Phase Diagrams and the Mass Law -
The J f m s Law Calculation of P - T X Phare Diagrams for Binary System Consisting of Vapor and a Congruently Dbsociating Solid Compound such as Ammonium Chloride or Ammonium Carbamaie. Diagram of Ammonium Carbamate in the system COrNH3 T . R. BRIGGS Cornell University, Zthaca, New York
T
HE USEFULNESS of the Mass Law in the field of heterogeneous chemical equilibrium may be shown in a most interesting and effective manner by applying the principle to the calculation of phase diagrams of systems consisting of vapor and a dissociating solid compound such as ammonium chloride, ammonium hydrosulfide, or ammonium carbamate. The method of calculation is a good example of the application of elementary principles, and the phase diagrams, which are typical of systems of this kind, should be of interest to teachers and students of the phase rule. Such diagrams are seldom found in the textbooks. Compounds of the class mentioned above--decomposing in the vapor state to give the gaseous constituents A and B and therefore representable by the empirical molecular formula A,B.-may be said to dissociate congruently since the vapor produced by any such compound has the same empirical composition as the compound itself. Because of the dissociation, a compound of this type, when in contact with gaseous mixtures of its two dissociation products A and B in which the ratio between A and B is a variable quantity, may be treated from the point of view of the phase rule as a binary solid phase of fixed composition in equilibrium with gas or vapor in the two-component system A-B. It follows, accordingly, that the equilibrium relationships between the dissociating solid compound and the binary gaseous mixtures may be represented effectively by means of a P-T-X space model or by means of separate P - X , T - X , and P-T phase diagrams of the system A-B. It is a well-known fact that the dissociation mentioned above is governed by a simple form of the Mass Law. This being the case, it should not be difficult to calculate the data needed for the coustruction of the P-T-X space model or the separate phase diagrams by an application of the Mass Law to the dissociation equilibrium, the only experimental quantities required being the dissociation pressures of the compound A,B, and the degree of dissociation of the compound in its own saturated vapor. The general method of calculation will be presented directly below, and this will be followed up with an application of the method to ammonium carbamate in the system C02-NH3. The required data will be taken entirely from the paper on the ammonium carbamate equilibrium published nearly 20 years ago by Briggs and Migrdichian (I). A brief reference will also be made to ammonium chloride in
the system NHhHCl on the basis of data obtained by Rodebush and Michalek (3). Both systems, of course, are typical of the class under consideration. THE GENERAL MASS LAW EQUATIONS
We shall begin by considering a compound, C, having the molecular formula A,B, and dissociating incompletely in the vapor in accordance with the following scheme: A,B. (solid) e A,B. (gas) 5 mA (gas)
+ nB (gas)
When equilibrium is established between the solid compound and any mixture of its gaseous components A and B, let the partial pressures of the three molecular species in the gas phase be p,, p,, and p,, corresponding to the mole fractions X A , xg,and xc, respectively. Since the compound is a solid phase of fixed composition, the partial pressure pc is fixed a t a given temperature regardless of the ratio between A and B in the gas phase; hence the dissociation is governed by the simple Mass Law expression = Kv From this -assuming that the gaseous constituents are ideal-we obtain:
in which P is the total pressure of the gas or vapor phase and g is written form n. Now, what we wish to calculate first with the aid of the Mass Law is an isothermal P-X phase diagram which will represent the solid compound, C, in equilibrium with the gas or vapor phase in the two-component system A-B. In such a diagram the composition is expressed in terms of the gross (i. e., analytical) fractious of the two components, A and B, and not in terms of the true fractions of all the molecular species which may be present in a given phase. Accordingly if we let X, and X , be the gross mole fractions--defined, of course, by the relation X , X, = l-we have for the gas phase: X , = x, (m/q)xc and X , = x, (n/q)xc,and since x, is equal to pc/P, we obtain :
+
+
+
+
These expressions, when combined with equation (1: give the desired isothermal P-X formula.
