1474
Anal. Chem. 1980, 52, 1474-1479
Automated Purity Determination by Stepwise Melting in Differential Scanning Calorimetry Arnold C. Ramsland Hoffmann-La Roche Inc., Research and Diagnostic Products Section, Quality Control Department, Nutley, New Jersey 07 1 10
Purity determinatlons are limited to systems which do not form solid solutions and to samples of hlgh purity by approximations implicit in the commonly used van7 Hoff equation. Thls paper describes purlty determination based upon a more nearly exact equation which uses a partitlon coefficient to describe the distribution of impurity between the solid and iiquld phases of the main component. A dedicated computer controls stepwlse melting and evaluates calorimeter data. Heat capacity contributlons of the solid and llquM phases and instrumental blanks are taken into consideration to yield accurate values of the fraction melted at various temperatures. An iteratlve, multiple regression analysis of these values yields an accurate determination of the mole fraction of the major component, the melting point of the pure major component, and the partition coefficient. Accurate purity determination of samples doped to be 90 mol % pure have been obtained wtthout any arbitrary data manipulation.
Most previous work in purity determination using thermal analysis is based upon the following "van't Hoff" equation:
See Table I for definition of all symbols. Three approximations used in deriving this equation are as follows:
A. X z s = 0
B. In X l l = - X i C. TsTo= To2 These approximations determine limits of applicability and accuracy of the "van't Hoff' equation. Approximation A limits purity determination to systems which do not form solid solutions. Approximation B is in general very poor at the eutectic composition where the mole fraction of the major and minor component in the liquid phase would be approximately equal. Approximation B limits investigation to high purity samples and low 1 / F values. Approximation C is good except at cryogenic temperatures. T h e objectives of this work have been to derive a more nearly exact equation which does not use the three above approximations, to devise a method of solving the equation, and to test the predictions of the equation experimentally in various cases.
Table I. Definition of Symbols component 1 = major component component 2 = total of all minor components N o = total number of moles in sample N , = number of moles of component 1 in original sample N, = number of moles of component 1 in liquid phase N, = number of moles of component 1 in solid phase N,' = number of moles of component 2 in liquid phase N, = number of moles of component 2 in solid phase x, = mole fraction of component 1 in liquid phase x, = mole fraction of component 1 in solid phase X,'= mole fraction of component 2 in liquid phase x,S = mole fraction of component 2 in solid phase X,t = mole fraction of component 1 in original sample x,t = mole fraction of component 2 in original sample F = fraction melted s * = chemical potential of pure solid component 1 p il* = chemical potential of pure liquid component 1 Ts= sample temperature after temperature equilibration (K) To= melting point of pure component 1 ( K ) AHF = heat of fusion of component 1 Equations similar to Equation 1 have been developed by others ( I ) ; however, at least one of the approximations listed in the introduction have been used in the derivation. Applying approximations A, B, and C to Equation 1yields the following equation traditionally used in purity determinations:
Appendix I1 develops the following new equation for purity determination which does not use approximations A, B, or C:
" ' 1 L X l- t - l + K + F - K F 4I
K is a partition coefficient defined by van't Hoff (2) to be K = X2s/X21 Solving for
T,in Equation 3 yields
THEORETICAL More E x a c t Equation for P u r i t y Determination. Purity determination in this paper is based upon the following equation which is derived in Appendix I:
RT,To XIB T , = To - -In --i m f
x 1
''
[
KXlt - K F
+F
1\ (4)
Determination of P u r i t y by Iterative, M u l t i p l e ReA H f , XIt,F , and gression. Equation 4 predicts T,when To, K are known. The equation, however, is not suitable for
0003-2700/80/0352-1474$01.00/00 1980 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 52, NO. 9, AUGUST 1980
w
h
0 FGACT:CY
1
(X I@>
VELTEO
1475
3
2
4
5
6
A 7
8
9 1 0
FGACTiON MELTED (X 10)
Figure 1. Three dimensional representation showing the error correction variable ( L ) as a function of the fraction melted and the partition coefficient ( K ) . Equation 6 used to calculate L with X , ‘ = 0.95
Figure 2. Three dimensional representation showing the error correction variable ( L ) as a function of the fraction melted and the mol % purity. Equation 6 used to calculate L with K = 0
determining X1‘, K , and To when AHf, F , and T , are experimentally determined. Equation 3 has therefore been rewritten in the following equivalent form:
The value of L a t any F is not known a priori; therefore, it is initially estimated to be a constant, L’, which can be combined with J to give a new constant J‘ such that:
T , = To +
J‘=
RT,To In Xlt AHf(J + F - L )
(5)
L
Substitution into Equation 5 yields
T , = To +
where
= L(Xlt, F ,K)
In XIt 5
KXlt - KF + F In X I t - 1 + K + F - KF In Xlt
J = J(XC, K) E .
