A theoretical contribution to the analysis of DTA-peaks of rapid reactions

99

Thermochimica Acta, 83 (1985) 99.-106 Elsevier Science Publishers B.V.. Amsterdam

A THEORETICAL

CONTRIBUTION

- Printed

in The Netherlands

TO THE ANALYSIS

OF DTA-PEAKS

OF RAPID REACTIONS

K.-R. Loeblich Potash Research

Institute

of Kali und Salz AG, D-3000

Hannover

(F.R.Germany)

ABSTRACT The paper introduces the idea of the signa 1 potentia 1 0 having the dimension of an electromotive force. lhis signal potent .ial is given by a differential equation of the second order refered to the DTA-signal AU. Not AU, but @ is directly proportional to the momentary reaction rate. In the equation there are only two factors T and r,which are specific instrument parameters and easily determinable. A method is shown to transform a DTA-peak curve into the signal potential 0 as a function of time and temperature. @(t,T) is a true picture of the reaction rate course.

INTRODUCTION Since the investigations of quantitative about reaction DTA signal

by Borchardt

kinetics.

is generally

It is very important not directly

or to the rate of the reaction into account,

because

very slow reactions reactions cases,

(ref. 1) in the early days

to be a source of information to remember

equivalent

that the momentary

to the momentary

heat flux

in the sample. This fact is often not taken

the successful

as reliable

has frequently

misled

consideration

plots of the course scientists

of the peaks caused by of the rate of these

to use this proceeding

for other

as well.

Unfortunately

Borchardt

the peak interpretation which

& Daniels

DTA the peak curve is supposed

clearly

developed

The difficulties

arose,

magnitude

1 show the half-life

than the time constant

the heat capacity mentioned satisfying

accuracy.

easy to apply because

formula

The original

into the formulas

reactions

of the instrument. coefficient.

the conditions

The examples

unsimplified

reported

to be of a higher

T is given by dividing

Only

if t,,*

>) T the

the value of the rate constant

of its complexity.

in

the scope of reasonable

the simplifications.

reveals

the uncertainty

when they did not specify

by the heat transfer

simplified

within

of the chosen T

produced

simplifications

exactly

enough which are justifying

in reference

themselves

by introducing

they had originally

assumptions.

& Daniels

formula,

however,

kr with

is not

100 In order to avoid difficult the approach formation

to the problem

of the DTA-peak

function

arrangements

it is prefered

into the curve of the momentary

of time or of temperature

analysis

of the transformed

reaction

order,

respectively,

of methods

DTA-signal

describable

by means of regular

heat generation

caused

@ is introduced

chemical

by any process

rate and the

as a tool which

too, which are not

kinetics.

sensitivity,

kinetics

rate curve.

of the reaction

to include such phenomena

q = dQ/dt, and E is the caloric

such as the

The present

of chemical

a reaction

distinction

the idea of a signal potential

energy.

the science

which allow to analyse

generalizes the proceeding

characteristics

and the activation

For the purpose of a greater mental

heat flux as a

and the second one is the

curve for the kinetic

the rate constant,

paper deals only with the first step because is disposing

now to divide

into two steps. The first step is the trans-

Because

0 = qE, and

o is a true picture of the rate of

in the sample. 0 is equidimensional

to

the signal AU.

FUNDAMENTALS Because calorimeter Hoehne

of the relationship which has already

(ref. 2) reported

to the difference

between a DTA device and a heat conduction been shown up by S. Spiel (ref. 3), Hemminger

a differential

of temperatures

from or into a reacting

equation

The present

investigation

of temperatures

location

by Borchardt

of the substances

& Daniels.

+ CdAT/dt,

to be expected

Substituting

introducing

the

thermometer The fundamentals the

but ATr means only the

and not that of the thermocouples. between

which

the

is governed

by

TS and TR in AT by an arrangement

like

for q, becomes

= ATrm/Rw + (7,/Rw + C)dAT/dt + T,Cd2AT:dt2 qr Now nTrm means the difference of measured temperatures but not that of the substances.