Before the isothermal P-X formula can be used, however, it is necessary to evaluate the constants K, and 9,. These constants may be obtained from the dissociation pressure, Po, of the solid compound and the degree of dissociatiou, a",of the saturated vapor which is produced from the compound. The required formulas are:
the last equation being a working formula which would enable us to calculate the phase diagrams for solid ammonium chloride and vapor in the system NH,-HCI between the temperature limits mentioned aboye. THE
+
in which equations No is equal to 1 (q - 1)ao. Given the dissociation pressure POand the degree of dissociation ao,one may calculate the P-X isothermal for various assigned values of the' composition X by employing the numbered equations listed above. After obtaining a sufficient number of such P-X isothermals a t different temperatures, one may interpolate T-X isobars and P-T isopleths by graphical or other methods. The amount of labor involved in obtaining the diagrams is, of course, considerable.
CALCULATED
AMMONIUM
CARBAMATE
DIAGRAMS
The Mass Law equations will now be appliedin detail to ammonium carbamate in the system COZ-NHs on the basis of the dissociatiou pressure data which Briggs and Migrdichian (1) published as part of their Mass Law investigation (g. 8.) of the ammonium carbamate equilibrium. Since the carbamate is practically completely dissociated in the vapor and the reaction may COz (gas) 2NHs be written, NHPNH2C02(solid) (gas), we obtain from equation (5) the following isothermal P-X formula:
=
+
THE SPECIAL EQUATIONS FOR COMPLETE DISSOCIATION
In the analysis of the problem up to this point it has been assumed that the dissociation of the compound C is definitely incomplete in the vapor phase. If we assume complete dissociation+. e . , the limiting case in which au = l-the Mass Law equations are greatly simplified and it is possible for us to obtain not only a convenient P-X equation but also an extremely useful combined P-T-X equation which is valid, at least as a good approximation, over a more or less wide range of temperatures. These equations follow. For complete dissociation (i. e., oro = I), p, in the general Mass Law equations becomes zero, No becomes equal to p, and we obtain by combination the single expression:
an isothermal P-X equation which, in a somewhat different form, was first derived by Briner ( 2 ) . The combined P-T-X formula is obtained from equation (5) by introducing the well-known empirical relationship: log Po = A ( B I T ) . The result may be written as follows:
+
log P = A
1 B + -1 log mmn" - - - log ( X I - X B " ) + PP P ¶
(6)
Rodehush and Michalek (3) have 'shown that ammonium chloride is completely dissociated between 501' and 557' Abs. in accordance with the reaction, N H O (solid) $ NHs (gas) HCI (gas), and that the dissociation pressures (in mm.) are represented between these limits by the expression:
+
4402 log PO = 10.1070 - -
T
We therefore obtain, from equations (5) and (G), respectively,
A critical examination of the dissociation pressure data of Briggs and Migrdichian has been made especially for the present study. I t shows that the dissociation pressure Pz (in mm.) may be accurately represented (see Figure 1) between lo0 and 4 5 T . by the equation : log PO = 11.1050
- 2730 T
Accordingly, one obtains from equation (6) the following working P-T-X formula:
We shall now discuss the phase diagrams, which are interesting, though simple and easy to interpret. The P-X diagram (see Figure 2) shows a number of isolog P = 10.8286 - I/alog (Xco,Xpa,~) - 2730 - ( 6 B ) thermals for the binary vapor saturated with solid ammonium carbamate. Each isothermal has been his formula has been employed throughout the drawn about a vertical solidus line a t X N H I = a/3 Present Paper to produce the Mass Law phase diagrams representing ammonium carbamate; hence there is and 3. Its that are shown in Figures produced a set of typical P-X phase diagrams of the of course, is uncertain if the total pressure is high system CO%-NH~ in a limited pressure range, each diaand the temuerature lies outside the limits given gram having a homogeneous field for the unsaturated above. vapor and two heterogeneous fields for saturated vapor and solid carbamate. The point of minimum pressure on the isothermals a t XNnI = 2/3 should be noted; it is, of course, the congruent dissociation point (i. e., dissociation pressure Po) of the solid carbamate a t the specified