In
KXlt-K+l
+F+J
-1
J-L’ RT,To In Xlt AHf(J’ + F)
Appendix I11 A shows that Equation 9 can be expressed as
T , = To + B’T, (7)
1
-
F
+ B’,
1
F
and that a multiple regression fit of T , vs. 1 / F will yield values of X:, J‘, and To from an analysis of the coefficients. An expression defining K’ analogous to Equation 8 is
Xlt Solving for K yields:
Substitution of Equation I into Equation 6 demonstrates that
L can be expressed as a function of only Xlt, F , and K . J is a constant in any system because K and X: are both constant. L is defined to be a function which corrects for errors associated with the removal of F and K from the logarithmic term in Equation 3. L is equal to 0 when F equals 1. Figure 1 shows L as a function of F and K when X : = 0.95. Figure 2 shows L as a function of F and X : when K = 0. The graphs demonstrate that despite the complicated formula, L can be simply approximated as: (1)decreasing almost linearly with increasing F , (2) decreasing almost linearly with increasing X:, and (3) being almost independent of K. As explained in the section entitled “Eutectic Formation”, L i s undefined when
XIt- 1 + K
+F -
KF C 0
This explains why the upper left portions of the graphs are missing . Because L is an error function, it can be neglected only when F >> L , Le., a t high values of F and Xlt. Consequently, if one neglects L as is done when Equation 2 is used, the error in purity determination will increase with increasing impurity and decreasing values of fraction melted. However, if one can correctly determine L , purity determinations will no longer be restricted to high purity systems and narrow ranges of low 1 / F values.
It is now possible to approximate L a t each F by inserting the values of X:, K’, and J ‘ into Equation 6. Appendix 111 B shows that a multiple regression fit of T , vs. 1 / ( F - L ) of the form
T , = To + BT, 1/(F - L ) + Bz 1 / ( F - L )
(12)
will yield values of XIt, J, and To from an analysis of the coefficients. Since the value of L a t each F is approximate, however, the initial estimates of XIt,J, To,and K will also be. Exact values of X:, To,L, J , and K can be obtained by an iteration which will improve the estimates of each variable in each loop. When a value of XIt is within a minimal range (Le., 0.005%) of a previous determination, the loop is stopped, and the final values of X:, To,and K are used in a purity determination. Effect of L a n d K on Plot of T,vs. I / F . Figures 3 and 4 show plots of T , vs. 1/F as predicted by Equation 4. T h e effect of L can be seen with eutectic systems (K = 0). L causes the negative slope of T , vs. 1 / F to become more negative with increasing 1/F. The effect of K can be seen on the plots of solid solutions where K = 0.05 and 0.1. Increasing K causes the slope of T,vs. 1 / F to become less negative with increasing 1 / F . One can see that a small change in K can greatly affect the plot of T , vs. 1 / F . Effect of Fusion Area Change upon Determination of To,Xlt, a n d K . In a purity determination, an iterative, multiple regression fit is done using T,, 1 / F pairs to determine X t , To,and K . T h e first and last T,, 1/F pairs used for determination are defiied to be TsM,l/F.,,, and Tend, l/Fe,+. The fusion area prior to the peak equilibrating a t T,, 1s
1476
ANALYTICAL CHEMISTRY, VOL. 52, NO. 9, AUGUST
1980
G
4E3 T
4a.
