Avogadro Constant Research Articles

Experimental determination of the Avogadro constant

Experimental determination of the Avogadro constant

Open questions in the physics of deformation of polymer glasses

Polymer glasses have very appealing properties for a number of optical and other applications and their high resistance to plastic deformation, expressed as the ratio of low temperature yield stress to elastic modulus of order 0.05, is an additional bonus. Their normalized resistance to crack propagation, as measured by their ratio of fracture energy Gc to 2γ, twice the surface energy, is more than 5000 for polymer glasses whose molecular weight exceeds 10 Me, where Me is their entanglement molecular weight. These are potentially very tough materials, which nevertheless have an important flaw, their propensity to deform plastically by crazing, which competes with shear deformation. Because crazes are highly localized zones of plastic deformation that can break down to form cracks at very low overall strains, crazing is a source of brittleness. Suppressing crazing and encouraging less localized shear deformation is thus an important molecular design goal. It has been known for some 20 years that two glassy polymer properties strongly influence whether crazing or shear deformation dominates, the physical aging of the polymer glass and its entangled strand density νe = ρNAv/Me, where ρ is the polymer strand mass density and NAv is Avogadro's number. Physical aging of the polymer glass increases the shear yield stress, as measured for example in compression, above the flow stress, the stress that is required to continue shear deformation at moderately larger strains. The flow stress has been shown to be independent of physical aging, giving rise to so-called strain softening, a decrease in (true) stress as the deformation continues past yield until the flow stress is reached. Recent experiments demonstrate unequivocally that a mild plastic predeformation by rolling erases the effects of physical aging (“rejuvenation”) so that if the polymer glass is deformed immediately after, even in tension, the yield stress and flow stress are the same. Shear deformation is favored over crazing even in polymers such as polystyrene (PS), which would normally craze in preference to deform by shear.1 Aging at room temperature for 48 h is sufficient to restore the normal yield and crazing behavior of PS. Obviously an understanding of the changes in structure of the glass during the initial stages of plastic deformation by shear is crucial to achieving a fundamental understanding of the competition between brittle (crazing) and ductile (shear) modes of deformation of polymer glasses, but to date very little is known except from molecular dynamics simulations of model Lennard-Jones nonpolymeric glasses under conditions of constant density at 0 K.2, 3 These simulations, viewed from the standpoint of energy landscape models, show that the rejuvenation by deformation is due to irreversible creation of structural states with shallower energy minima, that resemble, but are distinct from, those of the glass freshly quenched from an equilibrium melt, and that physical aging corresponds to the relaxation of these structural states toward those with deeper energy minima. This relaxation at constant density restores the yield point and strain softening behavior. Although previously this relaxation has been taken to be due to a decrease in free volume, the simulations show it occurs even in the absence of such a decrease. In addition, there is no experimental evidence of a volume increase during and immediately after moderate shear deformation that produces rejuvenation. In fact, a volume decrease has been reported.1 Recent positron annihilation experiments support this conclusion.4, 5 Simulations that treat the aging and rejuvenation of polymeric glasses, at constant pressure rather than constant density and at finite temperature, will certainly help us to clarify these important issues. Strain hardening modulus GR for PPO/PS blends and cross-linked PS as well as the rubbery plateau modulus G of PPO/PS and modulus Gx of rubbery cross-linked PS versus the strand density νe of the entanglement and cross-link density. Although both sets of moduli scale linearly with νe, the hardening modulus is two orders of magnitude larger than the true modulus of the entangled or cross-linked elastomer. (Data taken from Ref.6.) Although clearly experiments that can monitor the changes in strand dimensions with λ can help us understand how to fundamentally explain strain hardening in polymer glasses and its dependence on νe, I think that simulations can perhaps play an even more important role. In this regard the work of Rottler, Robbins, and coworkers suggests that it should now be possible to arrive at a molecular level understanding of this phenomenon.7 I therefore issue this as a challenge to the simulation community and look forward to a much improved physical picture for this important property of polymer glasses.

