Research: Science & Education
Analysis of Chemical Reactions by Means of Isoconversion Curves: Kx = Constant Valeriu Voiculescu* and Luminita Simoiu University of Craiova, Faculty of Science, A.I. Cuza, 13, Craiova, Romania Gavril Niac University of Cluj, Faculty of Chemistry, Cluj-Napoca, Romania For the chemical industry, equilibrium reactions involving only a gas phase or reactions involving gas–solid phases are of practical interest. Most often, the equilibrium constant Kp is used for these reactions. However, the equilibrium constant Kx also has some practical applications for these reactions, and some features can be discussed only by using Kx , as will be shown in this paper. The gas phase is considered to have ideal behavior. For a chemical reaction the degree of transformation of the reactants into products is indicated by the equilibrium conversion. The conversion depends on temperature, pressure, the proportion of reactants involved, and the presence of an inert gas. From this point of view, it is of practical interest to know the equilibrium conversion at different values of the indicated parameters. Tables or diagrams with such values can be realized, but they are difficult to use and do not offer a picture of the thermodynamic possibilities of a chemical reaction. The situation is greatly simplified if, instead of the equilibrium conversion, the equilibrium constant Kx is used. This depends on only two parameters: temperature and pressure. For convenient variations of the two essential parameters, the equilibrium constant Kx maintains the same value. If the logarithm of the pressure is plotted against temperature, a family of curves can be obtained. On each of them, Kx has a constant value, which differs from one curve to another. On such a curve, the conversion has a constant value, if the proportion of the reactants involved in the reaction does not change. As a consequence, every curve with Kx = constant is an isoconversion curve. Among the curves belonging to the family with Kx = constant, the curve with Kx = 1 has special importance. This curve may be called “the curve of normal null affinity” because the normal affinity (1), APT , is linked to the equilibrium constant according to the expression { APT = ∆GPT = { RT ln Kx
(1)
∆G PT
where is the Gibbs function for a chemical reaction at temperature T and pressure P. On the isoconversion curve with Kx = 1, the normal affinity APT is zero. On one side of this curve, APT > 0, Kx > 1, and the reaction has a favorable conversion of reactants into products, as is shown in Figure 1. On the other side of the curve, the situation inverts: APT < 0, 0 < Kx < 1 (crosshatched area), and either there is very little conversion of the reactants or the chemical reaction does not take place at all. Equation of the Normal Null Affinity Curve, Kx = 1: Thermodynamic Types of Chemical Reactions At constant temperature the normal affinity A PT is calculated from the well-known relation *Corresponding author.
P(atm)
P
A T = A°T –
1
∆VdP
(2)
where ∆V represents the change of the volume during the chemical reaction and A°T = {∆G°T , where ∆G°T is the standard Gibbs function for the chemical reaction.1 For a reaction involving the gas phase, ∆V = ∆nRT/P
(3)
If eq 3 is substituted into eq 2 and the second term is integrated, the following expression is obtained: APT = A°T – ∆nRT ln P (atm)
(4)
AP
On the condition that T = 0, the equation of the curve of normal null affinity is obtained immediately (2): log P (atm) = { ∆G°T / (2.303 ∆nRT)
(5)
where ∆G°T can be calculated from the relation
∆G°T = ∆H°298 +
T 298
∆C°P (T)dT – T∆S°298 T
–T 298
(6)
∆C°P T / T dT
or by using the approximate relation: o
o
o
o
o
∆G T = ∆H 298 + ∆C P (T– 298) – T∆S 298– T∆C P ln (T/ 298) (7) o
where ∆C P is the average of the difference in molar heat capacities for the temperature interval. The two integrals from the relation in eq 6 have opposite signs, and for most reactions they are largely compensated. Thus the curve of normal null affinity can be linearized by plotting log P vs. 1/T. If the molar heat capacities C°P(T) are not available, the following approximate expression can be used: ∆G°T = ∆H°298 – T∆S°298
(8)
The slope of the normal null affinity curve is obtained by differentiation of eq 5 with respect to temperature: d log P/dT = ∆H°T / (2.303 ∆n RT 2)
(9)
If the eq 5 is written in the form log P (atm) = { (∆H°T /T – ∆S°T) / (2.303 ∆nR)
(10)
when T → ∞ the following expression results: lim log P (atm) = ∆S°T / (2.303 ∆nR)
