Excluded volume

# Excluded volume

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The concept of excluded volume was introduced by Werner Kuhn
Werner Kuhn
Werner Kuhn is a Swiss physical chemist who developed the first model of the viscosity of polymer solutions using statistical mechanics. He is known for being the first to apply Boltzmann's entropy formula:S = k \log W \!...

in 1934 and applied to polymer
Polymer
A polymer is a large molecule composed of repeating structural units. These subunits are typically connected by covalent chemical bonds...

molecules shortly thereafter by Paul Flory
Paul Flory
Paul John Flory was an American chemist and Nobel laureate who was known for his prodigious volume of work in the field of polymers, or macromolecules...

.

## In liquid state theory

In liquid state theory, the 'excluded volume' of a molecule is the volume that is inaccessible to other molecules in the system as a result of the presence of the first molecule. The excluded volume of a hard sphere is eight times its volumeâ€”however, for a two-molecule system, this volume is distributed among the two particles, giving the conventional result of four times the volume; this is an important quantity in the Van der Waals equation of state
Van der Waals equation
The van der Waals equation is an equation of state for a fluid composed of particles that have a non-zero volume and a pairwise attractive inter-particle force It was derived by Johannes Diderik van der Waals in 1873, who received the Nobel prize in 1910 for "his work on the equation of state for...

. The calculation of the excluded volume for particles with non-spherical shapes is usually difficult, since it depends on the relative orientation of the particles. The distance of closest approach of hard ellipses
Distance of closest approach of ellipses and ellipsoids
The distance of closest approach of two objects is the distance between their centers when they are externally tangent. The objects may be geometric shapes or physical particles with well defined boundaries...

and their excluded area has been recently considered.

## In polymer science

In polymer science, excluded volume refers to the idea that one part of a long chain molecule can not occupy space that is already occupied by another part of the same molecule. Excluded volume causes the ends of a polymer chain in a solution to be further apart (on average) than they would be were there no excluded volume. The recognition that excluded volume was an important factor in analyzing long-chain molecules in solutions provided an important conceptual breakthrough, and led to the explanation of several puzzling experimental results of the day. It also led to the concept of the theta point
Theta solvent
In a polymer solution, a theta solvent is a solvent in which polymer coils act like ideal chains, assuming exactly their random walk coil dimensions...

, the set of conditions at which an experiment can be conducted that causes the excluded volume effect to be neutralized. At the theta point, the chain reverts to ideal chain characteristics. The long-range interactions arising from excluded volume are eliminated, allowing the experimenter to more easily measure short-range features such as structural geometry, bond rotation potentials, and steric interactions between near-neighboring groups. Flory correctly identified that the chain dimension in polymer melts would have the size computed for a chain in ideal solution if excluded volume interactions were neutralized by experimenting at the theta point.