickly frozen samples of solid solutions and found his impurity determinations invariably low. In some cases only 6% of the amount of impurity ilcldcd coulcl bc trncccl. rlriul. Chim. Ada, 24 (xgGr) 4x-45
IMPURITY
DETERMINATION
BY
THERMAL
ANALYSIS.
II.
45
It may be useful to note here that the comparative method is applicable to quickly frozen samples, provided that the function f of the two samples to be compared is the same. Then, as can be shown easily, eqn. (12) of part I is validl. Since it appears that the same equation of the comparative method is applicable to samples showing total equilibrium as well as to both gradually or quickly frozen samples, it is comprehensible that the comparative method yields acceptable results in so many casts. The comparative method only demands the same pretreatment of both samples to be compared. Any special treatment is not required. Nevertheless it should be stressed that cluick freezing is not recommended, even when the comparative method is applied. In addition to the risk of unforeseen variations in the behaviour of the quickly frozen sample, it should be remembered that quickly frozen mixtures show smaller depressions than any other mixture does. ACKNOWLEDCEMISNT
We wish to thank Mr. A.
VAN
KUYK
for his accurate mcasuremcnts.
A temperature-heat content relation is dcrivcd for a homogeneous solid solution of a binary system showing a very rcstrictcd rate of diffusion in thu solid. It is shown cxpcrinicntally that quickly frozen samples behave like homogeneous solid solutions. According to the new theory the dcprcssions dctcrmincd on pztrtly molten satnplcs arc usually small and tlcpcnd on the conditions; thus purity clctcrlllirl:ttir)rls Lxxx1 on hcatinycurvcs of quickly frozen samples arc of dubious value. The theory shows tll:rt supcrhcating of the solid may occur when a solid solution is melted; this is confirmed cxperinlt~nlully.
Lors dc lcur 6tudc sur la dbtcrmirurtion dcs impurctbs par analyse thcrrnique, lcs autcura ont constatd quc dcs &hantillons refroidis rapidcmcnt se comportaicnt commc cles solutions homog&ncs solidcs. ZUSAMMENFASSUNG Bci Bestimrnungcn des IZeinheitsgradcs durch thcrmische Analyst rasch abgckiihltc Proben wit homogcnc fcstc Ldsungcn vcrhaltcn.
wurdc gcfundcn,
dass sic11
REFERENCES 1 H. 2 W. 3 W. 4 W. 6 W. flJ.
1:. VAN WIJK AND W. RI. SMIT, And. Clrirn. Acta, 23 (1gGo) 545. M. SMIT, 12ec. Iruv. cltim., 75 (rg5G) I 309. M. Sr.rrr AND G. KATEMAN. Amal. Chirn. Acta. 17 (1957) 1Gr. hl. SMIT, Thesis, Free University, hmstcrdnm, 1946. M. SnlIT, J. H. RUYTER AND M. 1:.VAN WIJK, And. Claim. rlctu, l-1.BADLEY,J. Phys.Cherw., 63 (1959) Iggr.
22 (1960) 8.
A tral. Chi9n.
Acta,
24 (xgGx) 41-45