DAVID I.KERN Dartmouth College, Hanover, New Hampshire R A R E L Y has an important constant resisted accurate evaluation as successfully as the sublimation energy of carbon. In the last 40 years estimates of this elusive quantity have ranged from 70 to over 200 kcal. per gram atom. Since 1948 work on the problem has been especially active. New data have been appearing thick and fast in the journals, new techniques have been introduced, and it appears that a t long last the problem is approaching its final solution. In this article the experimental methods employed, together with the attendant difficulties, are briefly described and the course of the investigations is traced over the last 25 years with particular emphasis on the latest developments. A special interest is attached to the determination of the sublimation heat of carbon because of its relevance to structural theoriesin organic chemistry. Any theory of structure must account in the first place for the strength of chemical bonds. In organic chemistry, this means primarily the strength of bonds involving the carbon atom. The combined energies of the bonds in a compound can be defined as the energy required to break up the molecule into the constituent gaseous atoms; that is, it is numerically equal to the heat of formation from the gaseous atoms. The heat of formation from the elements in their standard states is readily obtainable from heats of combustion and other thermometric data. The missing link is the heat involved in converting the element from its standard state to the gaseous atom. For elements like oxygen and hydrogen this is given by the dissociation energy of the diatomic molecules. For carbon, it is given by the sublimation energy of graphite to the monatomic gas. There are two main paths by which the determination of the heat sublimation of carbon may be approached. They are (1) vaporpressure measurements over graphite a t high temperatures and (2) measurement of dissociationenergies of simple gas molecules containing C, like CO, CzH2,C1N2,etc. The results of both of these methods are open to various interpretations, so that there is still no universally accepted value.
THE HEAT OF SUBLIMATION FROM VAPOR PRESSURES The conventional way to determine heats of sublimation is by the measurement of vapor pressure. According to the familiar ClausiusClavevron eauation. "
dlnPd m ) 
AH,
R
A plot of in P against 1/T gives a line with slope proportional to the heat of sublimation, m,,a t the experimental temperature.
Preferable to this is the socalled third law relation,' by which a single vaporpressure measurement is sufficient to calculate the heat of sublimation a t absolute zero, denoted by the symbol AH:. (It is this heat which is significant in bond energy calculations. As it differs from the value a t the experimental temperatures (2500°K.) by only a kilocalorie or so, the distinction can be neglected; in later sections both heats will be designated by the symbol L.) The third law relation can be applied only if values of the function (Fa H,)/T are available for both the solid and its vapor a t the temperature of the experiment. The functions have been estimated for many substances, for solids from heatcapacity data, and for gases by the methods of statistical thermodynamics. Fortunately these functions vary slowly with temperature and accurate extrapolations can be made over hundreds of degrees. They have been tabulated for graphite, and for the gaseous carbon species CI, CZ,and C3. Employing the well known equality AFo =  RT In P for the free enerw of sublimation. we find
where Fo is the free energy a t unit activity, H : the heat content a t absolute zero, and P the equilibrium vapor pressure a t the temperature T of the experiment. The only unknown in this equation is AH,, the standard heat of sublimation a t absolute zero. The determination of the vapor pressures of highly nonvolatile substances is not easy. Thus it is necessary to go to very high temperatures (2400'280O0K.) to obtain an accurately measurable vapor pressure over graphite, and even then it is only of the order of lo' atmospheres. Two techniques commonly employed in this type of work are the Langmuir or evaporationrate method, and the effusioncell or equilibrium method. These will be discussed in the following sec,. "On. The Evaporationrate Method. When a substance is at equilibrium with its vapor, the rates of condensation and evaporation must necessarily be equal. The rate of condensation of a eas is related to its oressure bv an equation (see below) derived from the kinetic molecular gas theory. An identical equation must accordingly relate the evaporation rate and the equilibrium vapor pressure. In the Langmuir method, the solid is heated in a vacuum so complete that an atom, once evaporated, has a negligible chance of recondensing. The solidvapor

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For a dirreussion of the funotions described in this pmagaph, CHEM.EDUC., 32, 520 (1955).
see MARGRAVE, JOHNL.,J.
