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The van der Waals radius, r w, of an atom is the radius of an imaginary hard sphere representing the distance of closest approach for another atom. It is named after Johannes Diderik van der Waals, winner of the 1910 Nobel Prize in Physics, as he was the first to recognise that atoms were not simply points and to demonstrate the physical consequences of their size through the van der Waals ...
The Schwarzschild radius of a body is proportional to its mass and therefore to its volume, assuming that the body has a constant mass-density. [17] In contrast, the physical radius of the body is proportional to the cube root of its volume.
In applied sciences, the equivalent radius (or mean radius) is the radius of a circle or sphere with the same perimeter, area, or volume of a non-circular or non-spherical object. The equivalent diameter (or mean diameter ) ( D {\displaystyle D} ) is twice the equivalent radius.
For more recent data on covalent radii see Covalent radius. Just as atomic units are given in terms of the atomic mass unit (approximately the proton mass), the physically appropriate unit of length here is the Bohr radius, which is the radius of a hydrogen atom. The Bohr radius is consequently known as the "atomic unit of length".
The concept of atomic radius was preceded in the 19th century by the concept of atomic volume, a relative measure of how much space would on average an atom occupy in a given solid or liquid material. [3] By the end of the century this term was also used in an absolute sense, as a molar volume divided by Avogadro constant. [4]
Phase diagram of hard sphere system (Solid line - stable branch, dashed line - metastable branch): Pressure as a function of the volume fraction (or packing fraction) The hard sphere system exhibits a fluid-solid phase transition between the volume fractions of freezing η f ≈ 0.494 {\displaystyle \eta _{\mathrm {f} }\approx 0.494} and ...
The volume can be computed without use of the Gamma function. As is proved below using a vector-calculus double integral in polar coordinates, the volume V of an n-ball of radius R can be expressed recursively in terms of the volume of an (n − 2)-ball, via the interleaved recurrence relation:
Since the reduced mass of the electron–proton system is a little bit smaller than the electron mass, the "reduced" Bohr radius is slightly larger than the Bohr radius (meters). This result can be generalized to other systems, such as positronium (an electron orbiting a positron ) and muonium (an electron orbiting an anti-muon ) by using the ...