T →∞
(11)
Thus, for the reactions having ∆n ≠ 0, the temperature axis is not an asymptote for the curve Kx = 1. At sufficiently high temperatures, the curve has a nonzero intercept with a zero slope (eq 9). At the other extreme, for the same reactions,
JChemEd.chem.wisc.edu • Vol. 75 No. 2 February 1998 • Journal of Chemical Education
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Research: Science & Education the log P axis is an asymptote for the curve lim log P (atm) = ± ∞
(12)
T →0
Equations 5, 9, 11, and 12 are sufficient to draw the curve of normal null affinity for any chemical reaction. The shape of the curve depends of the sign of the terms ∆H°T , ∆S°T , and ∆n. Taking into account these signs, and the case ∆n = 0, chemical reactions can be of twelve types, as shown in Figure 1. If the sense of the chemical reaction is neglected, the twelve types can be grouped into six pairs because for the pairs of type I/XII, II/XI, III/X, IV/IX, V/VIII, VI/VII the curves are identical. Then, it can be concluded that the forward and the reverse reactions have the same curve for the normal null affinity. Moreover, by multiplying the chemical equation for the reaction by a constant, the curve remains the same. The curves of null affinity for the pairs of types III/X and IV/IX were drawn with dotted lines because reactions of this type are not encountered—that is, reactions in which ∆S°T and ∆n have opposite signs. An increase of the amount of gas in a chemical reaction (∆n > 0 ) is accompanied by an increase of the entropy (∆S°T > 0 ) and vice versa. Reactions of type I, such as N2 (g) + 2O 2 (g) = 2NO2 (g), have no technological applications because they need very high temperatures and pressures. In contrast, reactions of type XII, 2CH4 (g) + 2NH3 (g) + 3O2 (g) = 2HCN(g) + 6H2O(g), have industrial applications, the temperature and pressure being lower (very often P = 1 atm). Reactions of type II, such as 2HX(g) = H 2 (g) + X2 (g), where X is F, Cl, Br, I, are not possible or have a negligible conversion because ∆GPT > 0 for any value of the pressure and temperature. In contrast, the opposite reaction (of type XI) corresponds to chemical equilibria that are strongly displaced towards products or complete reaction. The processes
belonging to this type do not contain the normal null affinity curve, Kx = 1. For the pair of types V/VIII, where ∆n = 0 and ∆GPT = ∆G°T , the curve with Kx = 1 is reduced to a vertical line, with the slope (eq 9) infinite. A few examples are presented. Type V: N2 (g) + O2 (g) = 2NO(g); C2H 2 (g) + N2 (g) = 2HCN(g) Type VIII: CO (g) + H2O (g) = CO2 (g) + H2 (g) For the reactions or processes belonging to the pairs of types V/VIII and VI/VII, the curve with Kx = 1 intersects the axis of the temperature in the point where ∆G°T becomes zero and the equilibrium constant Kp has a unity value. Reactions of type VI/VII are often encountered, as shown in Figures 2 and 3. If ∆n = 1, eq 9 becomes identical with the Clausius– Clapeyron equation for phase transformation processes. Therefore the vaporization and the sublimation (∆S > 0, ∆H > 0, ∆n = 1) are of type VI, and the condensation is of type VII. For these processes, the curve of normal null affinity intersects the axis of the temperature at the normal boiling point or at the normal sublimation point, where the vapor pressure has the value P = 1 atm. Another feature of the curve with Kx = 1 refers to the total change of entropy, ∆S tot , on this curve. Taking into account the following identities (3–5): ∆Stot = { ∆GPT / T = ∆SPT – ∆HPT / T = R ln Kx
(13)
and substituting Kx= 1, one sees that the total change of the entropy is zero on the normal null affinity curve. On one side of the curve there are ∆Stot > 0 and Kx > 1, and on the other ∆S tot < 0 and 0 < Kx < 1. When eq 9 is differentiated with respect the temperature, d2 log P/dT2 = (∆C°P – 2 ∆H°T / T) /(2.303 ∆nRT2) (14) and taking into account Figure 1, the result for reactions of type I and VI is ∆C°P < 2 ∆H°T / T
(15)
and the result for the reaction of type VII and XII is
Figure 1. Normal null affinity curves (ATP = 0; Kx = 1) for different chemical reactions.
204
Figure 2. Family of curves with KX = constant for the reaction N2(g) + 3 H2 (g) = 2 NH 3(g).
Journal of Chemical Education • Vol. 75 No. 2 February 1998 • JChemEd.chem.wisc.edu
Research: Science & Education ∆C°P > 2 ∆H°T / T
(16)
Equations 15 and 16 have importance for checking the experimental C°P and ∆H°T data.
