NAT L INST. OF STAND & TECH NIST

PUBLICATIONSI

AlllDt 13St11

NBSIR 76-1 147

A Combined Least Sums and

Least Squares Approach to the

Evaluation of Thermodynamic

Data Networks

D. Garvin, V. B. Parker, D. D. Wagman and W. H. Evans

Physical Chemistry Division

Institute for Materials Research

National Bureau of Standards

Washsington, D. C. 20234

July 1976

Interim Report

)ared for fO 0 =ice of Standard Reference Data, NBS

NBSIR 76-1 147

A COMBINED LEASTSUMS AND

LEAST SQUARES APPROACH TO THE

EVALUATION OF THERMODYNAMIC

DATA NETWORKS

D. Garvin, V. B. Parker, D. D. Wagman and W. H. Evans

Physical Chemistry Division

Institute for Materials Research

National Bureau of Standards

Washington, D. C. 20234

July 1976

Interim Report

Prepared for Office of Standard Reference Data, NBS

U.S. DEPARTMENT OF COMMERCE. Elliot L Richardson, Secretary

Edward O. Vetter, Under Secretary

Dr. Betsy Ancker-Johnson, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director
I
Table of Contents

I. Introduction 1

II. The Type of Problem 5

Table I. Types of Thermodynamic Measurements 8

III. Data Evaluation 9

IV. The Sequential Method 10

V. Advantages of Using Computer Techniques 13

VI. Computer Assisted Data Evaluation 15

VII. Comparison of Methods: Goodness of Fit 18

Table 2, Statistics on Comparisons of Sequential and 21

Simultaneous Solutions. "Residuals" are Observed - Calculated Values on Measurements

VIII. Discussion 22

References 24

Figures 26

A Combined Least Sums and Least Squares Approach to the Evaluation

of Thermodynamic Data Net^.-^orks

David Garvin, Vivian B. Parker, Donald D. wagman and William H. Evans

National Bureau of Standards

Washington, D. C. 20234

Abstract

A description is given of a system for computer-based evaluation

of interrelated thermodynamic measurements of enthalpies of reaction,

equilibria and entropies. This system, is an extension of the CATCH

program developed by J. B. Pedley, University of Sussex. In the new system linear least sums and least squares techniques are used to

solve networks of thermodynamic equations to obtain the enthalpies

and free energies of formation and the entropies of chemical substances.

The least sums technique is shown to be useful in assessing the

consistency of the data. A method combining least sums and least

squares solutions, provides a weighted solution that reproduces

closely the solutions that are obtained by a detailed analysis of the

data using the customary sequential procedure. The results from tests

on four large networks involving compounds of B, U, Rb and salts of

Sn, Pb, Cd and Hg are discussed.

Keywords: Computerized Data Analysis, Daca Evaluation^ Least Squares ''L2),

Least Sums (LI), Thermochemistry, Thermodynamic Data Networks.

I . Introduction

Tables of thermodynamic properties of substances are used widely

for the prediction of the energetics of chemical processes and of the

yields at equilibrium. Typically these tables list enthalpy of

Presented at the 5th International CODATA Conference, Boulder, Colorado

June 28 -July 1, 1976.

1
formation, £hEf, Gibbs energy of formation, AjGf^ entropy, S, and heat

capacity, Cp, at a reference temperature, usually 298.15 K, the

enthalpy of formation at 0 K, ZXHf ° , and the change in enthalpy between

0 and 298 K, H° (298) -H° (0) . They may also list properties at

transition temperatures. Other tables give thermal functions, the

thermodynamic properties H, G, S, and Cp as a function of temperature.

Construction of these reference data tables is an exercise in the

art of data evaluation. Often the properties of a substance can be

determined using several different sets of measurements. The

experiments usually do not agree. Choices must be made in establishing

the "best" values to be used by technologists.

Two methods are now being used to construct these tables. One

is a sequential approach in which the data networks are solved piece

by piece according to a preset strategy. This method has been used for

decades. More recently simultaneous solutions of the data networks

have been made using computer techniques.

Our concern in this paper is with the quality of the computer

solutions of data networks. How well do the machine solutions compare

with the hand-crafted selections? l^/hat human controls are required to

assure high quality? \4hat parts of the problem can be standardized?

Are these techniques suitable for construction of tables now and in

the near future? In particular, are they suitable for use in setting

the CODATA Key Values for Thermodynamics?

