Question: What is one determinant of an elements atomic weight?
About the article "Atomic number", section "Abstract".

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The atomic mass and relative isotopic mass are sometimes confused, or incorrectly used, as synonyms of relative atomic mass (also known as atomic weight) and the standard atomic weight (a particular variety of atomic weight).
However, as noted in the introduction, atomic weight and standard atomic weight represent terms for averages of isotopic abundances, not for single nuclides.
As such, atomic weight and standard atomic weight often differ numerically from relative isotopic mass, and they can also have different units than atomic mass when this quantity is not expressed in unified atomic mass units (see the linked article for atomic weight).
The atomic mass (relative isotopic mass) is defined as the mass of a single atom, which can only be one isotope (nuclide) at a time, and is not an abundance-weighted average, as in the case of relative atomic mass/atomic weight.
The atomic mass or relative isotopic mass of each isotope and nuclide of a chemical element is therefore a number that can in principle be measured to a very great precision, since every specimen of such a nuclide is expected to be exactly identical to every other specimen, as all atoms of a given type in the same energy state, and every specimen of a particular nuclide, are expected to be exactly identical in mass to every other specimen of that nuclide.
For example, every atom of oxygen-16 is expected to have exactly the same atomic mass (relative atomic mass) as every other atom of oxygen-16.
In the case of many elements that have one naturally occurring isotope (mononuclidic elements) or one dominant isotope, the actual numerical similarity/difference between the atomic mass of the most common isotope, and the relative atomic mass or (standard) atomic weight can be small or even nil, and does affect most bulk calculations.
However, such an error can exist and even be important when considering individual atoms for elements that are not mononuclidic.
For non-mononuclidic elements that have more than one common isotope, the numerical difference in relative atomic mass (atomic weight) from even the most common relative isotopic mass, can be half a mass unit or more (e.g.
see the case of chlorine where atomic weight and standard atomic weight are about 35.45).
The atomic mass (relative isotopic mass) of an uncommon isotope can differ from the relative atomic mass, atomic weight, or standard atomic weight, by several mass units.
Atomic masses expressed in unified atomic mass units (i.e.
relative isotopic masses) are always close to whole-number values, but never (except in the case of carbon-12) exactly a whole number, for two reasons: protons and neutrons have different masses, and different nuclides have different ratios of protons and neutrons.
atomic masses are reduced, to different extents, by their binding energies.
The ratio of atomic mass to mass number varies from about 0.99884 for 56Fe to 1.00782505 for 1H.
Any mass defect due to nuclear binding energy is experimentally a small fraction (less than 1%) of the mass of a nucleon, and is an even smaller percentage compared to the average mass per nucleon in carbon-12, which is moderately strongly-bound compared with other atoms.
Since protons and neutrons differ from each other in mass by an even smaller fraction (about 0.0014 u), rounding the atomic mass of any given nuclide to the nearest whole number always gives the nucleon count, or mass number.
The neutron count (neutron number) may then be derived by subtracting the number of protons (atomic number). 
The atomic mass ("m"a) is the mass of an atomic particle, sub-atomic particle, or molecule.
It may be expressed in unified atomic mass units; by international agreement, 1 atomic mass unit is defined as 1/12 of the mass of a single carbon-12 atom (at rest).
When expressed in such units, the atomic mass is called the relative isotopic mass (see section below).
The atomic mass or relative isotopic mass refers to the mass of a single particle, and is fundamentally different from the quantities elemental atomic weight (also called "relative "atomic" mass") and standard atomic weight, both of which refer to averages (mathematical means) of naturally-occurring atomic mass values for samples of elements.
Most elements have more than one stable nuclide; for those elements, such an average depends on the mix of nuclides present, which may vary to some limited extent depending on the source of the sample, as each nuclide has a different mass.
By contrast, atomic mass figures refer to an individual particle species: as atoms of the same species are identical, atomic mass values are expected to have no intrinsic variance at all.
Atomic mass figures are thus commonly reported to many more significant figures than atomic weights.
The atomic mass of atoms, ions, or atomic nuclei is slightly less than the sum of the masses of their constituent protons, neutrons, and electrons, due to binding energy mass loss (as per "E"="mc"2). 
There is a one-to-one map from the set of normalized "n"×"n" {1, −1} matrices to the set of ("n"−1)×("n"-1) {0, 1} matrices under which the magnitude of the determinant is reduced by a factor of 21−"n".
This map consists of the following steps.
Subtract row 1 of the {1, −1} matrix from rows 2 through "n".
(This does not change the determinant.)
Extract the ("n"−1)×("n"−1) submatrix consisting of rows 2 through "n" and columns 2 through "n".
This matrix has elements 0 and −2.
(The determinant of this submatrix is the same as that of the original matrix, as can be seen by performing a cofactor expansion on column 1 of the matrix obtained in Step 1.)
Divide the submatrix by −2 to obtain a {0, 1} matrix.
(This multiplies the determinant by (−2)1-"n".)