t
399
t
3c7
I
v
6
-
' ,
-
- 4
e.1
\
/
\
....6 = 0.05
_ _
r:
I\
/ \
\
= 0.0
I
\
\
//-=--.-
.--
361 389
\ \
t 4 E
C' 2
4
6
8
:2
le
l/FRACTIOIU
14
18
15
20
I
=
__
0.1
K
=
+ F > KF + X Z t
K > X Z t for all F
O.a
Thus, if K < X2,, all values of F are not observable. If K = 0, F must be greater than Xzt. This is, in fact, observed in a eutectic melt where all the impurity plus some of the main component melts. In solid solutions, a eutectic is not necessary if K > Xzt; however, it is necessary if K < X;. The appearance of a eutectic with increasing X2tis observed in a Rozeboom Type V solid solution ( 3 ) . 1
2
3
4
5
6
7
8
9
10
l/FRACTIGh M E L T E D
Figure 4. Effect of Kand X,' (by comparison to Figure 3) upon T, as a function of 1/F. T, calculated using Equation 4 where T o = 403 K , AH, = 6000 cal/mol, and X,' = 0.95
defined to be H,,, and the fusion area subsequent to the peak equilibrating at Tendis defined as HWt The derivation shown in Appendix IV proves that changes in H,,, and/or H,, have no effect upon the determination of Xc and Toin the multiple regression fit of T, vs. 1 / F based upon Equation 10. The Results section shows that changes in Hpreand/or Hpost will also have little effect upon the calculation of Xlt and Toin the iterative, multiple regression fit of T , vs. 1 / ( F - L ) based upon Equation 12. Eutectic Formation. Equation 3 can be rewritten as
[
T , = To - RTsTo fi€ In
KXlt + F ( l - K ) X l t - 1 + K + F(l - K )
< K < 1 and F < 0 KXlt
+ F(1
-
9 1,
1
K ) > 0 for all X l t
Therefore, the numerator is always positive. In addition, since
KX2t < X i -KX2' > -XZt K - KXzt
XZt
+ F(l
-
K)
K ) > K - 1 + XI'
+ F(1
-
K)
+ F(l
-
K) > K
-
and
KXlt
straints are placed upon the values of Xzt,K , and F. Since the denominator in the logarithmic term can be expressed as K f F - K F - X;, the following must hold: or
I
When 0
D
Figure 5. Method of peak integration using C'ABC as t h e base line
K -K
...._ r: = 0 . 0 5
389 4 0
E
i
VEL'Eg
Figure 3. Effect of Kupon T, as a function of 1/F. T, calculated using Equation 4 where T o = 403 K, AH, = 6000 cal/mol,and X,' = 0.99 403
A
I
+ F(l
-
Thus, the numerator will always be larger than the denominator and melting point depression will occur. Because the numerator is always positive in the logarithmic term, the denominator must also be greater than zero so that the logarithmic term does not become undefined. Therefore, con-
EXPERIMENTAL Instrumentation. The instrumentation used consists of three subsyskms. The fvst subsystem consists of a Perkin-Elmer DSC-2 differential scanning calorimeter equipped with a Lauda K-2/R heating/cooling circulating bath for subambient work and for better thermal stability of the sample holder block. A Hewlett-Packard 17505A strip chart recorder displays the output of the DSC. The second subsystem consists of a Princeton Applied Research Model 131 data acquisition system. A 1-s quartz clock has been substituted for the clock provided. The third subsystem consists of a Hewlett-Packard 9830A computer which controls temperature steps, integrates peaks, and evaluates data. The computer is programmed in Basic to either heat or cool in temperature steps entered before at the beginning of a run. Experimental Conditions. All samples are run under nitrogen atmosphere to minimize decomposition. The samples are tightly packed into volatile sample pans under nitrogen to provide good contact throughout the sample. The sample is weighed before and after each run to ensure that no weight loss has occurred. The range setting is 0.5 mcal/s for the DSC and the range/accuracy setting for the PAR is 1 V/O.l%. A 50-mV recorder range has been used t o ensure that all peaks are kept within range: feedback from the recorder causes peak areas to change when the recorder pen is off scale. It has been found necessary to heat to a desired initial temperature because a slightly higher temperature is obtained (due to instrument design) by cooling to a lower limit. Peak Integration. Peak integration is carried out during an analysis using Simpson's rule. Prior to the start of heating at points A or C in Figure 5 , the slope of the previous 30 points must be within a desired minimal range. In addition, the computer will not determine the peak ending at point C until after a desired equilibration time. In Figure 5 , line AD is used as a temporary base line for area determination. Then, after point C has been established, area ABCDA is subtracted to give the peak area. It has been found that the much more accurate and reproducible results are obtained by subtracting ABCDA instead of ACDA as is traditionally done. Sample Preparation. The sample is prepared by first melting a weighed (ca. 400 mg) mixture of major component and minor component(s)under a nitrogen atmosphere. The sample is melted in a 1 X 1 / 2 inch stainless-steel grinding vial (Spex Industries) which has been placed in a heating block. A glass slide glued to