2
We present here a novel method for a computer-based evaluation

of thermodynamic data that combines the use of least sums and least

squares techniques in the simultaneous solution of data net-works.

This is compared to the sequential method in four experiments. The

new method appears to be suitable for general use.

Starting in the mid 1960 's two groups in the USA

and the USSR, have been preparing and publishing large up-dated

tables. These tables are the National Bureau of Standards (US) "Selected

Values of Chemical Thermodynamic Properties" [1] and the Institute

for High Temperature (USSR) "Thermal Constants of Compounds" [2].

At the present time both of these programs are nearing completion.

These tables and all earlier ones have been constructed using

the sequential method.

During the past ten years an alternative method for preparing

tables of self consistent thermodynamic properties has been developed.

This is the simultaneous solution of all of the measurement data

using computer techniques. Guest, Pedley and Horn [3] used linear

least squares analysis on the enthalpy measurements for boron

compounds. They pointed out the advantages of the computer system:

(a) a data base of evaluated measurements is accumulated for

future use, (b) new measurements may be incorporated easily in a

solution, and (c) the selected values may be revised quickly as new

CODATA Key Values for Thermodynamics [4] are established.

3
Pedley expanded the scope of this effort with the establishment

of the CATCH (Computer Analysis of Thermochemica 1 Data) series

of tables [5]^ of which five have been issued covering enthalpy

data on compounds of nitrogen, phosphorus, the halogens, silicon,

chromium, molybdenum and tungsten. Two features of the CATCH system

are important: (a) cooperative data evaluation, the data for each

table being assessed by thermochemists familiar with the particular

subject, and (b) the possibility of issuing new, updated tables at

frequent intervals.

Slightly earlier Syverud and Klein had developed a

similar simultaneous solution system [6] at the Dow Chemical

Company on contract from the National Bureau of Standards. It had

the capability for handling mixed sets of enthalpy, free energy

and entropy data and provided solutions based on either linear

least squares or least sums techniques. Their system has been

used in evaluation of data for the JANAF Thermochemica 1 Tables [7]

and for the alkali metal ions in the CODATA Key Values for Thermodynamics [4 ] .

Since 1972 there has been active cooperation between the

CATCH program at the University of Sussex_, England, and the

Chemical Thermodynamics Data Group at NBS . Jointly a single

system is being developed for the manipulation of thermochemica 1

data bases and the machine -based preparation of selected values.

4
II. The Type of Problem

The mathematical problem to -which a netvork of thermochemica 1

measurements reduces is a set of linear algebraic equations in many

variables. The variables_, or some of them, are overde termined

where the a's are known coefficients, the x's are the variables to

be solved for, y is a measured value and u is its uncertainty.

The thermochemical experiments from which this set of equations

is derived all measure the change in a property of a system. These

changes are. by convention^, expressed in terms of properties of the

individual substances involved in the process.

For example, consider the enthalpy of solution of one mole of

rubidium oxide in water

Rb^OCcrystal) + H^OC liquid) - 2RbOH(aqueous)

m = -338 ± 3 kJ

which is expressed as

where each zlHf represents the molar enthalpy of formation of the substance

from the elements in its formula. In the mathematical problem,

this becomes

m» m > n

ZXH = 2-^f(RbOH) - ^(Rb^O) - (R^O)

u^ = -338 ± 3 = 2 x^ - x^ X

3

5
Similar equations relating the measured changes in Gibbs (free)

energies of formation, AGf, or the entropies. may also be written.

Very often in the solution of a thermochemi cal network the

properties of compounds of one particular element are considered

variables and those of other elements are held fixed. In the

example above, when solving for properties of rubidium compounds,

AHf(RbOH) and Z^HfCRb^O) are variables but ZfflfCH^O) would be held

fixed at a preassigned value and lumped with the measurement.

When networks involving compounds of only one element as

variables are examined several characteristics stand out. First

there are many different types of measurements to be combined.

They use different techniques and measure different classes of

properties. The principal types of measurements are listed in

Table 1. It is very difficult to compare the reliability of these

widely differing measurements; one is comparing apples and oranges.

Second, when the measurements are reduced to mathematical form almost

all have the following simple forms:

y = ax

y = a^x^ -

y ~ ^1^1 ^2^9 " ^3^3

with the third case being relatively uncommon. Third , the

networks, although containing 10 to 100 variables, are linked loosely.