Example: formula_1 In this example, the original matrix has determinant −16 and its image has determinant 2 = −16·(−2)−3.
Since the determinant of a {0, 1} matrix is an integer, the determinant of an "n"×"n" {1, −1} matrix is an integer multiple of 2"n"−1. 
The first scientists to determine relative atomic masses were John Dalton and Thomas Thomson between 1803 and 1805 and Jöns Jakob Berzelius between 1808 and 1826.
Relative atomic mass ("Atomic weight") was originally defined relative to that of the lightest element, hydrogen, which was taken as 1.00, and in the 1820s Prout's hypothesis stated that atomic masses of all elements would prove to be exact multiples of that of hydrogen.
Berzelius, however, soon proved that this was not even approximately true, and for some elements, such as chlorine, relative atomic mass, at about 35.5, falls almost exactly halfway between two integral multiples of that of hydrogen.
Still later, this was shown to be largely due to a mix of isotopes, and that the atomic masses of pure isotopes, or nuclides, are multiples of the hydrogen mass, to within about 1%.
In the 1860s Stanislao Cannizzaro refined relative atomic masses by applying Avogadro's law (notably at the Karlsruhe Congress of 1860).
He formulated a law to determine relative atomic masses of elements: "the different quantities of the same element contained in different molecules are all whole multiples of the atomic weight" and determined relative atomic masses and molecular masses by comparing the vapor density of a collection of gases with molecules containing one or more of the chemical element in question.
In the 20th century, until the 1960s chemists and physicists used two different atomic-mass scales.
The chemists used a scale such that the natural mixture of oxygen isotopes had an atomic mass 16, while the physicists assigned the same number 16 to the atomic mass of the most common oxygen isotope (containing eight protons and eight neutrons).
However, because oxygen-17 and oxygen-18 are also present in natural oxygen this led to two different tables of atomic mass.
The unified scale based on carbon-12, 12C, met the physicists' need to base the scale on a pure isotope, while being numerically close to the chemists' scale.
The term "atomic weight" is being phased out slowly and being replaced by "relative atomic mass", in most current usage.
This shift in nomenclature reaches back to the 1960s and has been the source of much debate in the scientific community, which was triggered by the adoption of the unified atomic mass unit and the realization that weight was in some ways an inappropriate term.
The argument for keeping the term "atomic weight" was primarily that it was a well understood term to those in the field, that the term "atomic mass" was already in use (as it is currently defined) and that the term "relative atomic mass" was in some ways redundant.
In 1979, as a compromise, the definition was refined and the term "relative atomic mass" was introduced as a secondary synonym.
Twenty years later the primacy of these synonyms was reversed and the term "relative atomic mass" is now the preferred term; however the "standard atomic weights" have maintained the same name. 
The Russian chemist Dmitri Mendeleev was the first scientist to make a periodic table similar to the one used today.
Mendeleev arranged the elements by atomic mass, corresponding to relative molar mass.
It is sometimes said that he played 'chemical solitaire' on long train journeys, using cards with various facts about the known elements.
On March 6, 1869, a formal presentation was made to the Russian Chemical Society, entitled "The Dependence Between the Properties of the Atomic Weights of the Elements".
In 1869, the table was published in an obscure Russian journal and then republished in a German journal, "Zeitschrift für Chemie".
In it, Mendeleev stated that:  The elements, if arranged according to their atomic mass, exhibit an apparent periodicity of properties.
Elements which are similar as regards to their chemical properties have atomic weights which are either of nearly the same value (e.g., Pt, Ir, Os) or which increase regularly (e.g., K, Rb, Cs).
The arrangement of the elements, or of groups of elements in the order of their atomic masses, corresponds to their so-called valencies, as well as, to some extent, to their distinctive chemical properties; as is apparent among other series in that of Li, Be, B, C, N, O, and F.  The elements which are the most widely diffused have small atomic weights.
The magnitude of the atomic weight determines the character of the element, just as the magnitude of the molecule determines the character of a compound body.
We must expect the discovery of many yet unknown elements – for example, elements analogous to aluminium and silicon – whose atomic weight would be between 65 and 75.
The atomic weight of an element may sometimes be amended by a knowledge of those of its contiguous elements.
Thus the atomic weight of tellurium must lie between 123 and 126, and cannot be 128.
Certain characteristic properties of elements can be foretold from their atomic masses.
Scientific benefits of Mendeleev's table  It enabled Mendeleev to predict the discovery of new elements and left spaces for them, namely eka-silicon (germanium), eka-aluminium (gallium), and eka-boron (scandium).
Thus, there was no disturbance in the periodic table.
It could be used by Mendeleev to point out that some of the atomic weights being used at the time were incorrect.
It provided for variance from atomic weight order.
Shortcomings of Mendeleev's table * The table was not able to predict the existence of the noble gases.
However, when this entire family of elements was discovered, Sir William Ramsay was able to add them to the table as Group 0, without the basic concept of the periodic table being disturbed.
A single position could not be assigned to hydrogen, which could be placed in either the alkali metals group or in the halogens group.