interferences of Cra+,Fea+,and C12+ are diminished to some extent by all three systems. The enhancement effect of Mn2+ is not appreciably diminished by any of the systems. At the 400-ppm level, most interferences are decreased by the chelating agents ; however, the inteferences of certain metal ions are increased. The interference of Pb2+ is significantly increased by the 0.5 % Nas EDTA-5.Oz sodium potassium tartrate system. These findings indicate an inherent weakness of the color development system, that of the metal ion interference. The use of EDTA or sodium potassium tartrate results in a significant reduction in interference with some metal ions; no change in the interference of others; and with certain other ions, enhances the interference. In this comprehensive method, chelating agents were not included because sample composition is often unknown or widely varied. This is consistent with the conclusions of others-(7, 8, IO)-which have elected not to include chelating agents in the color development system, although 0.5 % EDTA and 5.0 % sodium (10) J. R. Todd and J. H. Byars, Auromat. Anal. Chem. Technicon Symp., 1, 140 (1967).
potassium tartrate system in the 35 sodium hydroxide solution is now recommended by Uhl(9). CONCLUSIONS
Sensitivity has been substantially increased as a result of optimizing reagent ratios and increasing the proportion of digested sample being analyzed. This eliminates the need for a specially built longer flow-cell and its related problems. The digestion procedure has been improved to eliminate the need for “similar matrix” standards and enables one to use the same digestion procedure for a variety of nitrogen compounds. Liquid samples containing up to 10% carbohydrate can be digested without interference or further dilution. In more than five years of routine laboratory use, we have found ammonium sulfate standards directly correlate with all types of proteins analyzed. This enables one to confidently analyze proteins of unknown types as well as mixtures of several proteins in one sample.
RECEIVED for review May 16, 1972. Accepted September 5, 1972.
On the Drawing of the Base Line for Differential Scanning Calorimetric Calculation of Heats of Transition Charles M. Guttman and Joseph H. Flynn Institute for Materials Research, National Bureau of Standards, Washington, D.C. 20234
IN THE DIFFERENTIAL SCANNING CALORIMETRIC method (DSC), the heat of transition of a pure substance is determined, from the area under the differential power-time curve. How one should draw the base line in determining the heat of transition has been either ignored (1-4, arbitrarily posited ( 5 , 6 ) , or based on the author’s conception of what is happening during the transition process (7-10). The uncertainty in how to draw the base line usually arises from the fact that the heat capacity of the sample after the transition is different from that before the transition. Generally since this change in heat capacity gives rise to heats quite small compared to the heat of transition, the method used to determine the base line may (1) E. S. Watson, M. J. ONeill, J. Justin, and N. Brenner, ANAL. CHEM.,36, 1233 (1964). (2) M. J. O’Neill, ibid., p 1238.
(3) A. P. Gray, “Analytical Chemistry,” R. S . Porter and J. R. Johnson, Ed., Plenum Press, New York, N.Y., 1968, p 209. (4) “Thermal Analysis News Letter” Analytical Division, PerkinElmer Corp.. Norwalk, Conn., Nos. 1-9 (1965-70). ( 5 ) H. G. McAdie, “Report of the Committee on Standardization,” International Confederation for Thermal Analysis, Abstracts of Third International Conference on Thermal Analysis, Davos, Switzerland, August 1971, p 11. (6) R. C. MacKenzie, Chairman, “Report of Nomenclature Committee,” ibid., p 3. (7) W. P. Brennan, B. Miller, and J. C. Whitwell, l i d . Eng. Clzem., Fundurn., 8, 314 (1969). (8) W. P. Brennan, PhD. Thesis, Princeton University, 1970. (9) G. Adam and F. H. Muller, Kolloid-Z. Z. Polym., 192, 29 (1963). (10) A. Engelter, ibitl., 205, 102 (1964). 408
not be critical within the limits of experimental accuracy for available instruments. However, advances in DSC instrumentation (11)should increase the precision of the heat measurements and thus make significant the correct determination of the base line. Furthermore, some cases do exist where the heat capacity of the sample before the transition differs considerably from its heat capacity after the transition. Generally these arise from a change in sample mass. Such transitions may involve the sudden evolution of a gaseous product, combination with a gaseous atmosphere, or vaporization of the sample itself (4). For such cases, the correct drawing of the base line is of importance. In this note we shall present arguments suggesting that the correct base line can be obtained by extrapolating the heat capacities of the initial and final temperature state to the thermodynamic transition temperature. The heat of transition is then the area under the differential power-time curve using this base line. It will be shown that this procedure gives heats which are largely independent of the kinetic processes of transition for transitions that occur at a single thermodynamic temperature. In the first part of the note we shall describe the traces from a DSC; then we shall describe the drawing of a base line when one does not have to worry about machine time constants. Finally, we shall include the effect of machine time constants in our discussion. (11) M. J. O”eil1 and A. P. Gray, “Design Considerations in Advanced Systems for Differential Scanning Calorimetry,” Proceedings, Third International Conf. on Thermal Analysis, Davos, Switzerland, 1971, to be published.