Using the formal

oAT = AU and uRw = E (E is the caloric presented

having

1 to 3. They described

there is also a difference

and the substance

TS - T:, = ATSM = T,dT,Jdt. this, the equation

result when

theory are shown by the figures

In the case of unsteadiness measuring

constants

dealing with the so-called

caused heat flux q, by q, = ATr/Ru

difference

(1) determinable

gives a very similar

equation

into the heat flux equation

of these authors' reaction

the heat flux qr

sample more precisely:

of time and the square of time, respectively,

first order differential problem

of the second order refered

in order to describe

= (AT + adnT/dt + bd2nT/dt2)lR qr w wherein a and b are complicated and difficultly the dimensions

&

in a more practicable

(2) of the thermocouples,

relations

sensitivity)

CRU=~2 and

the equation

form. ois the so-called

steepness

can be of the

101 thermocouple

I 1

u = dAU/dt.

Fig. 1.

I Plotter

arrangement:

Very simplified

AU f

model of a DTA device.

The heat flux F to R is in the steady

wherein

Rp is the heat transport

(reasonably

33 s

R

__-L___

R2

h

//I

state

case q, = (TF-TR)/R2=(CA+CC+CR)dTF/dt

ik

Furnace

resistance

should be R,=R2=Rw),

heat capacity, characterize

C is the

the indices R,C, and A

the reference,

the crucible,

and the participating

environment.

case without

a similar equation

describes there

reaction

In the

the flux from F to S. Probably

is a difference

in CS and CR.

That gives -ATC = (Cs - CR)RwdTF/dt

or

-AUc = (Cs - CR)EB

(3)

Fig.2.

T

Diagram

TF

showing

the temperatures

TF, TR

and TS. TF and TR are linear functions Ts TR

time, the onset phase excepted. linear

in so far as a reaction

transition

or a phase

does not occur. The example

an endothermic

phase transition

of

TS is also

of

is shown.

1

AU Fig. 3. Showing The distance

the AU-Plot.

of the base line being parallel

to time (in this case) is AUc. The distance

of the peak signal from the base

line is AU,_.

102 THE SIGNAL

POTENTIAL

0

When the equation

(2) is multiplied

side gives qroRw. For a shorter

by 3. 3 is called signal potential electric

by the difference

of temperatures q,, through

to an

force which would a steady heat

driving

the heat transport

Rw.

Now, using the abbreviations those which will be explained written

it is equidimensional

the electromotive

flux of q, being equal to the unsteady resistance

this product qoRw is substituted

because

force. Besides 0 represents

formally be generated

by dRw at either side, the left-hand

version

mentioned

in the precedent

later, the equation

in its final form including

paragraph

to calculate

only two instrument

and

Q can be

parameters

7 and 5

0 = aUr + rdfiU/dt + (r2/c)d2nU/dt2 The signal potential

(4)

0 is the sum of the momentary

base line and the first derivation

LITA signal nUr over the

of AU at this point of time multiplied

by the time constant

T and the second derivation

of the time constant

7 divided

multiplied

by a dimensionless

magnitude

by the square c. The calculation

of o point by point from the peak curve aUr = f(t) gives finally

the signal

potential

the sample

@ as a function

temperature

is calculable

of time t and of temperature from the time-linear

furnace or of the reference

respectively.

T because

increasing

Remembering

temperature

of the

7 to be the sum of

it may be written:

Tl + T2 T2/', = Y

T = (1 + Y)T, = T2(1 + 'i)/y i1T2

=

(7)

7 2/1

(6)

<=Z+y+l/y

Because

l/y<
yw<-

2

(9)

Because of the fact 0 being proportional This does not imply that the signal AUris

to q,, @ must be zero if q, = 0. to be zero in any case if qr is zero.

After a short heat impulse q, drops immediately not do so. Probably thereafter

the signal even increases

following

the function

impulse this gives the opportunity of MJ, against

equation

for a moment

AU, = exp[-(t-tm)/T2]. to calculate

at where At is running

The magnitude

to zero but the signal AU, does

of 5 may be estimated

and decreases

In the case of an

~2 from the slope of the plot

from the &Jr-maximum

to infinity.

by use of the condition

(4) has to give 0 = 0 when the impulse

that the

is terminated.