Each variable appears in only a small percentage of the equations.

6
Fourth,, thermodynamic laws place constraints upon the solution in

the form of interrelations bed:v-een variables. The most important of

these is the relation

^ = -

This is the only linkage between the three main classes of m.easurements shown in Table 1.

The mathematical approach is simple: solve an overdetermined

set of linear algebraic equations, using an accurate algorithm.

Preparation of the input data is more difficult. Computational

programs require substantial sections to handle chemical bookkeeping

and quality control. And. most important, very careful examinacion

of the experiments is required. This often involves extensive

pre 1 imina ry c ompu ta t i ons .

7
TABLE I. TYPES OF THERMODYNAMIC MEASUREMENTS

ENTHALPY (HEAT) CHANGES

DIRECT FORMATION OF A COMPOUND FROM THE ELEMENTS

COMBUSTION (OXIDATION) OF A COMPOUND .

REACTION BETWEEN TWO (OR MORE) COMPOUNDS

DECOMPOSITION REACTIONS

SOLUTION OF A COMPOUND

DILUTION OF A SOLUTION

PHASE CHANGES

IONIZATION AND APPEARANCE POTENTIALS

GIBBS ENERGY (FREE ENERGY) CHANGES

EQUILIBRIUM MEASUREMENTS (OFTEN AT HIGH TEMPERATURES)

SOLUBILITY

DILUTION

VAPORIZATION

REACTION

ELECTROMOIIVE FORCE (CELL DATA)

ENTROPY MEASUREMENTS

ABSOLUTE ENTROPIES FROM HEAT CAPACITY DATA

TEMPERATURE COEFFICIENTS OF EQUILIBRIA AND EMF DATA

8
ill . Data Evaluation

There are two fundamental elements that are common to both

of the methods discussed here for preparing tables of thermochemical

data. In our opinion they should underlie any method.

The first is the careful examination of each measurement by

the data evaluator. This is the heart of data evaluation and cannot

be automated. It is necessary to determine^ in the light of

present knowledge^ whether or not the interpretation of the

chemistry was correct and whether the technique was suitable.

Reinterpretation and correction of the data to standard conditions

may be necessary. The overall reliability of each data item must

be estimated if it is to be combined rationally with others.

This estimate is difficult to o^ike and often nay be subjective.

The second is the application of a set of criteria for a

reasonable solution for AHf^ Z^f^ and S° for interrelated compounds.

(1) The values selected must reproduce well those measurements

considered to be reliable. (2) The values selected must be as

consistent as possible with all other values in the tables and

should be in reasonable accord with the properties of similar

substances or with physico-chemical correlations. (3) A consistent

set of auxiliary data should be used throughout the entire set of

tables. These auxiliary data include both the values for physical

constants and the properties of substances ubiquitous to thermodynamic measurements.

If these criteria are satisfied^ one presumes that the

individual iXHf's etc. may be combined to predict the "best" value

of the enthalpy^ Gibbs energy or entropy change for any process,

measured or not.

9
IV. The Sequential Ifethod

How are the criteria for a reasonable solution applied in the

traditional sequential or hand-crafted method? Values for certain

compounds that are constantly and repeatedly needed^ the "core"

auxiliary data, such as those for H^O^ OH" (aq. std. state), common acids

and bases, CO2,, etc., are evaluated first. It is then customary

to evaluate all of the data related to compounds of one element at

one time.

To do this the data anal57st assembles all of the data on

compounds of an element and proceeds to evaluate and calculate

the properties £<Rf ^ Z^f, and 3 compound by compound, working through

the network. The analysis starts with compounds whose properties

can be determined independently, that is^ they depend only on known

auxiliary data but not on other compounds of the same element.

Then the properties of other compounds dependent on these first

selections are set. If several measurement paths lead to the same

compound a confirmation of the choice may be obtained. On the other

hand, some or all of the previous selections may have to be revised

to obtain a reasonable overall fit. A sample network is shown in

Figure 12. Compounds a, b, c and d are determined first. There are

direct connections involving a and c and also c and d. Then

compounds e to 1 are selected. Finally, m and n are set.

10
When discrepancies are noted, decisions have to be made as to

how to resolve them. Should a path be ignored as being a poor

measurement, or should a weighted average be taken? Is it possible

that the problem is not that of erroneous measurements but of a poor

value selected in an earlier step or in the auxiliary data?

