Radon partitions

Radon is slightly soluble in water, and obeys Henry s Law. At 20°C the partition coefficient (amount of radon per litre of water at equilibrium divided by the amount per litre of air) is 0.26. Despite the low solubility, water supplies derived locally from granite and metamorphic rocks can be an important source of airborne radon in dwellings (Nero Nazaroff, 1984 Hess etal., 1987). Radon is more soluble in fats and organic liquids, and the partition coefficient between air and human fat is about unity at 37°C. 

Early experiments in liquids were quite variable for many reasons. The conductivity technique, which was used in the gas phase to measure dose, was not applicable to the liquid phase. Reactions were measured using dissolved radium salts or radon gas as the ionization source. Some thought the chemistry was due to the reactions with radium however, it was soon recognized that it was the emitted rays that caused the decomposition. Both radium and radon could cause radiation damage. Because the radon would be partitioned between the gas and liquid phase, the amount of energy that was deposited in the liquid depended critically on the experimental conditions such as the pressure and amount of headspace above the liquid. In addition, because the sources were weak, long irradiation times were necessary and products, such as hydrogen peroxide, could decompose. 

US Environmental Protection Agency (EPA) (1999b) Understanding variation in partition coefficient, Kd, values Volume II. Review of geochemistry and available Kd values for cadmium, cesium, chromium, lead, plutonium, radon, strontium, thorium, tritium (3H) and uranium. Prepared for the EPA by Pacific Northwest National Laboratory. 

Naturally occurring radionuclides such as radium isotopes and radon-222 have gained popularity as tracers of SGD due to their enrichment in ground-water relative to other sources and their built-in radioactive clocks . The enrichment of these tracers is owed to the fact that the water-sediment ratio in aquifers is usually quite small and that aquifer sediments (and sediments in general) are enriched in many U and Th series isotopes while many of these isotopes are particle reactive and remain bound to the sediments, some like Ra can easily partition into the aqueous phase. Radon-222 (tia = 3.82 days) is the daughter product of Ra (G/2= 1600 years) and a noble gas therefore, it is even more enriched in groundwater than radium. 

Environmental Fate. Information is available on the environmental fate of radon in air and water and on the transport of radon in environmental media. Factors which affect the partitioning of radon from soil or water to air have been identified. Flowever, rates of flux from one media to another are rarely reported. The emanation rate of radon from soil is uncertain. Additional information on the behavior of radon at the soil-air interface, as well as soil-gas measurements, would facilitate a better understanding of the emanation rate of radon from soil. Movement of radon into and within homes and the influence of meteorological conditions on this movement should be investigated. Study of radon movement would enhance understanding of potential indoor exposures. Transformation of radon has been adequately characterized. There is limited information on the uptake of radon by plants. Additional research of this phenomenon is needed in order to determine the effects of exposures which might be incurred from ingestion of food. 

The measurement of the radon content of water is based on extraction processes that exploit the high partition coefficient of radon either between gas and water or between organic liquid scintillators and water. In particular, by introducing fine gas bubbles to water, it is possible to extract radon very efficiently. Radon is thus bubbled out from water and collected in a Lucas cell. The detection limit for this method is very low, 50Bqm . ... 

In this chapter we describe the unified generation of stereoisomers including conform-ers of a molecular structure [102,103,105]. This method has the potential to generate stereoisomers that cannot be described in terms of stereocenters, stereogenic double bonds or single bond rotations. Fundamentals such as the concept of a (partial) orientation function are discussed, and mathematical tools such as Radon partitions and binary Grassmann-Plucker relations are used to construct tests for abstract orientation functions. Some simple examples are treated in detail. 

A Radon partition (Johann Radon, Austrian mathematician, 1887-1956) is a pair A, B) of disjoint subsets of atoms, i.e. of points in 3D space, such that their convex hulls intersect. 

We call a Radon partition A, B minimal if and only if it is not possible to remove an atom from either set A or B without destroying the Radon partition property. In other words, the Radon partition A, B is minimal if and only if there is no Radon partition A, B A, B with A c A and B c b. 

Fig. 4.7. Radon partitions thin lines clarify geometrical relations, while the thick lines in (d) represent chemical bonds.