often, this is the curve of normal null affinity, Kx = 1. Such chemical reactions are, for example, disintegrations of carbonates and hydroxides and the dehydration of the crystal hydrates: CaCO 3 (s) = CaO(s) + CO2 (g) ;
Equation of the Family of Curves Kx = Constant From the eqs 1 and 4 there is obtained the equation
l = 1;
Kx = 1
2Al(OH) 3 (s) = Al2O3 (s) + 3 H2O (g) ; l = 1; Kx = 1
log P (atm) = { log K x / ∆n – ∆G°T / (2.303 ∆nRT) (17)
CuSO4?5H2O(s) = CuSO4?3H2O(s) + 2H2O(g); l =1; K x = 1
Equation 17 represents the equation of the family of curves with Kx= constant (6). In order to obtain these curves, ∆G°T is calculated initially at different values of the temperature. Then, with particular values for Kx using eq 17 there is obtained the family of curves with Kx = constant for a certain chemical reaction. The relation 17 is used for reactions in which ∆V ≠ 0. For reactions with ∆V = 0, the curves Kx = constant are obtained by using the well-known relation
For other reactions taking place in monovariant systems, the equilibrium is attained on another curve of the family:
∆G°T = { RT ln K x
(18)
For Kx = 1, the relations 5 and 17 become identical; then relation 5 represents a particular case of the general relation 17. Performing the differentiation of eq 17 with respect to the temperature and with Kx = constant, eq 9 is obtained. Thus, at the same temperature, all curves with Kx = constant have the same slope as the curve of normal null affinity. This is why the curves Kx = constant and Kx = 1 have the same shape, as is shown in Figures 2 and 3. On the curve with Kx = constant the normal affinity APT changes according to eq 1. An exception is represented by the curve with Kx = 1, for which APT = 0. The thermodynamic term, which has the same value on every curve with Kx = constant, represents the total variation of the entropy, ∆Stot, given by eq 13. The family of curves with Kx = constant have a real existence for chemical reactions having the variance l ≥ 2. 2 If the chemical reaction takes place in a monovariant system (l = 1), then chemical equilibrium is attained on a single curve from the family of curves with Kx = constant. Most
Figure 3. Family of curves with KX = constant for the reaction NH4 Cl(g) = NH3(g) + HCl(g). The chemical equilibrium is not possible for Kx > 0.25.
NH4Cl(s) = NH3 (g) + HCl(g); xNH3 = x HCl = 1/2; l = 1; K x = 1/4 (NH4) 2 CO3 (s) = 2NH3 (g) + CO2 (g) + H2O(g); xNH3 = 2xCO2 = 2xH2O = 1/2; l = 1; Kx = 1/64 Consequently, for reactions developed in a monovariant system the family of curves of Kx = constant is reduced to a single curve, on which Kx = 1 if the gas phase has a single component, or on which 0 < Kx < 1 in case two or more components are present. If one (or more) of the conditions previously mentioned is eliminated, then the system becomes bivariant or multivariant. It is then possible to obtain other curves from the family of curves with Kx = constant. For example, at the disintegration of NH4 Cl(s) with xNH3 ≠ xHCl , the system becomes bivariant. In this case it is possible to obtain other curves from the family of Kx= constant, on condition that 0 < Kx < 1/4 (Fig. 3), in the range of ∆GPT > 0. Examples and Discussion In Figures 2 and 3 are shown the family of curves with Kx = constant, for two chemical reactions belonging, respectively, to types VII and VI. The necessary data for calculations have been taken from reference 7, and ∆G°T is obtained by using eq 7. The family of curves with Kx = constant offers a general view of the thermodynamic possibilities for a chemical reaction. In Figure 2 the curves with Kx = constant are shown for the ammonia synthesis reaction. Also, the conditions of temperature and pressure for the reductive fixation of nitrogen according to the Haber–Bosch procedure and by bacteria are indicated. The natural fixation of nitrogen by bacteria can be realized in much more favorable thermodynamic conditions (Kx = 104 –108) than the direct synthesis from elements by the Haber–Bosch procedure. The synthesis of ammonia is possible, from the thermodynamic point of view, even at 1 atmosphere—but in this case the temperature must be lower than 190 °C. Because the Haber–Bosch catalyst is active at higher temperature (450–500 °C), it is necessary to increase the temperature to have a satisfactory rate of the reaction. The increase of the temperature for kinetic reasons does not favor the reaction from the thermodynamic point of view. In such a condition, ∆G PT remains negative only if the pressure is increased. From Figure 2 it can be seen that for isobaric conditions, the influence of temperature is greater in the range of low pressure, when Kx has a faster variation with the temperature. In an analogous way, one can discuss the thermodynamic and kinetic factors that can influence various other chemical reactions by using the isoconversion curves with Kx = constant. Notes 1. Standard state is of the pure component at 1 atm pressure and at the temperature of interest, T.
JChemEd.chem.wisc.edu • Vol. 75 No. 2 February 1998 • Journal of Chemical Education
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Research: Science & Education 2. The variance can be defined as the number of parameters (T, P, the concentrations of the substances) that can be arbitrarily modified in such a way that the system can maintain all its phases.
Literature Cited 1. Carapetiant, M. H. Termodinamica Chimica (trans. Russian); Ed Tehnica: Bucharest, 1953.
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2. 3. 4. 5. 6. 7.
Voiculescu, V.; Niac, G. Rev. Chim. 1978, 29, 1026. Craig, N. C. J. Chem. Educ. 1970, 47, 342. Strong, L. E.; Halliwell, H. F. J. Chem. Educ. 1970, 47, 347. Nash, L. K. J. Chem. Educ. 1970, 47, 353. Voiculescu, V. Rev. Chim. 1985, 36, 136. Baron, N. M.; Kviat, E. I.; Padgornaia, E. A.; Ponomareva, A. M.; Ravdeli, A. A.; Timofeeva, Z. N. Kratkii spravatcinik fizikokhimiceskih velicin; Izd. Himia: Leningrad, 1967.
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