temperature. Since the isothermal curves are also isothermal contour lines on the dissociation surface of the P-T-X space model (i. e., the surface representing the saturated vapor in equilibrium with solid carbamate), the general shape of the space model can be seen from the diagram in Figure 2. The contours are spaced a t approximately equal temperature intervals. The points (small circles) shown along the calculated isothermal curves in the P-X diagram are experimental P-X determinations that have been derived from the Mass Law tests of Briggs and Migrdichian ( I ) , of which mention has already been made. These tests (p. o.) were made by measuring-at or very close t i the I 1 several temperatures of the isothermals shown on the sR.o.#om **, . OZ *"I o.rraam&w.r & . u l u . r ~ 3 (saimr.s&G* P-X diagram-the total pressure P of the vapor phase when the latter, containing a known quantity of carbon RCURE~.-MASSLAWP-XAND T-XDIAGRAMSOPAMMONIUM CAREAMATE IN THE SYSTEM COa-NHs AT 10' To 50- C. AND 0dioxide or of ammonia in excess of the stoichiometric 400 MM. carbamate ratio, was brought into equilibrium with solid ammonium carbamate. Many such tests were made; and since it is an easy matter-assuming complete dissociation of the carbamate molecules and ideality for the gas-to compute from the published data (seeTable I1 of the Briggs and Migrdichian paper) the mole fraction X N Hin ~ the saturated vapor for each test measurement of P, we have available a large number of independent experimental P-X determinations which may be plotted for comparison along the corresponding calculated Mass Law curves. That most of the experimental points do in fact lie on, or close to, the theoretical curves may be seen from the diagram in Figure 2, which, as presented, should therefore be of special interest to teachers of elementary physical chemistry. In addition to showing the vapor phase relationships of the carbamate in the binary COrNHa system, the diagram gives a good picture of a typical experimental Mass Law test of a case of heterogeneous chemical equilibrium. The calculated T-X diagram-also shown in Figure 2-requires little comment in view of what has been The temuerature maxsaid about the P-X diagram. " ima on the saturated vapor isobars at X N H l= 2/3 are the '08 IS' Zb 8 30' W e congruent dissociation points of solid ammonium car3.-MAss P-T ''AMMoNIUM bamate at the specified pressures. The complete BAMATE IN THE SYSTEMCO1-NHs AT 10' TO 50' C. AND 0similarity of form between these vapor isobars and the 400 MM. -I
"OLE
J,!uAl! MOLC S U . , O " S
'* '=-
solubility curves of congruently melting binary compounds should be noted; this matter should be of interest to students in courses dealing with the phase rule or with chemical thermodynamics. The P - T diagram appears separately in Figure 3. I t consists of a set of typical P - T sections (isopleths) at constant total composition X calculated by means of equation (6B). The composition corresponding to each section is given on the diagram in terms of the mole percentage, 1 0 0 X ~ ~ ,If. we except the special case of the isopleth a t the carbamate composition 100XN~,= 662/3 (in which each field represents a single phase), every P - T section shows a homogeneous field for unsaturated vapor below the P - T curve and a heterogeneous field for solid carbamate and saturated vapor above the P - T curve. As the pressure is increased a t constant temperature in the latter field, more and more solid carbamate is produced a t the expense of the saturated vapor, which becomes steadily richer in the component that is in excess of the carbamate ratio in the system as a whole. The points (small circles) shown along the P - T curve a t the carbamate composition are the actual Podeterminations of Briggs and Migrdichian. It seems worth while to mention one or two other matters in connection with the P-T curves. If we differentiate equation (6B) with respect to T for any assigned value of the composition X and transform the result into the Clausius-Clapeyron equation applied to one mole of solid ammonium carbamate, we obtain, d In P/dT = 37,480/3RT2, and since the same result is obtained for all values of X, the heat of dissociation (expressed above in gram-calories) is independent of the C02/NH3 ratio in the vapor phase. It is also evident from equation (5B) that the ratio Po/Pfor any one of the P - T curves should be independent of the temperature. THE PROBABLE F O M OF THE EXTENDED DIAGRAMS
.