Protofibril Formation of Amyloid β-Protein at Low pH via a Non-cooperative Elongation Mechanism

Deposition of the amyloid beta-protein (Abeta) in senile or diffuse plaques is a distinctive feature of Alzheimer's disease. The role of Abeta aggregates in the etiology of the disease is still controversial. The formation of linear aggregates, known as amyloid fibrils, has been proposed as the onset and the cause of pathological deposition. Yet, recent findings suggest that a more crucial role is played by prefibrillar oligomeric assemblies of Abeta that are highly toxic in the extracellular environment. In the present work, the mechanism of protofibril formation is studied at pH 3.1, starting from a solution of oligomeric precursors. By combining static light scattering and photon correlation spectroscopy, the growth of the mass and the size of aggregates are determined at different temperatures. Analysis and scaling of kinetic data reveal that under the studied conditions protofibrils are formed via a single non-cooperative elongation mechanism, not prompted by nucleation. This process is well described as a linear colloidal aggregation due to diffusion and coalescence of growing aggregates. The rate of elongation follows an Arrhenius law with an activation enthalpy of 15 kcal mol(-1). Such a value points to a conformational change of peptides or oligomers being involved in binding to protofibrils or in general to a local reorganization of each aggregate. These results contribute to establishing a clearer relation at the molecular level between the fibrillation mechanism and fibrillar precursors. The observation of a non-cooperative aggregation pathway supports the hypothesis that amyloid formation may represent an escape route from a dangerous condition, induced by the presence of toxic oligomeric species.

Quantum effects in many-body gravitating systems

A hierarchy of equations for equilibrium reduced density matrices obtained earlier is used to consider systems of spinless bosons bound by forces of gravity alone. The systems are assumed to be at absolute zero of temperature under conditions of Bose condensation. In this case, a peculiar interplay of quantum effects and of very weak gravitational interaction between microparticles occurs. As a result, there can form spatially-bounded equilibrium structures macroscopic in size, both immobile and rotating. The size of a structure is inversely related to the number of particles in the structure. When the number of particles is relatively small the size can be enormous, whereas if this numbder equals Avogadro's number the radius of the structure is about 30 cm in the case that the structure consists of hydrogen atoms. The rotating objects have the form of rings and exhibit superfluidity. An atmosphere that can be captured by tiny celestial bodies from the ambient medium is considered too. The thickness of the atmosphere decreases as its mass increases. If short-range intermolecular forces are taken into account, the results obtained hold for excited states whose lifetime can however be very long. The results of the paper can be utilized for explaining the first stage of formation of celestial bodies from interstellar and even intergalactic gases.

Evidence of a Systematic Deviation of the Isotopic Composition of Neon from Commercial Sources Compared with Its Isotopic Composition in Air

Results are reported of a study concerning the variation in isotopic composition of a limited number of neon samples of commercial origin and the resulting influence on the temperature of the triple point of this element. All seven neon samples investigated were found to contain more 22Ne than neon in air, and the amount fraction of 22Ne varied by as much as 0.2% from sample to sample. This variation corresponds to a range of triple-point temperatures (Ttp) of more than 200 microK, much larger than the state-of-the-art uncertainty in the realization of this phase transition for metrological purposes. Deviations in the amount fractions of 21Ne were irrelevant, as far as their effect on T(tp) is concerned, though they may have relevance to other isotope studies. Ratios of amounts of neon isotopes at IRMM-Geel were obtained using the same measurement procedures, and instrumentation developed in the framework of the redetermination of the Avogadro constant and all significant sources of uncertainty were taken into account. The repeatability of the ion current ratio measurements on individual samples was 5 x 10(-5) relative. All uncertainty statements are made following the ISO/BIPM Guide to the Expression of Uncertainty in Measurements. Whereas these results proved unexpected, a more comprehensive study will follow incorporating a much wider range of samples of commercial origin.

Geometrically Frustrated Matter—Magnets to Molecules

AbstractGeometric frustration occurs when a set of degrees of freedom is incompatible with the space it occupies. A purely geometric example is the impossibility of close-packing pentagons in two dimensions. Another simple example is atomic magnetic moments with antiferromagnetic interactions. These moments lower their interaction energy by pointing antiparallel to their neighbors, an arrangement incompatible with the occupation of a crystal lattice of triangular symmetry. Other manifestations of frustration occur in ice, glass, liquid crystals, and correlated metals. Because frustration governs the rules of packing, examples are also found in biological materials, as in the self-assembly of liposomes that form nanotubules. Geometric frustration is essentially “many-body” in nature: the basic concept is trivial on the scale of three particles, but complex and anharmonic for an Avogadro's number of particles. In fact, geometrically frustrated systems are so anharmonic that no general theoretical framework exists to explain their collective behavior. This article will explore the basic concepts of geometric frustration and illustrate these concepts with examples from magnetism, crystal structures, and molecular systems.