In any case, the

evaluator must retrace his steps, find the suspect value and modify

that selection, and all the subsequent values dependent upon it.

Measurements that originally appeared to be reliable become

suspect and are downgraded if they are highly inconsistent with

values arrived at from other paths in the over-determined set.

The skeleton of values for key compounds is built up carefully^

taking care to reproduce well the measurements on which they are

based. Because of these various factors, this manual sequential

method is^ in reality, iterative. More than one pass is involved

in establishing the final values for the key network.

The major advantage to this way of evaluating data is that

the evaluator works from positions of strength, that is . he

emphasizes the definitive measurements and builds a framework with

them, fitting and adjusting small networks until he arrives at

a set of stable values.

This very advantage can be a disadvantage. If new and significant experimental data become available it becomes difficult

11
to incorporate or to modify selections without going through

another complete pass for the network. This is time consuming.

In addition, once the "selected" values are incorporated into

other tables within the series^ they cannot be changed without

considering the effect upon all compounds evaluated after them.

As a result it is a major problem to update. The system is static

in between major revisions.

12
V. Advantages of Using Computer Techniques

There are advantages to machine manipulation of data that have

led us to attempt to adapt them, with certain reservations^ to

thermochemica 1 data evaluation. They include:

1. Machine techniques are ideal for manipulating large masses

of data. Many calculations are routine and tedious. Let the computer

do them.'

2. Analyses can be made with the computer to determine the

effect of particular items of data. This is particularly important

for new measurements. For an extensive network these analyses would

be too time consuming to do by hand

3. A systematic diagnosis of the fit of the data can be

obtained

4. An analysis of the network can be made to be used as a

guide to the strategy of data evaluation, and to indicate the

importance of various substances.

5. Solutions can be made for the same data base with different

sets of auxiliary data, such as those required for the GODATA Key

Values^ the NBS Technical Notes or the Institute for High Temperatures

tables

6. Most important, the evaluated measurements can be stored

in easily reusable form for a future calculation.

13
But with these advantages come some potential disadvantages

that may lead to loss of control by the data analyst. They are:

1. Large masses of data are treated at one time. They may

overwhelm the data analyst and he may lose the "feel" for the

measurements that is an important part of the sequential method.

This could be particularly important when a stored data base is used

some time in the future.

2. The evaluation of individual measurements separately

followed by a solution of the entire network can conceal systematic

errors in a series of related measurements that would appear early

in the sequential solution.

3. It may be difficult to identify measurements that are

suspect and require reexamination or reinterpre ta t i on

4. The solutions may be mathematically acceptable but not

consistent with physical chemical correlations and experience.

In the next section we present an approach that takes advantage

of the machine capabilities and appears to m.inimize these disadvantages.

14
VI. Computer Assisted Data Evaluation

A multistage simultaneous solution technique is presented in

this section. It has been designed to take into account the two

features of the data emphasized by data analysts: reliability and

consistency with other measurements. The procedure is outlined

and discussed below. Then the results of some tests are given.

Step 1 . The measurements are examined by a data analyst.

An overall uncertainty is assigned to each one that is tentatively

accepted. This step is common to all evaluation procedures.

Step 2 . An equally-weighted simultaneous solution is made for

the entire data set.

This is used to assess the overall consistency of the data and

as a diagnostic tool to identify measurements that may require

reexamination. The solution may or may not confirm the original

judgement of reliability made by the data analyst.

CXir preference for this first solution is the least sums

technique in which the sum of the absolute values of the residuals

is minimized. This technique selects a set of equations and

solves them exactly. This set is analogous to the median of a group

of numbers. This technique is much less sensitive to outlying values

than is least squares.

Step 3 . The residual on each measurement in the first

solution, i.e. the difference between observed and calculated value,

is combined with the preassigned uncertainty to give an overall

15
measure of goodness. We call this the "average fit". It is a simple

average. It can be considered as an uncertainty modified to allow

for consistency with other measurements.

Step 4 . A second solution is made using a weighted linear

least squares technique. The final answers are the output of this

solution.

The weight used for each measurement is the reciprocal of the

"average fit" developed in step 3. In addition maximum and minimum

limits are imposed on the weights to keep them within a 200 to 1

range . _^

This weighting procedure departs from that commonly used in

least squares analysis, i.e., assignment of a weight proportional to the square

of the reciprocal of the uncertainty. This new procedure has been

adopted to de-emphasize the wide range of uncertainties found when

measurements made using different techniques are to be combined.

