constant; only the relative quantity of the two phases changes. Until all the gas has passed into liquid a further decrease of volume will not require increase of pressure. But as soon as the liquefaction is complete a slight decrease of volume will require a great increase of pressure, liquids being but slightly compressible.
The pressure required to condense a gas varies with the temperature, becoming higher as the temperature rises. The highest pressure will therefore be found at Tc and the lowest at T3. We shall represent the pressure at Critical pressure. Tc by pc. It is called the critical pressure. The pressure at T3 we shall represent by p3. It is called the pressure of the triple point. The values of Tc and pc for different substances will be found at the end of this article. The values of T3 and p3 are accurately known only for a few substances. As a rule p3 is small, though occasionally it is greater than 1 atmosphere. This is the case with CO2, and we may in general expect it if the value of T3/Tc is large. In this case there can only be a question of a real boiling-point (under the normal pressure) if the liquid can be supercooled.
We may find the value of the pressure of the saturated vapour
for each T in a geometrical way by drawing in the theoretical
isothermal a straight line parallel to the v-axis in such a way
that ∫v2
v1pdv will have the same value whether the straight
line or the theoretical isothermal is followed. This construction,
given by James Clerk Maxwell, may be considered as a result
of the application of the general rules for coexisting equilibrium,
which we owe to J. Willard Gibbs. The construction derived
from the rules of Gibbs is as follows:—Construe the free energy at
a constant temperature, i.e. the quantity – fpdv as ordinate, if the
abscissa represents v, and determine the inclination of the double
tangent. Another construction derived from the rules of Gibbs
might be expressed as follows:—Construe the value of pv − ∫pdv
as ordinate, the abscissa representing p, and determine the point
of intersection of two of the three branches of this curve.
As an approximate half-empirical formula for the calculation of the pressure, −log10 ppc = f (Tc – T)T) may be used. It would follow from the law of corresponding states that in this formula the value of f is the same for all substances, the molecules of which do not associate to form larger molecule-complexes. In fact, for a great many substances, we find a value for f, which differs but little from 3, e.g. ether, carbon dioxide, benzene, benzene derivatives, ethyl chloride, ethane, &c. As the chemical structure of these substances differs greatly, and association, if it takes place, must largely depend upon the structure of the molecule, we conclude from this approximate equality that the fact of this value of f being equal to about 3 is characteristic for normal substances in which, consequently, association is excluded. Substances known to associate, such as organic acids and alcohols, have a sensibly higher value of f. Thus T. Estreicher (Cracow, 1896) calculates that for fluor-benzene f varies between 3.07 and 2.94; for ether between 3.0 and 3.1; but for water between 3.2 and 3.33, and for methyl alcohol between 3.65 and 3.84, &c. For isobutyl alcohol f even rises above 4. It is, however, remarkable that for oxygen f has been found almost invariably equal to 2.47 from K. Olszewski’s observations, a value which is appreciably smaller than 3. This fact makes us again seriously doubt the correctness of the supposition that f = 3 is a characteristic for non-association.
It is a general rule that the volume of saturated vapour decreases when the temperature is raised, while that of the coexisting liquid increases. We know only one exception to this rule, and that is the volume of water Critical volume. below 4° C. If we call the liquid volume vl, and the vapour vv, vv – vl decreases if the temperature rises, and becomes zero at Tc. The limiting value, to which vl and vv converge at Tc, is called the critical volume, and we shall represent it by vc. According to the law of corresponding states the values both of vl/vc and vv/vc must be the same for all substances, if T/Tc has been taken equal for them all. According to the investigations of Sydney Young, this holds good with a high degree of approximation for a long series of substances. Important deviations from this rule for the values of vv/vl are only found for those substances in which the existence of association has already been discovered by other methods. Since the lowest value of T, for which investigations on vl and vv may be made, is the value of T3; and since T3/Tc, as has been observed above, is not the same for all substances, we cannot expect the smallest value of vl/vc to be the same for all substances. But for low values of T, viz. such as are near T3, the influence of the temperature on the volume is but slight, and therefore we are not far from the truth if we assume the minimum value of the ratio vl/vc as being identical for all normal substances, and put it at about 13. Moreover, the influence of the polymerization (association) on the liquid volume appears to be small, so that we may even attribute the value 13 to substances which are not normal. The value of vv/vc at T = T3 differs widely for different substances. If we take p3 so low that the law of Boyle-Gay Lussac may be applied, we can calculate v3/vc by means of the formula p3v3T3 = kpcvcTc, provided k be known. According to the observations of Sydney Young, this factor has proved to be 3.77 for normal substances. In consequence v3vc = 3.77pcp3 T3Tc. A similar formula, but with another value of k, may be given for associating substances, but with another value of k, may be given for associating substances, provided the saturated vapour does not contain any complex molecules. But if it does, as is the case with acetic acid, we must also know the degree of association. It can, however, only be found by measuring the volume itself.
E. Mathias has remarked that the following relation exists between the densities of the saturated vapour and of Rule of the rectilinear diameter. the coexisting liquid:—
ρl + ρv = 2ρc 1 + a(1 – TTc),
and that, accordingly, the curve which represents the densities at different temperatures possesses a rectilinear diameter. According to the law of corresponding states,a would be the same for all substances. Many substances, indeed, actually appear to have a rectilinear diameter, and the value of a appears approximatively to be the same. In a Mémoire présenté à la société royale à Liège, 15th June 1899, E. Mathias gives a list of some twenty substances for which a has a value lying between 0.95 and 1.05. It had been already observed by Sydney Young that a is not perfectly constant even for normal substances. For associating substances the diameter is not rectilinear. Whether the value of a, near 1, may serve as a characteristic for normal substances is rendered doubtful by the fact that for nitrogen a is found equal to 0.6813 and for oxygen to 0.8. At T = Tc/2, the formula of E. Mathias, if ρv be neglected with respect to ρl, gives the value 2 + a for ρl/ρc.
The heat required to convert a molecular quantity of liquid coexisting with vapour into saturated vapour at the same temperature is called molecular latent heat. It decreases with the rise of the temperature, because at a higher Latent heat. temperature the liquid has already expanded, and because the vapour into which it has to be converted is denser. At the critical temperature it is equal to zero on account of the identity of the liquid and the gaseous states. If we call the molecular weight m and the latent heat per unit of weight r, then, according to the law of corresponding states, mr/T is the same for all normal substances, provided the temperatures are corresponding. According to F. T. Trouton, the value of mr/T is the same for all substances if we take for T the boiling-point. As the boiling-points under the pressure of one atmosphere are generally not equal fractions of Tc, the two theorems are not identical; but as the values of pc for many substances do not differ so much as to make the ratios of the boiling-points under the pressure of one atmosphere differ greatly from the ratios of Tc, an approximate confirmation of the law of Trouton may be compatible with an approximate confirmation of the consequence of the law of corresponding states. If we take the term boiling-point in a more general sense, and put T in the law of