ANALYTICAL CHEMISTRY, VOL. 52, NO. 9, AUGUST 1980
a 'l2-inch i.d. washer is used as a cover for visual examination of melting. The melt is stirred by placing the heating block on top of a magnetic stirrer to spin the '/,-inch steel ball. Next, 0.3 to 0.4 mL of 95% ethanol is added to the cooled melt and the sample is ground for 2 min in a Spex Mixer/Mill (Model 5000). The sample is dried under vacuum a t 25 " C and then reground for 1 min to form a fine crystalline powder. This method of sample preparation has two advantages: (1) The error associated with taking a very small sample to represent a large lot is reduced because a 5-mg sample is, in effect, a sampling of a homogeneous 400-mg sample. (2) This method of preparation forms a homogeneous melt which eliminates the crystalline history of the sample. Therefore, after melting, doped samples prepared in the laboratory will become identical to actual production lots containing the same minor components. Determination of Heat of Fusion. The heat of fusion of a sample is the summation of the fusion area contribution of the individual temperature steps. Experimentally, the area under the curve of a single step is the sum of the following three contributions: (A) The difference in heat capacity between the sample and reference containers. (B) The heat capacity of the sample. (C) The heat of fusion of the sample. To determine the energy requirement for fusion in each step, the first two contributions must be subtracted out. The value of contribution A is determined by actually heating the empty sample and reference pans in large temperature steps &e., 30 "C) over the range in which the sample will be investigated. The sum of the heat capacity contributions of both the solid and liquid phases determines the value of contribution B. As a first approximation, the heat capacity of the solid a t any temperature is calculated and assumed to be equal to the heat capacity of the liquid. The difference between the first blank area and the first sample area divided by the step interval is the heat capacity of the solid a t the middle of the first temperature step. A linear heat capacity change as a function of temperature is then applied to obtain the heat capacity of the solid at any other temperature. Preferably, one should determine the heat capacity correction beforehand using a pure standard to ensure no fusion occurs; however, this value can be estimated in a purity determination if the sample is sufficiently pure. The mean temperature in each step is used for all heat capacity calculations. Thus, knowing the contribution of the empty pans and the solid heat capacity in each step, one can make an estimation of the fusion area in each step and, consequently, 1/F for each T,. Next, the difference in heat capacity between the solid and liquid phases is taken into account by heating the sample an additional step after it has completely melted. The calculated area divided by the final step interval is the liquid heat capacity One can calculate the correction in any previous correction, H,, temperature step using the following integral:
SHcorrF d T The integral can be approximated in each step interval, A T , by
AT
Hcorr i=1 X Fi n
Estimates of Fi are made by assuming a linear relation between T. and 1/F. The final calculation of the fusion area in each step is the initial estimate minus the above corrections.
RESULTS AND DISCUSSION Effect of Early Melting upon Purity Determination. Small errors in the measurement of early melting exhibit a very large effect upon the determination of XItand Toin a linear regression fit of T,vs. 1/F. The effect of changing H,,, can be seen by comparing part B-1 in Table I1 with part B-2 in Table 111. The calculated T,,1/F pairs listed in Table I1 have been altered by a change in H,,, (-300 cal/mol) to give new T,,1/F pairs listed in Table 111. This change, which represents only 5 % of the total heat of fusion results in a 5.05% change in purity and a 6.23 "C change in To. The reason for this is that large values of 1/F are greatly altered by small changes in F; hence, the plot is markedly curved. Previous investigators have used arbitrary data manipulations,
T A B L E 11.
