4302
CHRISTIANE GUIRAOAND F. A. WILLIAMS
RThO(ca1cd) = ( 1 / 4 ) R T h L i ~ C 1 (21) Table I11 shows a comparison of the experimental and calculated values of RTho. The experimental data needed for this calculation are the same as those needed for the calculation of eq 11 (see Table 11) and the same sources of data were used.’-l‘ As expected, the differences between experimental and calculated values in Table I11 are the same as in Table 11,except for round-off errors. Two direct experimental measurements of the heat of mixing alkali metal chlorides in the presence of a common concentration of alkaline earth chloride are also reported in Table 111. The direct experimental value of R Tho for the LiCl MgC12-KCl- MgClz mixing (RTho = -15 =t2) is in excellent agreement with the value from the thermodynamic cycle @Tho = -16) and this serves as a check on the accuracy of all of the measurements in the cycle. The direct experimental value of RTho for the KaCl.BaC12-1 1, the regime of transition between continuum and free molecule flow, uo/A N 1,cannot be analyzed accurately. Currently unresolved questions concerning the appropriate exact definition of A, the character of the nonequilibrium molecular velocity distribution function, etc., arise in the transition regime. A theory including transition contains aspects that are inherently less precise than the aspects of a theory restricted to the limiting cases. Moreover, for existing AP sublimation experiments, it is not obviously essential to consider the transition regime, because it is not clear that the experiments enter it. For vacuum sublimation experiments, at, temperatures below 270", u,/A < 1 and vacuum theory should suffice, while for sublimation experiments at pressures above 7 Torr, UO/A > lo2, and departures from continuum flow should not become observable before 6 reaches 0,999, a condition which is not achieved experimentally. For these reasons we have chosen to restrict our theoretical considerations to the two limiting cases. I n interpreting vacuum sublimation experiments, we neglect gas phase diffusion processes. For the more general theoretical considerations presented in subsequent sections, we are concerned only with interpreting experiments performed at pressures above 7 Torr and therefore assume that ao/A >> 1.
6. Equilibrium Vapor Pressure and Rate Observations at Pressures above 7 Torr Although general theoretical considerations may be of some intrinsic interest, we are concerned principally with AP. During the theoretical deliberations, it is therefore desirable to keep in mind experimentally observed facts about AP sublimation. These facts are summarized here. The equilibrium sublimation pressure measurements of Inami, Rosser, and Wise12yield a heat of dissociative 2 kcal/mol. Under the assumpsublimation of 58 tion that the partial pressure of NH3 is equal to that of HC104, their sum, the sublimation pressure P , was found experimentally to be given in mm by the equation
For the isothermal weight-loss experiments of Russell-Jones, a rate constant k may be defined as
If the contracting volume equation were valid, then IC would be the rate constant in that equation; since experimentally the contracting volume equation is not valid under 1 atm of air in the temperature range 304-375", k is merely a constant indicative of the early time rate of weight loss. Figure 1 shows the values of k obtained from the data of Jacobs and Russell-J~nes~~ for sublimation a t atmospheric pressure and at temperatures between 304 and 375". The first and most important observation to be made from Figure 1 is that the activation energy is approximately 30 kcal/mol, just as for vacuum sublimation. The second observation is that the value of k depends on the initial weight of the pellet: k decreases as the sample size is increased. Since only three sizes were used, it is difficult to draw an accurate conclusion concerning the dependence of IC on a,although k a ao-n, with a//z 5 n 5 3, would apparently be consistent with the data, Third it has been stated20that the formula
1 - (1 -
S)"Z
=
k't
(4)
(k' = constant) fits this atmospheric pressure data 6 0.93 in one over a wider range of weight loss (0 case) than does any other simple power formula. Seven data points a t 270" for pressures between 7,75 and 760 Torr produce the rate constant curve shown in Figure 2. The value of k appearing here is that defined in eq 3. The formula
<
> nA, n B where R is Boltzmann's constant. Two kinds (i = 1, 2 ) of adsorption sites will be permitted to exist on the surface, and each site will be permitted to adsorb both species A and B. Our belief is that real surfaces contain practically an infinite number of different kinds of sites, and our hope is that variations in site characteristics can be modeled with sufficient accuracy by considering only two kinds. We might remark that models considering surface diffusion necessarily require at least two kinds of sites. We do not consider surface diffusion in the present theories. The number of molecules of species k per second leaving the surface of the sphere from sites of type i is Qki
=
4na2fi[Wk.leki
-
Vkakt(1
IC
- 8r)nkI; = A, B,
C; i = 1, 2 ( 6 )
where fz is the fraction of the surface identifiable as sites of kind i (fz = 1 - fl), wki is the number of molecules of type k per unit area per second that would enter the gas from a surface of kind i completely covered The Journal of Physical Chemistry
&A(
=
Q B ~ ;
i
=
(8)
1, 2
(9 )
Equilibrium for the first step in the reaction requires eAieBl/(l -
e,)2
=
K ( ;i = 1 , 2
(10)
where the equilibrium constants K i depend only on temperature T . The constants K , can be related to the equilibrium constant for concentrations for the over-all sublimation process
K,
=
nAOnBO
= (P/2RT)'
(11)
through the equilibrium identities
K, =
i
1, 2
K ~ w A ~ w B ~ / v A v B ~ A ~ ~ B ~ ;
(12)
The subscript 0 on nk identifies equilibrium, and the equilibrium vapor pressure P appears in eq 2 for AP. A mass balance for the sphere yields Qk
= -(p/m)4aa2(da/dt);
IC
=
A, B
(13)
where p/m is the molecule (AB) number density of the solid. Equations 6 through 10 can be thought of as 14 equations in the 14 unknowns Q k i , &.I, n ~ and , nB; solution of these equations followed by substituting into eq 13 yields an explicit expression for da/dt in terms of u and constants that are presumed known. By eliminating nk between eq 6 and 7, we obtain
AMMONIUM PERCHLORATE
SUBLIMATION OF
4307
When solutions for elcahave been obtained, eq 14 can be substituted into eq 13 to yield an expression for the rate of weight loss. To understand the kinds of results that can ensue, it is instructive to investigate limiting cases.
8. Diffusion-Controlled Limit with One Kind of Site If f2 = 0 ( i e . ,fl = 1 ) and if Dkc/avkaki(I - e,)