Time-Reversal Test for Stochastic Quantum Dynamics

The calculation of quantum dynamics is currently a central issue in theoretical physics, with diverse applications ranging from ultracold atomic Bose-Einstein condensates to condensed matter, biology, and even astrophysics. Here we demonstrate a conceptually simple method of determining the regime of validity of stochastic simulations of unitary quantum dynamics by employing a time-reversal test. We apply this test to a simulation of the evolution of a quantum anharmonic oscillator with up to 6.022x10(23) (Avogadro's number) of particles. This system is realizable as a Bose-Einstein condensate in an optical lattice, for which the time-reversal procedure could be implemented experimentally.

Diameter Determination of Avogadro Spheres #1 and #2

For the redetermination of Avogadro's constant, N/sub A/, the spheres interferometer of Physikalisch-Technische Bundesanstalt (PTB) was used to measure the volume of two of the Avogadro spheres. With spherical reference plates and spherical beams of rays not only one diameter in the center is measured, but also covering a 60/spl deg/ field of view a topography with a large number of diameter measurements. About 40 different orientations of this spherical caps result in 500 000 diameter values for the sphere. Avogadro sphere #1 of PTB, Germany, and Avogadro sphere #2 of IMGC, Italy, have been measured with different setups. For the last measurement the calculation of the radius is based on a new algorithm taking the spatial distribution of the measured data into account. As the spheres have different diameters the absolute density of both spheres were calculated with the aid of the mass measurement. The agreement is within 4/spl middot/10/sup -8/.

Comparison of Density Difference Measurements at PTB and NMIJ

Silicon single crystals are used as density standards and to determine the Avogadro constant. Density comparisons by the flotation method can check the homogeneity of the silicon crystals and determine the absolute density of silicon samples by comparison to primary density standards. Results of the two flotation apparatuses at NMIJ (Japan) and PTB (Germany) were compared with each other and with density determinations from other methods. For this purpose, three small silicon samples from the crystals WASO 04 and NRLM4 were measured in both institutes. These measurements show that the flotation method is able to detect and measure density differences with uncertainties of 0.02/spl middot/10/sup -6//spl rho/. Additionally, the density of the 1-kg silicon sphere AVO#1 was determined by flotation comparisons to primary standards of the institutes and compared with density values obtained by different methods. With one exception, all results are consistent within the combined uncertainties, the largest relative density difference being only 0.18/spl middot/10/sup -6/.

An absolute replacement for the standard kilogram

This article revisits the proposal that the mass of any particle or atom can be defined absolutely as its rest-frame de Broglie frequency—a definition which requires no arbitrarily chosen mass standard, which has the physical basis that this frequency does indeed display the property of inertia, and which also makes electric charge dimensionless. The mass of any macroscopic object in kilogram could then be specified as a defined multiple of its total absolute mass, which is sufficiently close to the conventional mass for all present-day practical purposes, without having to wait for the determination of Avogadro's number or Planck's constant with few-parts-per-billion precision.

Estimating Avogadro's number from skylight and airlight

Naked-eye determinations of the visual range yield order-of-magnitude estimates of Avogadro's number, using an argument of Rayleigh. Alternatively, by looking through a cardboard tube, we can compare airlight and skylight and give another estimate of this number using the law of atmospheres.

Relation between surface tension and isothermal compressibility: application of statistical mechanical equation of state

A statistical mechanical equation of state is used to establish relation between liquid isothermal compressibility κ and surface tension γ with length parameter L, order of surface thickness, in the form of functional κ f· γ f = CL, where subscript f stands for freezing point. The proportionality constant C depends on the structure and molecular shape of a fluid, and for liquid hydrocarbons varies by a factor of 2–3. Pervious universal relation between second virial coefficient and surface tension at freezing point is used to derive the above relation. The length parameter is a measure of surface thickness and found as the size of ( ρ f N) −1/3, where ρ f is the liquid density at freezing point and N is the Avogadro's number. Pure theoretical treatment for numerical calculation of the proportionality constant is in reasonable agreement with literature, for which experimental data on isothermal compressibility and surface tension have been used.

Determining the crystallographic orientation of Avogadro silicon spheres

Highly polished spheres, manufactured from silicon single-crystal material, are used in the X-ray crystal density method (XCDM) to determine the Avogadro constant. If the measurement uncertainty associated with this method can be reduced to 0.01 p.p.m., it would be possible to redefine the SI unit of mass, the kilogram, in terms of a fixed number of atoms of a definite species. The spheres are manufactured with a nominal mass of 1 kg and nominal diameter of 90 mm and a surface roughness of 0.5 nm (peak to valley). A goniometer has been constructed to enable the crystallographic orientation of these spheres to be determined using the back-reflection Laue technique. Two spheres have been successfully orientated in this manner by identifying two orthogonal 〈100〉 directions.