Without this de-emphasis a solution tends to favor strongly a few

selected experiments.

The multistage automated procedure described above is performed by a single computer program. But, in our experience,

application of it alone to a set of measurements is not sufficient.

Intervention by a data analyst is desirable. This consists of

checking the input data (measurements) for errors of interpretation

and transcription, reanalysis of individual items, particularly those

16
that fit poorly with others . examination of the core netvork of

measurements emphasized in the solution, and testing the answers for

thermodynamic and physical acceptability. These interventions lead

to revisions in the input data and reapplica tion of steps 2 through 4.

In order to avoid small scale adjustments , we try to restrict the

interventions (other than error correction and reinterpre ta tion) to

removal of a measurement from the set, adding new data, or requiring

that a measurement be fit exactly. These major but simple (and

easily apparent) changes appear to be sufficient.

17
VII . Comparison of Methods: Goodness of Fit

Several tests of the sequential (manual) and computer solutions

have been made for four test problems. These are crude statistical

tests designed to determine how well each solution reproduces the

measurements and the extent to which they agree.

The tests were:

(a) Calculation of the differences between the answers

for each variable and comparing them to the uncertainties assigned

by the data analyst. These are presented in Figures I, 2, and 3

which show comparisons only for variables that are crucial to the

solutions. The solid line (of unit slope) in these Figures is the

locus of points for which the difference in values for a variable

obtained by the two methods would be equal to the total uncertainty

in the value as estimated by the data analyst. These tests are

indicative of performance but are not as strong as those that follow.

(b) The fit of each solution to the measurements were examined.

The patterns of residuals were developed using stem and leaf

histograms [8]. The questions to be answered were "Is this solution

a reasonable fit and are the outliers understood?". The patterns for

the two solutions to each test problem are presented in figures 4, 6,

8^ and 10.

(c) The correspondence of the pair of solutions to each

problem was examined by studying the differences between corresponding

residuals. The questions asked were "Are these solutions fitting

the same data the sameway and what is the significance of mismatches?".

Figures 5, 1, 9 and 11 display these differences.

18
The sca-istics developed in these tests are given in Table 2.

In general, the sequential and simultaneous solutions are

indistinguishable. They fit the r.easureir.ents equally vjell. as shown

by the pattern of residuals. Most of the data are fitted well^ but

some are not. This is to be expected --experiments disagree. In each

case the solutions are strongly correlated, as shown on the differenceof -residuals histograms. But in each case there are data thaz were

treated differently. Most of these outliers turn out to be

unimportant cases. The answers agree to within the expected accuracy

in the vast majority of cases. The four nen-Jorks are discussed below.

Boron compounds (enthalpy data) . This problem was used to develop

the multiphase method. The network, Figure 12, is strongly crosslinked. The "sequential" solution, Figure 4a, was not prepared Dy hand but is

a least squares solution with arbitrary weights^ assigned by the

data analyst, that reflect the emphasis given to each item in an

earlier manual solution. The high correlation between solutions, Figure 5,

was the basis for adopting the method.

Rubidium, compounds (enthalpy, Gibbs energy and entropy data) . The

network has m^any replicate measurements but is alm.ost a tree, i.e.,

has few cross-links. Even within the cross-linked region the

feasible calculation paths are restricted. (In effect, this problem

is based on a directed graph). The high degree of correlation,

shown in Figure 7, between the solutions is not surprising.

19
Uranium compounds (enthalpy data) . The network here is large with

several regions within which there is strong cross linkage but

between which the coupling is limited. The spread in the fits^

Figure 8, is more apparent here than in other cases^ but this is

due to poorer data and larger problem size. The correlation

between the solutions is strong.

Key compounds of Sn, Pb, Cd. and Hg (enthalpy, Gibbs energy and

entropy data) . This is an entirely different comparison than the

other three. For them the simultaneous and sequential solutions

were made concurrently. In the present case the sequential

solution, Figure lOa, is composed of selected values from NBS

Technical Note 270, as chosen in 1968, solving data for each element

separately. The simultaneous solution (in 1976), Figure 10b,

is based on the measurements used to set CODATA Key Values and the

network includes data on compounds of all four elements. About

one-fourth of the measurements are new. Residuals for the sequential

solutions were calculated using this updated set of measurements.