All Radon partitions shown in Figure 4.7 are minimal except for Figure 4.7b. A minimal Radon partition in Figure 4.7b is a, d, b, e. Note that a minimal Radon partition contains at most five atoms. Radon s theorem [63] states that any set of five atoms can be partitioned into a Radon partition. Furthermore, each set of five atoms that are not all in one plane contains exactly one minimal Radon partition. 

One fascinating and very useful feature of minimal Radon partitions is that they can be obtained from the orientation function, i.e. on a purely combinatorial level, without considering coordinates. For this purpose we introduce abstract minimal Radon partitions for an abstract orientation function. It will turn out that a pair A, B is a minimal Radon partition of a conformation if and only if it is an abstract minimal Radon partition of the corresponding orientation function. Before giving a formal definition of abstract minimal Radon partition, we first consider an example ... 

We obtain an abstract minimal Radon partition as the pair A, B of sets of atoms where set A consists of all atoms a whose two signs agree, and set B consists of all atoms with opposite signs. In the exeunple, we have... 

Definition (Abstract minimal Radon partition) An abstract minimal Radon partition of an abstract orientation function over atoms 0,...,n I) is a pair (A.B) of disjoint subsets of the set of atoms,... 

Note that if x(cio> > i i> i+i> . 4) = 0, then a, is neither in A nor in B, i.e. an abstract minimal Radon partition A, B may contain fewer than five atoms. Still, as for non-abstract minimal Radon partitions, exactly one abstract minimal Radon partition is assigned to each set of five atoms. Now, considering a conformation and its orientation function (which is also an abstract orientation function), we can formulate the following... 

Remark (Characterization of minimal Radon partitions) A pair A,B of disjoint... 

The implication of an abstract minimal Radon partition to minimal Radon partition is still true as long as the abstract minimal Radon partition consists of exactly five atoms. For abstract minimal Radon partitions with fewer than five atoms, a quadruple of atoms with assigned zero-orientation may not lie exactly in a plane. However, as the convex hulls nearly intersect , we can interprete this as some kind of tolerance for Radon partitions. For more details and a proof of Remark 4.16 refer to [105]. However, the relevance of the statement given there is not obvious at a first glance, so we note the following ... 

Oriented matroids are introduced in [25] on page 103 using the structure of (signed) circuits, here named abstract minimal Radon partitions. On pages 104f in [25] (example The CUBE ), it is demonstrated that minimal Radon partitions of a sequence of points correspond to the circuits of an oriented matroid. 

Theorem 3.5.5 in [25] finally states that the pairs of functions consisting of a chirotope X and its negative x are in one-to-one correspondence to oriented matroids. lemma 3.5.7 in [25] explains how to obtain the signed circuits (i.e. the abstract minimal Radon partitions) from the chirotope. See also Remark 3.7.3 in the same reference. 

Example (Figure 4.2c, cont.) We already mentioned that a central carbon atom and its four neighbors form a minimal Radon partition, such as in exeimple 4.2c. Thus, A, B = 0, (1,2,3,4 is a minimal Radon partition of conformation Ic. According to Remark 4.16, A, B is also an abstract minimal Radon partition of its orientation function. Knowing only the partial orientation function shown in the subsection on partial orientation functions, we have the foUowing situation, corresponding to the quintuple (0,1,2,3,4) ... 

As we know that the minimal Radon partition separates oq from the remaining four atoms, and as the signs in the table corresponding to uq eire equal, it follows that the signs for atoms a -a need to be opposite in the table. Thus we conclude ... 

Further tests based on minimal Radon Partitions... 

A conformation cannot have a minimal Radon partition of the form A, 0, as the convex hull of the empty set is empty, and thus, trivially, its intersection with the convex hull of A cannot be nonempty. However, abstract PDFs may have abstract minimal Radon peutitions of the form A, 0. Such abstract PDFs are not realizable because of Remark 4.16. This reasoning is correct even for zero-tolerances. Thus we can restrict our candidate abstract POFs by the following test ... 

Test 5. Ensure that x has no abstract minimal Radon partition of the form A, 0. ... 

Fig. 4.8. Two chemically forbidden minimal Radon partitions full lines represent chemical bonds, while broken lines clarify geometrical relations.

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