Partitions with parts in a finite set

Let be a nonempty finite set of relatively prime positive integers, and let denote the number of partitions of with parts in . An elementary arithmetic argument is used to prove the asymptotic formula

     

Evaporation of water and uptake of HCl and HBr through hexanol films at the surface of supercooled sulfuric acid

Vacuum evaporation and molecular beam scattering experiments have been used to monitor the loss of water and dissolution of HCl and HBr in deuterated sulfuric acid at 213 K containing 0 to 100 mM hexanol. The addition of 1-hexanol to the acid creates a surface film of hexyl species. This film becomes more compact with decreasing acidity, ranging from approximately 62% to approximately 68% of maximum packing on 68 to 56 wt % D(2)SO(4), respectively. D(2)O evaporation from 68 wt % acid remains unaltered by the hexyl film, where it is most porous, but is impeded by approximately 20% from 56 and 60 wt % acid. H --> D exchange experiments further indicate that the hexyl film on 68 wt % acid enhances conversion of HCl and HBr into DCl and DBr, which is interpreted as an increase in HCl and HBr entry into the bulk acid. For this permeable hexyl film, the hydroxyl groups of surface hexanol molecules may assist uptake by providing extra sites for HCl and HBr hydrogen bonding and dissociation. In contrast, HCl --> DCl exchange in 60 wt % D(2)SO(4) at first rises with hexyl surface coverage but then drops back to the bare acid value as the hexyl species pack more tightly. HCl entry is actually diminished by the hexyl film on 56 wt % acid, where the film is most compact. These experiments reveal a transition from a porous hexanol film on 68 wt % sulfuric acid that enhances HCl and HBr uptake to one on 56 wt % acid that slightly impedes HCl and D(2)O transport.     

Molecular beam studies of HCl dissolution and dissociation in cold salty water

Gas-liquid scattering experiments are used to explore collisions and reactions of HCl and DCl with 12 mol% LiBr solutions of H(2)O and D(2)O at 208-218 K. These ～6 M aqueous salt solutions have vapor pressures just below 0.01 Torr, requiring special consideration of the effects of gas-vapor collisions. We find that impinging HCl molecules readily equilibrate on the surface of the solution even at incident energies of 90 kJ mol(-1). Approximately 90% of the thermalized HCl molecules dissolve and dissociate for long times in the cold salty solution, while the remaining 10% desorb from the surface intact. There is no evidence for rapid, interfacial conversion of HCl into DCl, in striking contrast to previous observations of distinct submicrosecond DCl→HCl exchange in collisions of DCl with salty glycerol at 292 K. These results indicate that cold salty water efficiently captures impinging HCl molecules and suppresses interfacial proton exchange, most likely because of the long interaction times of the HCl molecules in contact with the cold surface and because of facile transport of H(+) and Cl(-) from the interfacial region into the bulk solution.     

The inhibition of N2O5 hydrolysis in sulfuric acid by 1-butanol and 1-hexanol surfactant coatings

Gas-liquid scattering experiments are used to measure the fraction of N2O5 molecules that are converted to HNO3 after colliding with 72 wt % H2SO4 containing 1-hexanol or 1-butanol at 216 K. These alcohols segregate to the surface of the acid, with saturation coverages estimated to be 60% of a close-packed monolayer for 1-hexanol and 44% of a close-packed monolayer for 1-butanol. We find that the alkyl films reduce the conversion of N2O5 to HNO3 from 0.15 on bare acid to 0.06 on the hexyl-coated acid and to 0.10 on the butyl-coated acid. The entry of HCl and HBr, however, is enhanced by the hexanol and butanol films. The hydrolysis of N2O5 may be inhibited because the alkyl chains restrict the transport of this large molecule and because the alcohol OH groups dilute the surface region, suppressing reaction between N2O5 and near-interfacial H3O+ or H2O. In contrast, the interfacial alcohol OH groups provide additional binding sites for HCl and HBr and help initiate ionization. These and previous scattering experiments indicate that short-chain alcohol surfactants impede or enhance sulfuric acid-mediated reactions in ways that depend on the chain length, liquid phase acidity, and nature of the gas molecule.     

Density of sets of natural numbers and the Lévy group

Let N denote the set of positive integers. The asymptotic density of the set A⊆N is d(A)=limn→∞|A∩[1,n]|/n, if this limit exists. Let AD denote the set of all sets of positive integers that have asymptotic density, and let SN denote the set of all permutations of the positive integers N. The group L^? consists of all permutations f∈SN such that A∈AD if and only if f(A)∈AD, and the group L^* consists of all permutations f∈L^? such that d(f(A))=d(A) for all A∈AD. Let be a one-to-one function such that d(f(N))=1 and, if A∈AD, then f(A)∈AD. It is proved that f must also preserve density, that is, d(f(A))=d(A) for all A∈AD. Thus, the groups L^? and L^* coincide.