Both solutions appear to fit the data reasonably well. The

correlation between the solutions, Figure 11, is about the same as

in the other tests. Three fourths of the answers agree to better

than their uncertainties. The remainder show that new selections

are indeed desirable,

20
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21
VIII. Discussion

We have shown that computer solutions can be obtained that

agree with the solutions from the sequential method, at least to

the level of testing used here. It is possible that more detailed

testing would indicate a clear preference for one method or the

other. But we do not believe that this will happen, because of

the high degree of agreement in the treatment of individual measurements and in the answers. A definitive choice must be based on

other than statistical considerations. '

The advantages of the computer solution^ stated in Section V,

tip the balance in its favor for the long term. Already we have exploited

them to study the same database, using both GODATA and NBS Technical

Note 270 auxiliary data. We have also used the computer technique

to update selections in light of important new data. And^ most

important^ the method has helped us to identify measurements for

which reinterpre tation was necessary. The application to compounds

of Sn, Pb , Cd and Hg has shown that the restriction of analysis to

compounds of one element at a time is not necessary or even desirable.

These computer solutions do not replace the data analyst.

They are aids. At least three quarters of the work is in the

careful analysis of the individual measurements. This will remain_,

and will become increasingly important because it will be the

principal contact that the analyst has with the data. Additional

22
quality control measures vill be needed to assure that errors in

the data can be spotted at this stage. But, on balance^ we recommend

the use of the computer algorithm given here as an aid to improving

the efficiency of thermochemica 1 data evaluation.

23
REFERENCES

Wagman, D. D.;, et al . , "Selected Values of Chemical Thermodynamic

Properties", Nat. Bur. Stand. (U.S.) Tech. Note 270-3 (1968),

270-4 (1969), 270-5 (1971), 270-6 (1971), 270-7 (1973), U. S.

Government Printing Office, Washington, D. C. [Parts 1 and 2 are included in part 3].

Glushko, V. P. (ed.) et al., "Termicheskie Konstanty Veshchestv",

Vol. 1 (1965), Vol. 2 (1966), Vol. 3 (1968), Vol. 4. part 1 (1970), part 2 (1971), Vol. 5 (1971), Vol. 6, part 1 (1972),

part 2 (1973), Vol. 7, part 1 (1974), part 2 (1974), VINITI,

Moscow

.

Guest, M. F., Pedley, J. B., and Horn, M. , J. Chem. Thermodynamics

I, 345 (1969).

CODATA Task Group on Key Values for Thermodynamics, CODATA Bull.

No. 17 (Jan. 1976), and CODATA Special Report No. 3 (Sept. 1975),

CODATA, Paris. [These publications include all recommendations

in earlier reports.] Parker, V. B., Wagman, D. D. , and

Garvin, D., "Selected Thermochemica 1 Data Compatible with the

CODATA Recommendations", Nat. Bur. Stand. (U.S.) NBSIR 75-968

(1976).

Pedley, J. B., (ed.), "Computer Analysis of Thermochemica 1 Data

(CATCH Tables) ", University of Sussex, Brighton. Data selectors

and tables issued:

Cox, J. D., "Halogen Compounds" (1972)

Pilcher, G., "Nitrogen Compounds" (1972)

Pedley, J. B., and Iseard , B. S.. "Silicon Compounds" (1972)

Head, A. J., "Phosphorous Compounds" (1972)

Barnes, D. S., "Cr, Mo and W Compounds (1974)

Syverud, A. N. and Klein, M. [Periodic Reports on USDC-NBS Contract

CST 1165], Dow Chemical Co., Midland, Mich., 1964-65.

Syverud, A. N., "Computer Analysis of Interrelated Data: Least

Sums and Least Squares Applied to Thermodynamic Data", typescript,

Dow Chemical Co., Midland, Mich., 1965.

Syverud, A. N. , "Simultaneous Adjustment of Thermochemica 1 Data.

I. Enthalpies of Formation of Hydrofluoric Acid and Five Closely

Related Fluorides", (Jan. 1969), "II. Enthalpies of formation

of C2F4, CF3 and six CF3X molecules", (Aug. 1969), typescript,

Dow Chemical Co., Midland, Mich.