ANALYTICAL CHEMISTRY, VOL. 45, NO. 2, FEBRUARY 1973
Detailed reviews and critiques of the operation of the DSC appear elsewhere (1-4, 8, 12, 13). Here we wish only to develop the simplest equations for the operation of the DSC. The heat supplied to each DSC cup can be divided into two terms, the heat going into the sample itself and the rest of the heat supplied to the cup. Thus we have for the sample cup
while for the reference cup we have
bq, where - and - are the rates of heat put into the sample
bt
bt
container and reference container, respectively, from the bh, bh, heaters imbedded in the cups. - and - are the rates of bl at heat input into the sample or reference material (heating or changing of phase). L, is the term which includes the power to heat the sample container and cup as well as the heat lost through the heat leak. L, is the corresponding term for the reference cell. The DSC recorder traces the difference in power supplied - - bqr With no sample in to the heaters in each cup, bqs bt at’ the sample or reference cup, we obtain the trace described by the quantity 61
from some arbitrary origin. We assume in this paper that 61 is a smooth reproducible function of the experimental conditions, Le., heating rate, temperature range of interest, etc. Upon the addition of material to the sample and reference cups, we have 62 for a DSC trace
A
Area (ABCDE)
=
K
where K is the constant of proportionality. K is obtainable by calibration agftinst enthalpies of known substances. From here on in the paper, we assume all areas divided by K . Now let us define the steady state true temperature of the sample at t1to be Tl, and that at tz to be T2s. TIS and Tzs as well as T,, are true thermodynamic temperatures ; generally they are not known to the experimentalist. By contrast Tlcand Tz,,the calibrated temperature (discussed later) are known via temperature calibration of the instrument. The enthalpy difference between the system at these temperatures is by thermodynamic arguments Area (ABCDE)
=
hs(f2)- h,(tl)
=
C,I dT
+
JTT
CPz dT
+ h,
(5)
where Tt, is the transition temperature, Cpl and Cpzdesignate the heat capacity of the two arbitrarily different states, 1 and 2 and h f is the heat of transition. Notice we chose TISand Tzs far enough away from the transition temperature so that we are away from any kinetic phenomena associated with the transition (machine or intrinsic). With Equation 5 and a knowledge of the heat capacity of state 1 and 2 as weil as of , TzS)we can easily get the sample temperatures (TI,, T Z sand heat of transition. Usually these quantities are not exactly known; the rest of the paper will be devoted to exploring why we may not need to obtain them. Let us first consider the problem of obtaining the temperatures of the samples, Tl,, T2,, and TtS. The DSC machine reads an average temperature which usually is not the sample temperature. The machine temperature scale is calibrated using the melting points of known substances in the temperature region of interest at the heating rate of interest. If the calibration is careful, then we may expect that the calibrated temperatures, TI^, T2,, and T,,, are fairly close to the true thermodynamic temperatures of the sample Tl,, T2,,and T,,, respectively. Thus we have for Equation 5 Area (ABCDE) = hf
(12) J. H. Flynn, “Status of Thermal Analysis,” 0. Menis, Ed., N B S Spec. Publ., 338, 119 (1970). ( 1 3) J. H. Flynn, Proceedings, ‘Third International Conf. on Thermal Analysis, Davos, Switzerland, 1971, to be published.