COMPARISON OF L I N E A R REGRESSION ( 8 1).
REGRESSION (C 1).
AND I T E R A T I V E .
1477
MULTIPLE
M U L T I P L E REGRESSION (D 1)
l/F P A I R S CALCULATED FROM EOUATION 4.
U S I N G I D E A L TS.
PART A NUMBER
T ZERO
MOLE F R A C T I O N I M P U R I T Y
420
1
K VALUE
0.1000
0.0000
PART 8 1 1 / F 6. 000
TEMPERATURE
372. 523 1 383.8684 392. 1383 400. 1487 407. 3569
5.000 4.000 3.000 2.000 HEAT OF F U S I O N (CAL/MOLE)
PURITY (MOLE X )
6000.080
85. 549
CORR.
TEMP.
CORRECTION
1. 1744 -0. 6961 -1.0912 -0. 4268 1. 0348
373.6975 382. 3723 391. 0471 399. 7218 488. 3966 STANDARO T ZERO OEVIATION 1. 1997 425.75
PART C 1 1 / F 6. 000
TEMPERATURE
372.5231 383. 8684 392. 1383 400. 1487 487.3569
5.000 4.000 3.000 2.080 HEAT OF F U S I O N (CAL/MOLE)
PURITY (MOLE X )
6000. 000
91.081
CORR.
TEMP.
CORRECTION
372. 5494 383.0122 392. 1477 400. 1932 407. 3329
0. 0263 -0. 0562 0. 0094 0. 0445 -0. 0240
STANDARO T ZERO OEVIATION
0. 0570
K PRIMEO
419. 44
-0. 07134
PART D 1 1/(F
-
L)
TEMPERATURE
8.697 6.579 4.848 3.385 2.118
372.5231 383. 0684 392. 1383 400. 1487 407.3569
HEAT OF F U S I O N (CAL/MOLE>
PURITY (MOLE X )
6000. 000
90. 000
T A B L E 111.
CORR.
TEMP.
CORRECTION
372. 5231 383. 0684 392. 1383 400. 1487 407. 3568
0.0000 -0.0000 0.0000 0.0000 -0.0000
STANDARO T ZERO OEVIATION
0. 0000
K VALUE
420. 00
0. 00000
EFFECT (BY COMPARISON TO T A B L E 11) UPON L I N E A R
REGRESSION ( 8 2 VS. AND I T E R A T I V E .
8 1).
M U L T I P L E REGRESSION (C 2 VS.
M U L T I P L E REGRESSION (0 2 VS.
D 1)
C 1).
DUE TO
SUBTRACTING 300 CAL/MOLE FROM EARLY MELTING.
PART
B 2
1 / F
TEMPERATURE
8. 143 6.333 4.750 3.353 2. I l l
372.5231 383.0684 392. I383 480. 1487 407.3569
HEAT OF F U S I O N (CAL/MOLE)
PURITY (MOLE XI
5708. 000
90.601
CORR.
TEMP.
372. 5654 383. 0006 392. 1315 488. 1882 407. 3496
CORRECTION
0. 0423 -0. 0678 -0. 0068 0.0395 -0.0072
STANOARU T ZERO OEVIATION
8. 0518
419. 52
PART C 2 l / F
TEMPERATURE
8. 143 6. 333 4. 750
372.5231 383.0684 392. 1383 480. 1487 407.3569
3.353 2.111 HEAT OF F U S I O N '(CAL/MOLE)
PURITY (MOLE X )
5780. 000
91. 098
CORR.
TEMP.
372. 5436 383. 0159 392. 1529 400. 1949 407. 3276 STANOARU T ZERO DEVIATION
0. 0566
419. 42
CORRECTION
0.0208 -0. 0525 0. 0147 0. 0463 -0.0292 K PRIMEO
-0.81538
PART 0 2
l/(F
-
L)
14. 381 9. 235 6. 053 3. 863 2. 249
TEMPERATURE
372.5231 383. 0684 392. 1383 400. 1487 407.3569
HEAT OF F U S I O N