24
Syverud;, A. N, "Simultaneous Adjustment of Enthalpy of

Formation, Gibbs. Energy of Formation and Entropy by Least

Squares", typescript, Dow Chemical Co., Midland, Michigan

Stull, D. R. and Prophet, H. ^ (eds,), "JANAF Thermochemica 1 Tables, 2nd edition, Nat. Stand. Ref. Data Series NSRDS-NBS 37,

U. S. Government Printing Office, Washington, D. C. 1971.

Statistical Engineering Laboratory, "Otanitab II News Letter

No. 15_", Nat. Bur. Standards (U.S.), unpublished (1975),

citing Tukey, J. W. , "Exploratory Data Analysis", AddisonWesley (1970).
FIGURES

Figure 1. Absolute values of the differences between the

enthalpies of formation of boron compounds in two

solutions. The abscissa is the analyst's estimate of

the reliability of the value in the sequential solution,

Figure 2. Absolute values of the differences between thermodynamic properties of rubidium and uranium compounds

in two solutions. Abscissa as for Fig. 1.

Figure 3. Absolute values of the differences between thermodynamic properties for key compounds of Sn^ Pb^ Cd^

and Hg, Abscissa based on CODATA Tentative Key

Values, Part VI (1976) .

Figure 4. Residuals on measurements , boron enthalpy data.

(a) Sequential solution, (b) Simultaneous.

Figure 5. Differences between residuals in two solutions, boron

enthalpy data.

Figure 6. Residuals on measurements, rubidium data.

(a) Sequential solution, (b) Simultaneous solution

Figure 7. Differences between residuals, rubidium data.

Figure 8. Residuals on measurements, uranium enthalpy data,

(a) Sequential solution, (b) Simultaneous

Figure 9. Difference between residuals in two solutions,

uranium enthalpy data.
Figure 10. Residuals on measurements, Sn, Pb, Cd and Hg data.

(a) Sequential solution, NBS Tech. Note 270 (1968),

(b) Simultaneous solution (1976)

Figure 11. Difference between residuals in two solutions, Sn,

Pb, Cd and Hg data.

Figure 12. Network for enthalpy data on key compounds of boron.

a = B(amorph), b = H3B03(c), c = BN(c), d = '5'F^(g),

e = B2Hg(g), f = BCl3(g), g = 6013(1) h = H3B03(soln, aq),

i = B203(am), k = 6303(0), 1 = HBF^(aq), m = H3303(HC1 soln),

n = (CH3)3NBH3(c)
IS Ul Kl h I (3}N3yjjji?)5aH
Figure 2 XX

E3

m

m

m

bj

m

c

Ln

Ld

LJ

12

1— ID

LJ

CD c2j

LJ m CD I

z:

a ZD

LJ 12

q: i~ 11 11 LJ

Lu. X

Ll_

P)i / (3}N3y3JJiO) 5SH
Figure 3

ID

V. m1 nLU

LTl lD

LL

v. —-r m

Ui

Ul.

1 1 LJ LJ

11

ID LD

-J

m

in

m

GO

Eum

IS

E3

!S3 SI

"3

I

d::

UJ

n I (3}N3H3jji<]) sea
Figure 4

residuals/kJ

-3

-2

-1

0

1

2

3

5

6

+++

BORON

+++++

+++++++

++++

+++++++++++++T

++

+ ++++

++

+ +

++

+

+

+

++ +

+

++++ +++++++++++

+++++++++++++

+++++++

++

+

++

+++

+++

(a) SEQUENTIAL (b) SIMULTANEOUS

DIFFERENCE BETWEEN RESIDUALS/kJ

-1

+1

++++++++++

++++++++

++++++++++++++++++-

++++++

+++++

+

BORON

Figure 5
Figure 6 (a)

residuals/kJ rubidium

3

++

2 +

+

1 ++++

+++++

0

+++

1 ++

++

2 ++

J. 3

(a) SEQUENTIAL (9 NOT PLOTTED)

5

H

3

2

1

0

1

2

3

residuals/kJ rubidium

+++

+++

+++++

+++++

+++

+++++

++++++++

++

(b) simultaneous NOT PLOTTED)
Figure 7

DIFFERENCE BETWEEN RESIDUALS/kJ

-5 +

-3 +

-2

-1 ++

++++++++

Q +++++^+ + + 4. + x+4.^+ + + ++4, + 4.+4, + + + + + + + + ++++ + + + + + + +++

++++++

1 +

+

2 ++

+

3 +++

4

5 ++

RUBIDIUM (1 NOT PLOTTED)