In a first run

T is set equal ~2, and 5 is found by trial. Then 7 can be estimated equations

(9), (5) and (6). The values are to be improved

be taken into account Another

paper which

i will be published

that T and 5 are functions is dealing

by

by recurrence.

It mus

of temperature.

with the exact determination

of y and then of

together with the results of the investigation

of the

103 temperature @(t,T)

dependency

of

has interesting

being parallel (peak onset)

In the case of a straight

properties.

to the time axis,

integrating

existing

integration

propably

results

The connection

gives the same result as

the identical

in shape between

are identical,

base line

lb(t) within the limits from t = 0

to t = t,,,(peak end practically) the peak AU,_ itself within

integrating difference

t,y,c. and E.

limits

in spite of a great

the two. The dimensions

of the

too.

of the signal potential

(11to the reaction

rate y is given

by the simple equation r

=

$ =-

t

a> J “’ d,dt t=o As the change of enthalpy AHr belonging o/AHrE

sample and E, the caloric picture

of the reaction

PEAK ANALYSIS

are constants,

O(t,T)

of the

represents

a real

rate.

of transforming

the peak curve

riaby hand, the differential

of differences. is calculable f'&)

sensitivity,

to the total conversion

IN PRACTICE

For the purpose potential

(10)

=

Setting

quotients

into the curve of the signal

can be replaced

p>Ur = y = f(t) the value of dM/dt

with satisfactory

accuracy

by quotients

at the point of t2

by

= [(YS - Y,) + (Y, - y,)1/2at

The points equation method

(11) is also applicable

to estimate

to calculate

paragraph.

The sum f(t2) + Tf'(t2)+ is applied

The transformation

d2~U/dt2

(T'/i)f"(t,)

f'(t) = y

correspondingly.

The

in the precedent

is the required

O(t2). This

point by point. The curve of a(t) is the final result. can easily be performed

device fitted with the DTA-instrument,

have the option of checking has to be recorded

Now setting

values of -I and of 5 has been shown

proceeding

similar

(11)

t,, t2. and t3 have to be equidistant.

by means of a computer but it is always

up the single steps,

in any case. The chart

or a

necessary

i.e. the original

speed should correspond

to

peak curve to the

problem. The more or less distinct the peak curve nUr(t)

deviation

of the shape of Q(t) from the shape of

gives an idea of the kind and the rate of a process

going on in the sample. The experimental fitted with a heating

investigation element

was carried

means. A heat impulse was given by a constant of duration.

The resistance

E = E,/n(T)=17.4 The DTA-record

out by using a Netzsch-DSC

at the sample cell for calibration

of the heating

current

element

uV/mW. Then 0 should theoretically is a peak of regular

444

by electric

of 27 mA and of 5 seconds

at 254'C was Rn = 135.8 Q be @qrE=i2RnE=l,726

shape. Fig. 4 shows this peak and the

uV.

104 @-curve

calculated

The impulse

from this peak by the methods

is a short one compared

to be 19 sec. The estimated with the theoretic

. . ---_

impulse

of this paper.

with the time constant

value of 5 is 23. The calculated rectangle

T which

is found

D(t) fits in

satisfyingly.

.l

I

I

500-

AU,

I

I

5

15

10

25

20

set

Fig. 4 showing the principal section of a peak caused by a rectangular heat impulse, the impulse itself (its height in units of qE) and the values of @ calculated from the peak point by point. Heating rate B = 5 degrees C per minute. The signal curve AU, exhibits apex.

Especially

in these ranges the measurements

of the corresponding should

satisfied

because

by

as narrow as possible.

the measuring

considered.

line of values

at the onset range and at the of the signal height AU, and

time t must be very exact ones. Here the measurement

have a distance

magnitudes

intense bendings

Therefore

This demand

error weighs with the smallness the trial is recommended

means of a polynomial.

points

is often not to be of the

to approach

the

105 In the case of the example describable conformity

by an expression

with the peak curve

more precise

approach

demonstrated

above,

the peak onset range was

AUr = a(t2 - t3/4), showing satisfying up to t < 1 s. By dimensional

is expected

analysis

the

to be AUr = a(t2 - t3/bT,) where b g 4.

