ANALYTICA CHIMICA ACTA
IMPURITY
DETERMINATION
BY THERMAL
545
ANALYSIS
I. T H E M E L T I N G CURVE OF A G R A D U A L L Y F R O Z E N SAMPLE
H. F. VAN W I J K AND W. M. SMIT
Institute/or Physical Chemistry T.N.O., Utrecht (The Netherlands)
(Received June 24th, 196o)
INTRODUCTION
The interpretation of melting curves is based on the assumption that thermodynamic
equilibrium prevails throughout the solid and the liquid phase during the determination of the curve. A number of experimental circumstances resulting in insufficient thermodynamic equilibrium have been discussed in a previous paper 1. Attention
has been paid to the limited mass transfer in the liquid and it has been shown that
its influence m a y be neglected under suitable experimental conditions%
Thus far, no attention has been paid to the influence of the rate of mass transfer
in the solid. As is obvious a limited rate of mass transfer in the solid has no effect
on the attainment of thermodynamic equilibrium, when no solid solutions occur in
the solid-liquid system. However, when the system contains mixed crystals the
approach of thermodynamic equilibrium between the entire solid and the liquid is
seriously affected when the rate of mass transfer in the solid is limited.
Since mass transfer in the solid can hardly be promoted by external means and
the diffusion constants in solids lie in the range of I0 -l~ to Io -l~ cm -1 sec -1 (those
of liquids are 10 -4 to 10 -5 cm -1 sec-1), it even must be concluded that mass transfer
within the solid is negligible under the experimental conditions usually applied in
the determination of heating curves. Consequently only the surface layer of the solid
participates in the thermodynamic equilibrium and the bulk of the solid is not in
t h e r m o d y n a m i c equilibrium during the determination of the melting curve.
These conclusions are used as the starting point for the interpretation of heating
curves which is developed in this paper. On the basis of the new interpretation some
discrepancies mentioned in the literature can be solved.
A further consequence of the lack of mass transfer in the solid is that the pretreatment of the sample influences the shape of the melting curve. This influence is absent
when cooling curves are determined which start with the liquid sample.
However. since rates of crystallisation m a y be slow compared to rates of melting,
serious deviations from the equilibrium temperatures m a y occur on cooling ~. Therefore
a heating curve obtained b y starting with a solid sample is preferable.
Two cases of pretreatment of the sample m a y be distinguished:
i. The solid sample has been obtained b y slow/reezing of the liquid.
2. The solid sample has been obtained by sudden/reezing of the liquid.
The present paper deals with gradually frozen samples. The heating curves of
samples obtained b y quick freezing will be discussed in another paper.
Anal. Chim. Acta, 23 (196o) 545-551
546
H.F. VAN WIJK, W. M. SMIT
Relation between temperature and/raction liquid o~ a binary solid-liquid system which
is slowly/rozen
Since only the surface layer of the solid participates in tile thermodynamic equilibrium, the amount of solid involved is negligible compared to the bulk of the solid.
Since diffusion in the liquid is sufficiently large, the total amount of liquid present
at a certain moment participates in the thermodynamic equilibrium,
Suppose that the total amount of liquid equals a fraction y of the total sample and
let the concentration of the liquid be XL (tool-fraction). The fraction of the sample
which is solid is (I - - y). Now suppose that the amount of solid increases with d(I - - y).
The concentration of this tiny deposit of solid is X , which is coexistent with the liquid
concentration XL. Owing to the law of conservation of matter:
X," d(r --y)
= --d
(XLy)
(I)
is valid, hence:
dy
y
dX~
XL - - X ,
(2)
According to thermodynamics (provided X~ and X , are small)
Q
(XL - - X , ) =
RT~
AT
(3)
where
Q
R
T
AT
=
=
=
=
h e a t of I n s i o n i n c a l / m o l .
gas constant in cal/degree mol.
temperature in ~
d e p r e s s i o n of m e l t i n g p o i n t i n ~
Further, if XL is small, the following equation may be considered as a fair approximation:
X ~ = K~ A T
(4)
where KL is a constant. Substitution of eqns. (3) and (4) in eqn. (2) yields:
dy
K L R T 2 d(AT)
y
~2-ar
Integration of the above equation results in:
RT 2
--lny
~
KLlnAT
Q
+ C
(5)
The concentration in the liquid Xr~ will be equal to X (the concentration in the total
sample) when y = I. Since, according to eqn. (4), (AT)u=1 = K~/X, the integration
constant C will be equal to:
C
RT2
KL
-KL i n - - -
Q
(6)
x
Combination of eqns. (5) and (6) results in:
RT 2
in y ~
Q
X
KL In - KLAT
(7)
A n a l . Chim. Acta, 23 (196o) 5 4 5 - 5 5 1
IMPURITY DETERMINATION BY THERMAL ANALYSIS. I.
