394
ELI S. FREEMAN AND BENJAMIN CARROLL
to Be elution with salicylates. Acknowledgments.-We are indebted to Mr.
VOl. 62
Roman V. Lesko for technical assistance in the early stages of this investigation.
THE APPLICATION OF THERMOANALYTICAL TECHNIQUES TO REACTION KINETICS.' THE THERMOGRAVIMETRIC EVALUATION O F THE KINETICS O F THE DECOMPOSITION OF CALCIUM OXALATE MONOHYDRATE BY ELI S. FREE MAN^ AND BENJAMIN CARROLL Chemistry Department, Rutgers University, Newark 2, N . J., and The Pyrotechnics Chemical Research Laboratory, Picatinny Arsenal, Dover, N . J . Received July $8, 1967
The a plication of thermoanalytical techniques to the investigation of rate processes is discussed. Equations have been derived for non-reversing reactions, which may be used to calculate energy of activation and order of reaction from thermogravimetric and volumetric curves. An equation, recently presented in the literature, for evaluating these parameters by the technique of differential thermal analysis has also been considered, so as to eliminate the trial and error procedure. The thermal decomposition of calcium oxalate monohydrate which involves dehydration, decomposition of calcium oxalate and calcium carbonate, is used to illustrate the applicabdity of the derived relationships.
Introduction Thermoanalytical methods, such as thermogravimetry, thermovolumetry and differential thermal analysis, are being employed increasingly in the investigation of chemicaI reactions in the Iiquid and solid states a t elevated temperatures. These techniques involve the continuous measurement of a change in a physical property such as, weight, volume, heat capacity, etc., as sample temperature is increased, usually at a predetermined rate. In this article, equations are derived for non-reversing reactions so that rate dependent parameters such as energy of activation and order of reaction may be calculated from a single experimental curve. For this purpose a relationship between specific rate and temperature is assumed = ze-E*/RT
A general derivation is presented and applied to thermogravimetry. For the method of differential thermal analysis, the derivation of Borchardt and Daniels3 has been expanded upon, so that the trial and error procedure now required for evaluating order of reaction and activation energy, may be replaced by a graphical or analytical solution. It should be kept in mind that the treatment may be applied to the measurement of any physical property which is unaffected by sample temperature. The advantages of evaluating reaction kinetics by a continuous increase in sample temperature are that considerably less experimental data are required than in the isothermal method, and the kinetics can be probed over an entire temperature range in a continuous manner without any gaps. In addition, where a sample undergoes considerable reaction in being raised to the tem(1) Thin paper has been presented i n part st the North Jersey Meeting i n Miniature of the A.C.S. i n Jan., 1957. and before the Division of Physical and Inorganic Chemistry at the National Meeting of the A.C.S. i n April, 1957. (2) Pyrotechnics Chemical Research Laboratory, Bldg. 1512, Picstinny Arsenal, Dover, New Jersey. (3) H. J. Borchardt and E'. D. Daniels, J . A m . Chem. Soc., 79, 41 (1957).
perature of interest, the results obtained by an isothermal method of investigation are often questionable. Theory and Derivation Consider a reaction, in the liquid or solid states, where one of the products B is volatile, all other substances being in the condensed state. a A = bB(g) + CC The rate expression for the disappearance of reactant A from the mixture is where X
=
concn., mole fraction or amount of reactant, A
k = specific rate x = order of reaction with respect to A
It is assumed that the specific rate may be expressed as k = Ze-E*/RT
(2)
Solving for k in (1) and substituting (2) for k gives Ze-E*/RT
=:
- (dX/dt) X"
(3)
where 2 = frequency factor E* = energy of activation
R = gas constant T = absolute temperature
The logarithmic form of equation 3 is differentiated with respect to, dX/dt, X and T, resulting in equation 4. E*dt = d In (-dX/dt) R T
- x d In X
(4)
Integrating the above relationship gives
Dividing (4)and (5) by d In X and A In X , respectively, one obtains equations 6 and 7.
APPLICATION OF THERMOANALYTICAL TECHNIQUES TO REACTION KINETICS
April, 1958
-
E* d T - d In (-dX/dt) RT2 d In X d In X
-
A
(>) - A In (--'dX/dt)
~ 1 n X
-
A In
-
X
(6)
(7)
From (6) and (7) it is apparent that plots of
Let us consider the case of differential thermal analysis. I n a recent article3 an equation was derived from which order of reaction and energy of activation was determined using differential thermal analysis. The expression given was C , dAT dt
d log ( -dX/dt) d log X
dT
KAV
+ KAT
( A - a) - C , AT). (K (7 )
=
and
395
(14)
where A (1/T) A log
A log (-dX/dt) A log X
x '"
K = heat transfer coefficient
+
should result in straight lines with slopes of or -E*/2.3R and intercepts of -x. Let us consider the cases where X refers to mole fraction of A, molar concentration and amount of reactant. 1. Mole fraction of A, X = na/M = N A where: ns. = no. of moles of A at time t M = total no. of moles in reaction mixture
(a). Total number of moles is constant during reaction. Substituting for X in (3) results in the relationships
+ In (-dn,/dt)
In k = In Ai*-'
- x In nn
(8)
and -(E*/R)A(l/T) = A 1n n,
-
+ A
In (-dn,/dt) A In n,
(9)
Equation 9 also may be written in differential form as equation 4. (b) Total number of moles is not constant, For this case In IC = (z - 2) In Jd - x In na +
A = area under curve AT = differential temp. a t a particular time dAT/dt = rate of chanee of differential temp. at the point where AT ia measured Y = volume of solution no = initial number of moles of reactants Cp = total heat capacity of reactant solution or liquid a = area under curve up to time where A T and dATldt is taken 2 = order of reaction with respect to one component
The method used by the authors3 to determine x and E* is as follows. A value of x is chosen and used to calculate k over an entire temperature range using equation 14. A graph of log k vs. T-1 was then plotted. If a linear relationship was obtained, it was assumed that the value of x was valid and the energy of activation could then be calculated from the slope of the line. A method of evaluating x and E" which eliminates this trialand error procedure becomes apparent if equation 2 is substituted in (14). The resulting expression written in logarithmic form is 111 Z
- E-*- = (z - 1) In KAV -- - x In (Z