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reaction is thus entirely unidirectional; its rate is determined by the rate of evaporation alone, which can be calculated directly from the rate of weight loss of the sample. The evaporation rate is a constant a t a given temperature, unaffected by a departure from equilibrium conditions. Hence the rate measured in vacw is the same as the rate a t equilibrium and can be used to calculate the equilibrium vapor pressure. The combined weight of the gas molecules which collide with or pass through unit area in unit time can be derived from kinetic theory and is giveu by P ~ M RT. It is not a priori certain that all gas molecules colliding with the surface of the corresponding solid actually stick. A socalled sticking or accommodation coefficient a is therefore introduced when the above expression is applied to the condensation rate. We have

rate of condensation (g./sec.) = r, =
~ A P ~ M J ? I ~ I R(2) T
equilibrium pressure can build up, and the vapor is allowed t o leak into vacuum through a small orifice in the cell wall. It follows from the theory discussed above that rate of effusion =
r.1, =
ap
(4)
where a is the area of the orifice. This method avoids the uncertainty introduced by the accommodation coefficient a in equation (2), for if the orifice is properly designed all gas molecules entering it will pass through. The question remains whether the pressure so measured is the trne equilibrium pressure, for the leakage of vapor may be enough to prevent the equilibrium pressure from ever building up. The flu balance in the effusion cell is represented by the equation evaporation rate = condensation rate 1..
= I.
++ effusionrate
(5)
The condition for solidvapor equilibrium, and hence for P = P., in equation (4), is that re. = 7,. To the rate of evaporation (g./sec.) = 7.. = ~AP~.T (3) extent that r+, disturbs this equality the pressure determined from the effusionrate will he less than theequilibwhere A is the surface area of the sample. Evidently a must be known if P., is t o be calculated rium pressure. The importance of rq, relative to r. from the evaporation rate. I n most evaporations it is in equation ( 5 ) is thus the critical factor. Comparison taken as unity. A coefficient less than one can be of equations (2) and (4) shows the ratio ra/r. is given interpreted as being due to an energy harrier in the by a/Aa or u/a, where u is the ratio of orifice size t o the evaporationcondensation process. I n fact it can be area of the sample: shown that a exp(AH*/RT), where LW* is the a = a/A (6) height of the harrier. Condensation processes in which The condition that P = P., is satisfied if /a (< 1. the gas species is closely related to the solid species would be expected to have no barrier; the gas molecules I n other words, if the effusion rate is to be a measure of or atoms can simply "fall into place" on the solid the equilibrium pressure, the limitation on orifice size surface. Thus every molecule that hits condenses, is the more stringent the smaller the value of a. The restriction that a places on u can be explained alternaand a = 1. If the condensation requires the formation of high tively as follows. It has been shown that a small energy intermediate states, as might happen, for in a corresponds to an energy barrier in the evaporation stance, if the gas species were quite diierent from the process, i . e., to a sluggish evaporationcondensation solid species, a barrier would he predicted. Thus red equilibrium. Such a sluggish system will be much less phosphorus shows a very slow rate of condensation in able to withstand the constant draining away of vapor relation t o its equilibrium vapor pressure, because the by effusion, than will a highly resilient system with gaseous P4molecules must be rearranged to form the Pt a = 1. Hence the necessity for smaller orifices for molecules of the solid phase. For red phosphorus systems with small a. Measurements with too large an orifice will yield vapor pressures less than the equiliba is about lo%, indicating that only one millionth of rium value, corresponding to an erroneously high heat the atoms hitting the surface have sufficient energy to surmount the barrier.% It will be shown belowthat of sublimation. The above discussion shows that the value of the the value of a in the carbon vapor equilibrium has accommodation coefficient is a key to the interpretabeen a major bone of contention between workers in the field. Suggested values vary from 1 to 104. From tion of both evaporation and effusion experiments. equation(3) it is seen that proponents of a small value I n both cases high heats of sublimation are associated 1, and low heats with would obtain high vapor pressures from their experi with the assumption that a a