P-X
AND
T-X
The discussion will be concluded with a few paragraphs of a more or less speculative nature concerning the probable form of the P-X and T-X diagrams of Figure 2 when tbese diagrams are extended to high pressures and low temperatures, respectively. At 19.9OC., for example, carbon dioxide condenses to the liquid form a t approximately 56 atm. and ammonia condenses to the liquid at approximately 8.3 a h . I t seems fairly certain, accordingly, that the 19.9'C. carbamate-vapor isothermal in Figure 2 will continue upward as shown until it meets two isothermally invariant systems a t the high pressures-one in the vicinity of 56 atm. representing solid carbamate, binary liquid (i. e., solution), and binary vapor rich in C02, and the other in the vicinity of 8.3 atm. representing solid carbamate, binary liquid, and binary vapor rich in NH8. The phase diagram above tbese pressures will show fields for COz-rich binary liquid, carbamate and COz-rich binary liquid, carbamate and
NH3-rich binary liquid, and NHs-rich binary liquid. Judging from the form of the vapor isothermal shown in Figure 2, it seems likely that the composition of the binary vapor is very close to that of a pure component in each of the invariant systems; hence the same thing is probably true of the binary liquid in these systems and one may therefore hazard the guess that, a t 19.g°C. and the necessary high pressures, the solubility of ammonium carbamate is extremely small both in.liquid carbon dioxide and in liquid ammonia. The construction of the extended P-X diagram for 19.9°C., in schematic form, is suggested as an instructive exercise; but if the extension is applied to others of the group of isothermals shown in Figure 2, one should remember the fact that the critical point of carbon dioxide (31.2"C.) lies within the temperature range of the group, while that of ammonia (132.g°C.) lies well outside. One may speculate in the same way about the form of the T-X isobars extended to low temperatures. The extended diagram for 300 mm. (see Figure 2), for example, presumably would show an isobarically invariant system representing solid carbamate, solid carbon dioxide, and binary vapor rich in COzon the COa side of the diagram in the vicinity of -90°C. (the sublimation temperature of solid carbon dioxide a t 300 mm.), and a second isobarically invariant system representing solid carbamate, binary liquid, and binary vapor rich in NH3 on the NH3 side of the diagram not far from -50°C. (the boiling point of liquid ammonia a t 300 mm.). A third invariant system+. e . , the eutectic system consisting of solid carbamate, binary liquid, and solid ammonia-would also be expected in the diagram in the vicinity of -7S°C., the melting point of ammonia crystals. The schematic T-X phase diagram for 300 mm.-drawn, let us say, so as to cover the temperature range between -100° and 5O0C.would thus show 10 different phase fields, two of which would be homogeneous fields (i. e., for the binary vapor and the binary liquid, respectively), while the others would be heterogeneous fields representing various phase pairs. For the reason already given in connection with the extended P-X diagrams, it again seems likely that the solubility of ammonium carbamate in liquid ammonia is extremely small-probably over the whole temperature range between the eutectic point (ca. -78'C.) and the boiling point of the saturated solution (ca. -50°C.) The vapor in equilibrium with carbamate and solid carbon dioxide near -90°C. is presumably almost pure Cot. As we have been careful to point out throughout the discussion, the diagrams shown in Figures 2 and 3 are based entirely upon the dissociation pressure determinations of Briggs and Migrdichian and are therefore restricted to the temperature interval (lo0 to 45'C.) in which tbese investigators worked. Similar diagrams for the carbamate-vapor system may be calculated for higher temperatures (and correspondingly higher pressures) on the basis of the other dissociation pressure (Continued on page 510)
PHASE DIAGRAMS AND THE MASS LAW (Continued from page 487) ACKNOWLEDDMENT
data which are available in the published literature (cf. I , 2). If one wishes to consider the probable form of these W-temperature diagrams extended to very high pressures, one should take account of the fact that ammonium carbamate has a solid-liquid-vapor triple point a t approximately 152'C. and 83 atm. (2). The compound also becomes chemically unstable at high temperatures.
The present article has been written largely as a to the paper of Briggs and Migrdichiau on the ammonium carbamate equilibriuni and, in closing, the hereby acknowledges it as such, LITERATURE CITED
( 1 ) BRIGGSANU MIGRDIC~IAN. J. P h w Chcm., 28, 1121 (1924)
E,","Gz ~ ~ (1929).
~
~
~
,
"
; Chcm, ; ~ Sot,. ' ~ 51,6 748 ~ ~
6
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