Homogeneous Chiral Autocatalysis: A Simple, Purely Stochastic Kinetic Model

A continuous-time, discrete-state stochastic approach was used to study a simple, chiral autocatalytic model that was composed of the following three reactions: A → 0.5BR + 0.5BS (v1 = ku[A]), A + BR → 2BR (v2 = kc[A][BR]), A + BS → 2BS (v3 = kc[A][BS]). It is shown that the final distribution of enantiomers BR and BS is described by the one-parameter probability function Cxδ(1 − x)δ, where x is the molar fraction of BR, δ = 0.5/α − 1 (where α = kc/(kuNAV), NA is Avogadro's constant, and V is the volume of the sample), and C = Γ(1/α)/{Γ(0.5/α)}2 (where Γ is the gamma function). Comparison with two published examples shows that the probability function introduced here gives a reasonable interpretation of the experimental results.

How to Win the Nobel Prize: An Unexpected Life in Science (review)

Reviewed by: How to Win the Nobel Prize: An Unexpected Life in Science Barbara Culliton J. Michael Bishop . How to Win the Nobel Prize: An Unexpected Life in Science. The Jerusalem—Harvard Lectures. Cambridge: Harvard University Press, 2003. xiii + 271 pp. Ill. $27.95 (0-674-00880-4). J. Michael Bishop is a modest and generous and modestly famous man. On 9 October 1989, at 3:00 A.M., he received the call from Stockholm that leading scientists cannot help but covet: he had won the Nobel Prize in Physiology or Medicine for demonstrating that genes in otherwise normal cells can, when incited, cause cancer. Excited colleagues at the University of California, San Francisco, where Bishop (now chancellor) spent most of his academic life, scheduled a press conference for 11:00 that morning. But Bishop already had plans, and the Nobel Prize was not about to change them. He had tickets for a [End Page 752] "crucial playoff game between the Chicago Cubs and the San Francisco Giants baseball teams," he recounts in How to Win the Nobel Prize, a charming book about his life that, in turns, has been devoted to research, politics, institutional leadership, the public understanding of science, and baseball: "Let the record show that I am an ardent Giants fan. I was unflinching: the press conference was moved to 8:30 A.M. so that I could arrive at the ballpark in time for batting practice, an essential ritual for the cognoscenti" (p. 4). In an anecdote that accompanies his story about the baseball game, Bishop touches astutely on one of the things that keep people away from science and scientists—namely, complex facts: "The Giants won the game and, thus, the National League Championship when Will Clark drove a two-out single 'up the middle' off a no-balls and two-strike pitch from Mitch Williams. I will remember that piece of trivia long after I have forgotten Avogadro's number (a physical constant of some use to scientists, but that does not come readily to mind even now)" (p. 4). In a world in which scientists often seem remote or even fearsome, Bishop establishes by the fourth page of his book that it is possible to engage in highly original experiments with chicken viruses that eventually establish an important principal of cancer; to win the Nobel Prize; and to be a normal human being too. He writes with grace and ease, packing into a book of fewer than three hundred pages a comprehensive and readily accessible account of his research, the ideal social ethos of science, national politics, and the relationship between science and culture. Bishop shared the prize with Harold E. Varmus, who joined his UCSF laboratory in 1970 as a postdoctoral fellow but quickly became a colleague. As Bishop writes: "Our relationship evolved rapidly to one of equals," (p. 54) and before long "we became a hyphenated self that gave its name to a social organism—the 'Bishop-Varmus' laboratory," (p. 55) that was the center of study oncogenes. First they isolated a gene, called SRC (pronounced "sark," for sarcoma), from a chicken tumor virus. Subsequently they showed that oncogenes exist in many kinds of tumors, and established a genetic paradigm for cancer. Bishop's explication of the science of tumor genes is one of the best there is. And his description of the defining moment in the laboratory exemplifies his generosity and gentle spirit: he recalls a young scientist who, on the night of Saturday, 26 October 1974, completed a crucial experiment demonstrating that normal DNA contains gene sequences related to the SRC gene that transforms healthy cells into malignant ones. "Where was I at the moment of magic?" Bishop asks; probably home in bed, giving his student the satisfaction of not having to share the magic moment with anyone else, he says without a trace of envy. As his life in science progressed, Bishop embraced a lesson that many in the ivory tower resist: funding for science comes from Congress, members of the House and Senate are creatures of politics, and if scientists want continued support they had better become adept at speaking out in their own interest (which Bishop...

