CALCULATION OF CHEMICAL EQUILIBRIUM AT HIGH PRESSURES RAYMOND H. EWELL
Purdue University, Lafayette, Ind. ___
~
dynamics-i. e., equilibrium, of three reactions: the methanol synthesis reaction, the hydration of ethylene, and the reaction of nitrogen and acetylene to give hydrogen cyanide. The knowledge of reaction rates and catalysis in these reactions is a separate field and will be given little mention. Some information on this latter can be found in books on catalysis (27) and in some of the experimental papers quoted in the list of references.
The utility of the third law of thermodynamics. combined with modern thermal data, for calculating the equilibrium constants of chemical reactions is demonstrated, with applications to the methanol synthesis, the hydration of olefins, and the reaction of nitrogen and acetylene to give hydrogen cy anide. Practical methods of calculating fugacities and acti,ity coefficients of gases are deteloped, including t h e use of Berthelot's equation of state and of a new modification of van der Waals' equation. The complete quantitative statement of ZRChatelier's principle is discussed and applied to the calculation of somc chemical equilibria a t high pressures. The equilibrium concentrations of methanol, ethanol, and hydrogen cyanide in the above reactions are favored more than a simple calculation would indicate on account of their low activity coefficients. Raoult's law is applied to the calculation of highpressure equilibria involving liquid phases. The equilibrium concentration of ethanol from the hydration of ethylene dccreases with increasing temperature and increases rapidly with increasing pressure, so t h a t it should be possible to condense concentrated ethanol from the reaction gases after reaching equilibrium a t 150 ' C. and pressures as l o w as 100 atmospheres. The possible yield of liquid hydrogen cy anide from nitrogen and acetylene increases both with increasing temperature and with increasing pressure. 90 t h a t rommerciallj attractive yields are possible a t moderate opera tinp temperatures and pressures.
Equilibrium Constants The first step in the thermodynamic study of a chemical reaction is the determination of the equilibrium constant. The equilibrium constant is a number characteristic of a given reaction a t a given temperature. I t is a function of temperature only, and not of pressure or concentration. Of the possible methods of determining the equilibrium constant of a reaction, the two most applicable to industrially important reactions are ( a ) direct experimental determination and ( b ) calculation by thermodynamics, where the third law of thermodynamics and the methods of statistical mechanics are utilized particularly. The experimental method is laborious and not reliable in many cases unless extreme precautions are taken. This paper will be concerned with a demonstration of the usefulness of the second method. For the calculation of equilibrium constants the following relations are important: A F = AH - T I S A F o = A H o - TAS"
K,, = e = e
-AFo/RT
--lHo/RT
e
.!iSo/R
(1) (2)
(3) (4)
= free energy change of reaction A H = heat content change of reaction (negative of the
where LW
ordinary heat of reaction)
A S = entropy change of the reaction
A S Y modern syntheses involving some of the unit processes are operated a t high pressures-for example, hydrogenation, by which ammonia and methanol are synthesized commercially. I n this field of applied science the theoretical calculation of the equilibrium state a t high pressures should be a useful tool. Such calculations are somewhat involved, but are nevertheless straightforward and ucderstandable thermodynamics. This article endeavors to make a simplified presentation of the fundamentals of such calculations. I n a previous paper (6) the writer pointed out the necessity of separating sharply the two unrelated factors, equilibrium and reaction rate, in considering the yield to be expected from a chemical reaction. This paper will treat the thermo-
T
=
Keq
=
absolute temperature equilibrium constant
The superscript zeros refer to changes in the free energy, heat content, or entropy when all reactants and products are in the standard state of unit activity (approximately one atmosphere partial pressure for a gas). The calculation of AH O requires a knowledge of the heat of combustion or the heat of formation (from the elements) of all reactants and products. If AH O is to be reliable, these heat data must be known to a high state of precision. Such data are best exemplified by the recent accurate determinations of the heats of combustion of hydrogen, carbon monoxide, hydrocarbons, and alcohols by Rossini and Knowlton ( 1 1 , 147
INDUSTRIAL AND EKGINEERING CHEXIISTR1
148
The use of inaccurat,e heats of reaction lias frequently led to misunderstanding and a loss of faith in the reliability of thermodynamic calculations in some quarters. The calculation of A.7' requires a knowledge of the absolute entropy of all the reactants and products in their standard states (approximately one atmosphere pressure for a gas). One method of determining the entropy of a substance requires a knowledge of the heat capacity of its stable phases down to very loa temperatures (usually around 20" K., the boiling point, of hydrogen), and of the heats of the various phase changes. The following relations are used to give the standard entropy of a gas at any t'emperature, T o IC.: If the normal boiling point of the substance is below T, 20-25).
Thiq ic a uaeful relation and shoirs clearly the change of both AH and A S with temperature Equation 8 is equivalent to the more commonly used but lezs apparent equation,
the derivatiun of which can be found in any textbook on t~hermotlynamics(14, 15). AH," and I are empirical integration constants, and t'hey can be evaluated if any of the following are known: Two values of A F o (at different temperatures) One A H o and one A F o at the same or different temperatures One A H o and one A S o at the same or different temperatures One A F o and one &So at the same or different temperatures
If the normal boiling point of the substa~iceis above T ,
C, (liquid) dlnT where
AH +$ + Rlnpr + 2732 K Ep , T
(G)
S o = entropy of crystal at 0" I. = heat of vaporization at normal boiling point ~ H =T heat of vaporization at T o K. p~ = vapor pressure at T oK., a t m . T, = critical temperature p , = critical pressure, atm.
The last term in each equation is relatively small and amounts to about 0.1 E. U. (entropy unit) per mole in most cases, which corrects for the nonideality of the vapor a t 1 atmosphere pressure, The third law of thermodynamics comes into the picture at this point st'ating that
so = 0 The other principal method of finding absolute entropies is calculation by the methods of statistical mechanics. Even a brief discussion of these methods is beyond the scope of this paper, but several books describe them in fairly simple terms (8, 9 , 15). The data needed for such calculations are the momenta of inertia (or structure) and the vibration frequencies of the molecule. A number of entropies calculated in this manner were compiled by Kelley (IO). Such calculations are quite accurate for rigid molecules, and in general this is the best method of obtaining the gaseous entropy of this type of molecule, but the results are still of doubtful reliability in the case of molecules with internal rotations (nonrigid molecules). Since heats of combustion and heats of reaction are usually measured a t temperatures not far removed from room temperature, heats of formation are commonly given for 25' C., and AH" for a reaction will usually be known only for 25" C. or some near-by temperature. I t is cust,omary also to give Sofor various substances a t 25" C. (10). This enables AF&s to be calculated by Equation 2 : AF& = AH& - 298ASPrs
(7)
SF" can be calculated a t any temperature, if, in addition,
the heat capacities of all reactants and products are known as functions of the t,emperature, as follows:
together with ACp as a function of 2'. Equations 8 and 9 lead to exactly the same results. The former is more convenient if it is desired to calculate AF" a t just a few temperatures, and the latter if a large number of calculations are to be made. Equations 3 and 8 will be used to calculate the equilibrium constants of the following reactions a t a number of temperatures:
+
E ~ A J I P L E 1: CO(g) 2H2(g) = CHIOH(g) 0 48,500 Qf3S 26,840 62.46 56.66 S& 47.32 A H & = -21,660 cal. A S & = -53.12 E. I-,
+
2 : CQHa(g) HzO(g) = C,HaOH(g) - 11,700 57,800 57,070 S/"& 52.5 45.17 67.0 AH& = -10,970 cal. AS.:a = -30.7 E. U
l;S.k>\IPLE QJP3E
I