Then the factor a has the dimensions

of an electromotive

force divided

by

the square of time.

Aur dAU/dt

= a(t2 - t3/4 7,)

(12)

= a(2t - 3t2/4_;,)

(13)

d2AU/dt2

= a(2 - 6t/4~,)

(14)

At the peak onset

d2AU/dt2

(t = 0) both AU, and dAU/dt are zero, but in the case

heat impulse 0 is not zero at t = 0. @ = i2RnE. If

of a rectangular

= 2a. At the onset

t = 0, then

therefore:

is Q = (T2/<)d2AU/dt2,

(15)

-r2/5 = @(0)/(2a) The approach

given by equ. 12 is valid

t = tk the value of d2AU/dt2 time smaller

than tk by inserting

in the range of time 0 < t < tk. At

zero. The value of "a" is calculable

becomes

the time t and the measured

at any

AU,(t)

into equ. 12: a = AU(t)/(t2

- t3/4T,)

(16)

For the first run it is set T, = 1. By doing this, an approximative of a is got (ax). Setting calculating

of equations

calculated

gives improved

Example:

value of i,,y g 5 - 2 gives an approach

Then T, = ~~/y

application

and

T = T2(y

(15) and (16) using the values of y and T just results of 5

and y.

of figure 4 yields

T, is assumed

- 2 giVeS

AU(l) = 39.8 uV. In the first run of

17.9s,thus T, = ~~/y

= 18.0/17.9

= 19.9.

= 1.01 s and T = 18.0'18.9/

17.9 = 19.0 s. Using these values of T, and of T the calculationsby equations 5

peak curve.

to be 1, then equ. 16 gives a, = 53.1 uV/s2.

If Q(O) = 1 726 uV, then (-t2/<), = 16.3 s2 or 5, = &IxTz2/@ Y " 5,

y, to the

+ 1)/y. The repeated

'r2 = 18.0 s was found from the slope of the descending

At t = 1 s the example calculation

at first T = ~2, equ. 15 shows the way for

an approximative

value of y = T~/T,.

value

15 and 16 are repeated.

That gives finally

means of the

~~/r; = 15.2 s2. Then is

= 23, y = 21, T, " 0.8 s, and T = 18.8 s.

CONCLUSION By introduction connection

between

of the idea of a signal potential the curves

rate dx/dt = - @,'AH,E has been found parameters

T and T'/c.

Methods

@ a relatively

simple

of the signal AU (Peak) and of the reaction (see equ. 4). The formula

are shown for estimation

contains

the

of these parameters

106

and for the practical into the rate curve.

application

of the formula

It is even possible

to transform

to restore

impulse from the peak curve which was caused by that impulse. regular chemical

reaction

the equation

(ref. 1) is after placement

of AT by @ unrestricted k.

The greater

the reaction

rate the more differs

function

of time from the peak curve in thermal

scanning

technique.

The possibility curve

to be applied

enhances

analysis

by means of quantitative

the confidence

& Daniels

to

the rate curve as a

of exact lmodelling of a real process

into the a(t)cu

heat

In case of a

22 of the paper of Borchardt

calculate

the rate constant

the peak curve

a short rectangular

by Imeans of the

by transforming

in kinetic

a peak

investigations

DTA.

ACKNOWLEDGEMENT The author performing

is thankful to Miss S. Philipps

the experiments

and the computer

for her valuable

help in

work at the institute

of

Kali und Salz AG. It is grateful investigations to Dr. W.-D.

acknowledged

by important Emmerich

that Netzsch

informations

Geraetebau

about details

for the encouragement

to publish

assisted

the

of the DSC 444. Thanks this paper.

REFERENCES 1 2 3

79 (1957) 41-46. H.J.Borchardt and F. Daniels, J.Am.Chem.Soc. W.Hemminger and G.Hoehne, Grundlagen der Kalorimetrie, Verlag Chemie, Weinheim (FRG) and New York, 1979, pp. 181 - 191 S.Spiel, Rep.Inv.No. 3765, US Bur. of Mines, 1944