Homeopathically prepared dilution of Rana catesbeiana thyroid glands modifies its rate of metamorphosis

One strand of research on the scientific basis of homeopathy is based on inversion effects of dilutions and the biophysical properties of information transfer. A model developed by Endler, was the basis for the study of the influence of high-diluted solution (1:10 26 part by weight) of thyroid glands on the rate of metamorphosis of the frog Rana catesbeiana from the no legged to four-legged stage. The glands were obtained from tadpoles and prepared according by (dilution and succussion). Similar pure hydroalcoholic solution (unsuccussed) was used as control. In order to identify significant differences in the frequencies of four-legged tadpoles, in homeopathic and control group, we used a chi-square goodness-of-fit test ( P<0.01) and the cumulative risk for metamorphosis by Cox's Proportional Hazards model ( P<0.05). The number of animals that reached the four-legged stage is generally smaller in the treated group, than in the hydroalcoholic control group. It was postulated that thyroid hormones transmitted information’ specific to the molecules used to prepare the solution, even though the molarity was beyond Avogadro's number.

Radiative corrections to the polarizability of helium.

The complete alpha(3) QED correction to the helium atom polarizability is computed assuming an infinite nuclear mass and found to be equal to 0.000030666(3) a.u., with the contribution from the electric-field dependence of the Bethe logarithm amounting to 0.000000193(2) a.u. After including the alpha(2) and alpha(3) corrections for the nuclear recoil and the leading part of the alpha(4) QED correction, we find that the molar polarizability of 4He is 0.51725419(9)(4) cm(3)/mol. The first of the two error bounds is dominated by the uncertainty of alpha(4) and higher-order QED corrections and the second reflects the uncertainty of the Avogadro constant.

Determining the Carbon-Carbon Distance in an Organic Molecule with a Ruler

This article describes how to estimate the carbon–carbon bond distance for the naphthalene molecule. The fundamental idea of this procedure is based on the well-known experimental determination of Avogadro's number using an oil, such as oleic acid. The experimental results are good and the cost of the experiment is negligible. All the products and objects used can be obtained at home or purchased for little cost. The procedure is easily performed and can be done either at home or in classroom, with the restriction that the mass of the naphthalene must be determined using an analytical or, at least, a precise balance. If there is not an appropriate balance, a neighborhood pharmacist could help weigh the samples. Before performing the experiment, it is necessary for the students to be familiar with: the form and size of atoms and molecules; Avogadro's number; atomic and molecular masses; the chemical bond, with emphasis on the carbon-carbon bond; and liquid solutions and solution concentrations. For the stu...

Density comparison measurements of silicon crystals by a pressure-of-flotation method at NMIJ

A new density comparison apparatus based on a pressure-of-flotation method (PFM) is presented. Density comparison measurements are performed for a new silicon crystal of the National Metrology Institute of Japan (NMIJ) for the density standard and determination of the Avogadro constant. The density comparison measurement of silicon crystals by the new PFM apparatus has an estimated relative standard uncertainty of 4.0 × 10−8. Adjusted values of density are evaluated from the data of the PFM and absolute measurements using a least-squares algorithm, where correlations are taken into consideration.

Intrinsic Surface Reaction Constant in 1‐pK Model of Mg‐Fe Hydrotalcite‐like Compounds

AbstractThe relationship among intrinsic surface reaction constant (K) in 1‐pK model, point of zero net charge (PZNC) and structural charge density (σst) for amphoteric solid with structural charges was established in order to investigate the effect of σst on pK. The theoretical analysis based on 1‐pK model indicates that the independent PZNC of electrolyte concenmtion (c) exists for amphoteric solid with structural charges. A common intersection point (CIP) should appear on the acid‐base titration curves at different c. and the pH at the CIP is pHPZNC. The pK can be expressed as pK=‐pHPZNC+log[(l+2αPZNC)/(1–2αPZNC)], where α‐PZNC σ/eNANs, in which e is the elementary charge, Na the Avogadro's constant and NS the total density of surface sites. For solids without structural charges, pK=‐ pHPZNC. The pK values of hydrotalcite‐like compounds (HTlc) with general formula of [Mg1‐x Fex(OH)2](Cl, OH)x were evaluated. With increasing x. the pK increases, which can be explained based on the affinity of metal cations for H− or OH− and the electrostatic interaction between charging surface and H− or OH−.