t2
Figure 1. DSC thermogram of system with no samples in cups (AE) and of system with material in sample cup which has a phase transition and a change of heat capacity as one goes from state 1 to state 2 (BCD). (Times tl and tz correspond, respectively, at constant heating rate, to temperatures TI, and Tz~)
JT::
where we have assumed that the heat loss is unchanged (within experimental error) by the addition of the samples. In Figure 1 we show the thermogram for a substance which has a heat capacity change in the region of temperature T t (which is between Tl and TJ as well as a heat of transition at T 1 . We have assumed a rate of heating and an endothermic transition in this discussion and Figures 1-3. The same arguments and equations apply, mutatis mutandis, to cases in which the transition is exothermic or a constant rate of cooling is employed. The line designated AE in Figure 1 is the power level when the machine has no material in the sample or reference container and is proportional to 61. With some material in the sample container and no material in the reference container, one obtains the line designated BCD which is proportional to &. From the previous equations, the area (ABCDE) is
E
t-
tl
f
=
-AlCp1(Tls)
+
STY:
Cpl dT
+
+ AtCpl(Tts) AtCpdTts)
sT:lc + C p 2dT
f
+ A2CpdT~s) (6)
ANALYTICAL CHEMISTRY, VOL. 45, NO. 2, FEBRUARY 1973
409
ture calibration is off by an additive constant then
A = A,
=
A,
A2
(8)
and
tI
t--+
t2
t3
Figure 2. Drawing the correct base line on a DSC thermogram for a heat of transition, the cross hatched area is the heat of transition except for the small correction into the circled area. This correction is shown in Figure 3
which is generally a small quantity compared to h,. If the heat capacity is essentially constant in each of the states, f i s zero for this case. Equation 6 still requires that we know the heat capacity of the two states. In Figures 2 and 3 we have applied Equation 6 to obtain the h, (the cross hatched area) assuming that (a) the base line is constant and stable, (b) the heat capacity of the final state can be extrapolated as we have by line D H (assuming it is constant), (c) the temperature calibration was obtained by drawing lines similar to KC on Figure 2 for various calibrating compounds and have identified the point K with the transition point, and (d) the errors due to f i n Equation 6 can be disregarded. We have noticed in drawing our base line that area (EDGF) in Figure 2 is given by
and, by analogy to Equation 10, the area IHDE by AREA(1HDE)
=
and area ABKI by AREA (ABKI) = Thus, using Equation 6 with f AREA (ABCDE)
ST::
(1 1)
ST::
(12)
CPz dT
C,I dT
=
0, we have
- AREA (IHDE) - AREA (ABKI)
(13)
= hj
Figure 3. The correction to the heat of transition given in Figure 2 for two extreme cases. For case 1 we add the cross hatched area in Figure 3 to the cross hatched area in Figure 2 and subtract the grayed area. For case 2 we add the cross hatched area in Figure 3 to the cross hatched area in Figure 2
(7) and where Cpl(Tls) and Cpl(Tts) are the heat capacities of state 1 at T I Sand T,, respectively, and Cp2(Tls) and C,Z(T,,) are the heat capacities of state 2 at T,, and Tzsrrespectively. We have assumed that the change in the heat capacity of the sample for all the A’s is small. The reader should note that f i s the error due to the fact that the temperature scale is not thermodynamic. Some simple cases will be noted. If the tempera410
Thus except for a small correction due to the areas enclosed in the circle in Figure 2, the heat of fusion is given by the cross hatched area of Figure 2. If one does not obtain the base line AIEF (as is common in h , determination), one can obtain the heat of fusion simply by extrapolating back D G to the transition temperature (as long as one assumes D G is parallel to the base line). Figure 3 gives details on how to take into account the areas in the circle in Figure 2 correctly if such corrections are needed. The cross hatched areas should be added to h, and the grayed areas subtracted. In this approach we have tried to exploit as much as possible the fact that the heat of transition is a thermodynamic quantity as defined in Equation 4 and expressing the enthalpy change between initial and final states. It is determined in this paper through the extrapolation of the heat capacities of the initial and final states to the same, though perhaps arbitrary, transition temperature. We have avoided as much as possible any definition and determination of a “heat of transition” involving the kinetics of heat flow over a temperature interval since this will be subject to the vagary of experimental design and procedure. Goldberg and Prosen (14) are presenting a generalized development of the above ideas applied to other types of scanning calorimetry. RECEIVED for review May 17, 1972. Accepted September 14, 1972. (14) R . N. Goldberg and E. J. Prosen, Thermochimica Acta, in press.
ANALYTICAL CHEMISTRY, VOL. 45, NO. 2, FEBRUARY 1973