547
When no solid solutions are formed the constant KL equals Q / R T 2 and eqn. (7) reduces
to the simple and well-known equation of White obtaining to systems which show
no miscibility in the solid phase:
y
RT2
X
(8)
Q AT
I t m a y be useful to note t h a t eqn. (7) is not restricted to solid-liquid systems. I t is
valid for any binary system showing a phase transition under the condition that the
bulk of one of the phases is excluded from further exchange with the other phase.
Since on heating of a slowly frozen sample the reverse process occurs, eqn. (7) is valid
for the heating (melting) curve of a gradually frozen sample.
As was mentioned in the introduction, heating curves are preferred to cooling
curves, because of the lack of thermal equilibrium on cooling. I t is stressed t h a t a
limited lack of thermal equilibrium does not necessary imply perceptable deviations
in the concentration of the coexisting phases. This view is fully confirmed by the
experimental results mentioned below.
EXPERIMENTAL CHECK ON EQUATION (7)
An experimental check on eqn. (7) was carried out with two systems showing solid
solutions, viz. : diphenyl (main component)-phenanthrene and tin (main c o m p o n e n t ) bismuth. Each component was carefully purified beforehand. Then the constant KL
of each system was determined independently of the heating curve:
Several mixtures of each system were prepared b y weighing, followed by thorough
mixing in the liquid phase. The mixtures were frozen and then their/inal melting point
was determined as follows. The (macro) samples were slowly heated and a thermometer was inserted; heating was continued until a minute fraction of solid was
present and then the sample was kept under adiabatic conditions and the temperature
was read. At this moment the concentration of the liquid does not perceptably differ
from the total concentration of the sample. This method yields results which are more
reliable and reproducible than temperature readings taken at the moment the "last"
crystal disappears (at that moment the contacting surface of solid and liquid is too
small).
The final melting points thus found were plotted as a function of the concentration.
The constant KL of each system m a y then be read from the graph according to
eqn. (4).
The heats of fusion (Q) of the main components were derived from the literature 4.
Subsequently, melting curves were determined on samples of known composition
b y means of the thin-film apparatus 5 operating according to the so-called dynamic
method. The pretreatment of the sample consisted of melting the sample followed
b y gradual solidification within 30-60 rain. The results obtained are presented in the
first two columns of the Tables I, I I and I I I . The third columns of these tables mention the depressions calculated with the aid of eqn. (7). Agreement between the temperatures observed and those calculated is striking. (The accuracy of the temperature
readings is not better than o.oi~ For comparison, the depressions calculated on the
basis of the equation which is valid in case of equilibrium between the liquid and the
total mass of solid are listed in the fourth column of each table. This equation reads:
Anal. Chirn. Acta, 23 (I96o) 545-551
548
i~. F. VAN WlJK, W. M. SMIT
Y --
RT~
X
Q
AT
K,RT
+
2
(9)
Q
where K8 is the depression constant of the initial m e l t i n g point. This c o n s t a n t was
calculated from eqn. (3) and the already known value of Kz.
Clearly the difference between the depressions found and those calculated from
eqn. (9) must increase when the fraction liquid (y) decreases. This is confirmed by the
results.
TABLE
I
D E P R E S S I O N S F O U N D AND CALCULATED OF A SAMPLE OF D I F H E N Y L C O N T A I N I N G
O.181 r n o l % OF P H E N A N T H R E N E
Depression A T in ~
Fraction liquid
Y
Found
Calculated on
the basis of local
equilibrium,
(eqn. 7)
Calculated on
the basis of total
equilibrium.