Figure 8 (a)

-10 ++ residuals/kJ Uranium

++

-8 +++

-6

++

-4 +++

++

-2 ++++++++

++++++++++++

0 ++++++++++++++++++++++++++++++++++++++++++++++ CSo)

+++++++++

2 ++++++

++++

4 +

6 ++++++++

8 +++

+ 10 ++

(a) SEQUENTIAL (19 POINTS NOT PLOTTED)
-10 +++

++

o -0 ++

++++

r -0 +++++

-4 +++

+ + + + + -2 ++++ ++++++++++++

0 ++++++++++++

++++++++++++

2 ++++++++

.+++

+++

6 +++

++

8

++

10

residuals/kJ uranium

•+++

(b) SIMULTANEOUS (10 POINTS NOT PLOTTED)

-10

-8 ++

DIFFERENCE BETWEEN RESIDUALS/kJ

-6

-4

-2

0 +++++++++++++++1-++-!-+++++++++++++++++++++++++++-' (68)

2

4

6 ++++++

8 ++++

++

'^^ URANIUM (6 POINTS NOT PLOTTED)

Figure 9
Figure 10 (a)

++

+++

+ +T +

+

+

+ + +

resiluals/kJ

+++

+++++

+ + -r-f + -i-+- +++ -r + + + - + ++ + + + -r -r -r -f- -f -i- 4- -r -i- +++ + 4- T-r x+++ T- +++ T- T- 4- -r +++++++

++ +++

4.

- (73

++

+ (a) SEQUENTIAL (10 POINTS NOT PLOTTED)

(b)

+

++

residuals/kJ Sn, Pb, Cd, Hg

+

+ + + -r-f- + + +++++ ++++++T+

+H- + -r + + +-r-f +++-f-4-r-f-

-i"r + +-f- + + -r +

- (63)

+++

O) SIMULTANEOUS OINTS NOT PLOTTED)
Figure H

3 +++++

DIFFERENCE BETWEEN RESIDUALS/kJ

++

++

++++

+

-f + + + -r -[- + + + + + + - + + + + + + + 4- + + - + + + + + + ++ ++++-'( 81 )

+ +

+ + SHj PBj Cd, Hg POINTS NOT PLOTTED)
Figure 12

U .S. 0=PT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PURLICATION OR RF-:P0RT NO.

NBSIR 76-1147

2. Gov't Accession

No.

3. Recipient's •Accession .No.

4, TITLE AND SUBTITI.H

A COMBINED LEAST SUl-IS AND LEAST SQUARES APPROACH TO THE

EVALUATION OF THERMODYNAMIC DATA NETWORKS

5. Publication Date July 1976

6. Performing Organization Code

f—

7. AUTHOR(S) 8. Performing Orgaa. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/ •'orlc Unit No.

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP) 13. Type of Report Si Period

Covered

Interim

14» Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less (actual summary o( most significant in formation. I[ document includes a significant

bibliography or literature survey, mention it here.)

A description is given of a system for computer-based evaluation of

interrelated thermodynamic measurements of enthalpies of reaction, equilibria

and entropies. This system is an extension of the CATCH program developed by

J. B. Pedley, University of Sussex. In the new system linear least sums and

least squares techniques are used to solve networks of thermodynamic equations

to obtain the enthalpies and free energies of formation and the entropies of

chemical substances. The least sums technique is shown to be useful in assessing

the consistency of the data. A method combining least sums and least squares

solutions, provides a weighted solution that reproduces closely the solutions

that are obtained by a detailed analysis of the data using the customary

sequential procedure. The results from tests on four large networks involving

compounds of B, U, Rb and salts of Sn, Pb, Cd and Hg are discussed.

17. KEY 'X'ORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons) Computerized data analysis; data evaluation; least

squares (L2) ; Least sums (LI); thermochemistry; thermodynamic data networks

18. AVAILABILITY Unlimited

1 ! For Official Distribution. Do Not Release to NTIS

1 1 Order From Sup. of Doc, U.S. Government Printing Office

"'ashington. D.C. 20402. SD Cat. No. CM

&C * Order From National Technical Information Service (NTIS)

Springfield, Virginia 22151

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37

41

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