(eqn. 9)
o.I
0.2
0.3
0. 4
o.5
0.6
0. 7
o.41
0.22
o.17
o.14
o.I 3
o.io
o.08
o.4o
0.24
o.17
o.14
o.12
o.io
0.08
0.22
o.18
o.14
o.12
o.io
0.09
0.08
TABLE
II
D E P R E S S I O N S F O U N D AND CALCULATED OF A SAMPLE OF D I P H E N Y L C O N T A I N I N G
0 . 7 6 2 I B o l % OF P H E N A N T H R E N E
Depression AT in ~
Fraction liquid
Y
Found
Calculated on
the basis 03*local
equilibrium
(eqn. 7)
Calculated on
the basis of total
equilibrium
(eqn. 9)
o.I
0.2
0.3
0. 4
0.5
0.6
o.7
1.85
1.o 3
o.77
o.62
o.54
o.47
o.42
1.86
I.O2
o.77
o.61
0.53
0.48
o.43
0.95
o.77
o.64
0.55
0.49
0.43
0.39
TABLE
III
D E P R E S S I O N S F O U N D A N D CALCULATED OF A SAMPLE OF TIN C O N T A I N I N G 0 . 3 0 0 m o l %
OF BISMUTH
Depression A T in ~
Fraction liquid
Y
Found
o.I
0.2
0. 3
0. 4
0. 5
o.6
2.82
1.82
1.42
1.18
~ .02
o.92
Calculated on
Calculated on
the basis of local the basis of total
equilibrium
equilibrium
(eqn. 7)
(eqn. 9)
2.9 ~
1.83
i .41
1.15
i .oo
0.92
1.84
1.4o
1.22
1.o8
0.98
o.9o
Anal.
Chim.
Acta,
23 (196o) 545-551
IMPURITY
DETERMINATION
BY THERMAL
ANALYSIS.
I.
549
DISCUSSION
The above results show t h a t a formula derived on the assumption of local equilibrium
(eq n. 7) fits the experimental melting curves more closely than a formula derived
on the assumption of total equilibrium (eqn. 9). I t m a y now be worthwhile to discuss
the method by which the content of impurity in an unknown sample m a y be derived
from the heating curve on the basis of the local equilibrium equation. Two methods
of impurity determination are normally available, viz. : the so-called absolute and
comparative methods. The absolute method demands only one heating curve. The
data derived from the heating curve when combined with thermodynamic data (the
depression constant and the distribution coefficient which are usually determined
separately) m a y yield the amount of impurity. The comparative method demands
two melting curves, one of the sample as such and one of the same sample after
addition of a known amount of impurity.
Impurity determination by the absolute method
According to eqn. (7) a plot of in y versus In AT gives a straight line. The slope of
this line gives the value of
RT ~
- - - KL
Q
and the line intersects the In y-axis at
RT 2
In y =
Q
KL I n X / K L
These data combined with the value of RT2/Q (derived from the curve or from elsewhere) enable the value of KL and X to be calculated b y simple algebra. This procedure is possible provided that the melting point T8 of the pure main component is
exactly known. When T8 is unknown, its value m a y be derived from the curve in the
following way. Three values of y viz. yl, y2 and ya are chosen and the corresponding
temperatures T1, T2 and Ta are read from the curve. The pertaining values of AT are
T~ - - T1, T8 - - T2 and T~ - - Ta. F r o m eqn. (7) it m a y be derived t h a t :
i n yl/y2
---
I n y2/y~
I n (r8 - - T2)/(Ts - - T1)
=
I n (T8 - - T~)/(T, - - T~)
(lO)
The lefthand term of this equation is known; if its value is a then:
(r,-
r~lo _ r , - r 2
Ts--~]
(T8 - -
T , ) ~+1 =
Ts-(r, --
TI'
T3) a ( T , - -
or
T1)
(II)
Now a series of plausible values of T~ is chosen and the values of the left and right
terms of equation(r~) are plotted as a function of T~. The point of intersection
of the two plots yields the desired value of T,. This method of calculation, the determination of Ts included, has been applied to the data represented in the Tables I, I I
and I I I . The results are listed in Table IV. The results show that this method, although
somewhat laborous, can give satisfactory results.
A n a l . Chim. Acta, 2 3 ( 1 9 6 o ) 5 4 5 - 5 5 1
550
H.F. VAN WIJK, W. M. SMIT
TABLE IV
C O M P A R I S O N O F T H E A M O U N T S OF I M P U R I T Y A D D E D A N D F O U N D B Y C A L C U L A T I O N A C C O R D I N G TO AN
A B S O L U T E M E T H O D B A S E D ON E Q U A T I O N (7)
Substance
I m p u r i t y added
Diphenyt
Diphenyl
Till
o.181 mol %
o.762 mol %
0.300 tool %
Found
o.18 H101%
0.74 11101%
0.28 tool %
Impurity determination by the comparative method
Eqn. (7), rewritten in its exponential form, reads:
RT 2
- - - - .KL
X
Q
or
Q
m.~nr -- ___KL(T, - - T )
X
(Y)
In the following the temperatures obtaining to the heating curve of the sample are
denoted by Tv, whereas those obtaining to the sample to which a known extra molfraction p of the impurity has been added, are denoted as Ty'. The difference of two
temperatures obtaining to the first curve is:
Tel --
Tg"2 ~
KL
( y l ) RT2KL - -
(y2) /r/if, KL
The difference of the corresponding temperatures of the second curve (read at the
same values of yl and y2) is:
Tv 2 -- rv 1
(Yl)RT2KL- - (y2)RT2~L
KL
Thus :
Tv~ - - T v l
,
,
X
(I2)
x+p
Tv, -- Tv 1
from which x m a y be solved. The above equation is identical with the equation previously derived b y one of us 5 on the basis of the assumption of total equilibrium.
CONCLUSIONS
Tile equation obtaining to the comparative method of impurity determination, v i z .
eqn. (r2), is applicable to a gradually frozen sample showing local equilibrium as well
as to a sample showing total equilibrium. However, when the absolute method is
applie d, the equation valid for total equilibrium (eqn. 9) differs from eqn. (7) which
is valid for total equilibrium of gradually frozen samples. These conclusions suggest
a possible explanation of the discrepancy between authors who obtained satisfactory
results on the impurity determination of solid solutions 5 and those who did not6, 7.
Anal. Chim. Acta,
23 (196o) 545-551
IMPURITY DETERMINATION BY THERMAL ANALYSIS. I.
551
SUMMARY
A temperature-heat content relation is presented for binary systems which show solid solutions.
Contrary to previous equations, the present equation accounts for the very restricted rate of
diffusion in the solid. The equation is based on local equilibrium during freezing and melting.
The equation obtaining to the comparative method of impurity determination and applicable
to a gradualIy frozen sample is identical to the equation previously derived on the basis of total
equilibrium. The corresponding equations of the absolute method differ considerably. Experiments
confirm the local equilibrium theory and show the inadequacy of the total equilibrium theory.
R~SUME
Les auteurs ont effectu~ une gtude sur la d~terminafion des impuret~s par analyse thermique,
k l'aide de la courbe de fusion d'un 6chantillon refroidi lentement.
ZUSAMMENFASSUNG
Beschreibung einer Untersuchung fiber die Bestimmung des Reinheitsgrades durch thermische
Analyse der Substanzproben, die einer langsamen Abkfihlung unterworfen waren.
REFERENCES
1 W. M. S~alT, Anal. Chim. Ac/a, 17 (1957) 23.
2 W. M. S~IT, Purity Control by Thermal Analysis, Elsevier, 1957, P. 175.
3 W. M. SMIT, Thesis, Free University, Amsterdam, 1946.
Int. Cr~t. Tables, 2 (1927) 458; 4 (1928)177.
5 W. M. SMIT, Rec. tray. chim., 75 (1956) I3O9.
6 j. p. McCuLLOUGH AND G. WADDINGTON, Anal. Chim. Acta, 17 (1957) 80.
7 j. H. B~.DLEY, J. Phys. Chem., 63 (1959) 1991.
Anal. Chim. Acta, 23 (196o) 545-551
DETECTION AND COLORIMETRIC DETERMINATION
OF POLYNITRO AROMATIC COMPOUNDS
E U G E N E SAWlCKI AND THOMAS W. STANLEY
A i r Pollution Engineering Research Robert A. Taft Sanitary Engineering Center
Public Health Service U.S. Department o[ Health, Education, and Wel[are
Cincinnati 26, Ohio (U.S.A.)
(Received June 22nd, 196o )
INTRODUCTION
Some new tests for polynitro compounds are investigated in this paper. As the literature on the utilization of polynitro organic compounds in organic analysis is so extremely extensive, any new analytical methods for these compounds should be capable
of extensive modification and application.
Benzene, which interferes with hematopoiesis, has been determined in the air by
nitration to m-dinitrobenzene followed b y reaction with alkaline butanone or acetone
and measurement of the resultant violet color I 3.
In a similar fashion, toluene, xylene, and chlorobenzene have also been determined
b y nitration to a m-dinitrobenzene derivative4, 5. Recently a nitration test for polynuclear hydrocarbons was described in which pyrene, benzo[aipyrene, chrysene,
Anal. Chim. Acta, 23 (196o) 551-556