# virial theorem

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## Virial theorem

A theorem in classical mechanics which relates the kinetic energy of a system to the virial of Clausius, as defined below. The theorem can be generalized to quantum mechanics and has widespread application. It connects the average kinetic and potential energies for systems in which the potential is a power of the radius. Since the theorem involves integral quantities such as the total kinetic energy, rather than the kinetic energies of the individual particles that may be involved, it gives valuable information on the behavior of complex systems. For example, in statistical mechanics the virial theorem is intimately connected to the equipartition theorem; in astrophysics it may be used to connect the internal temperature, mass, and radius of a star and to discuss stellar stability. The virial theorem makes possible a very easy derivation of the counterintuitive result that as a star radiates energy and contracts it heats up rather than cooling down. See Statistical mechanics

The virial theorem states that the time-averaged value of the kinetic energy in a confined system (that is, a system in which the velocities and position vectors of all the particles remain finite) is equal to the virial of Clausius. The virial of Clausius is defined to equal -½ times the time-averaged value of a sum over all the particles in the system. The term in this sum associated with a particular particle is the dot product of the particle's position vector and the force acting on the particle. Alternatively, this term is the product of the distance, r, of the particle from the origin of coordinates and the radial component of the force acting on the particle.

In the common case that the forces are derivable from a power-law potential, V, proportional to rk, where k is a constant, the virial is just -k/2 times the potential energy. Thus, in this case the virial theorem simply states that the kinetic energy is k/2 times the potential energy. For a system connected by Hooke's-law springs, k = 2, and the average kinetic and potential energies are equal. For k = 1, that is, for gravitational or Coulomb forces, the potential energy is minus twice the kinetic energy. See Coulomb's law, Gravitation, Harmonic motion

## virial theorem

The energy of a system in equilibrium is distributed between the kinetic energy E K and the potential energy E P such that, when averaged over time, 2E K = –E P. Application of the virial theorem to a cluster of galaxies or stars allows an evaluation of its total mass from observations of its size and velocity dispersion. The mean temperature at which the system would satisfy the virial theorem is the virial temperature, and the system is said to be in virial equilibrium.

## virial theorem

[′vir·ē·əl ‚thir·əm]
(statistical mechanics)
References in periodicals archive ?
Specifically, SIC was used to measure the second virial coefficient, [B.
1] que incorpore la ecuacion virial de dos terminos, la cual se integrara numericamente para obtener la composicion del vapor.
The latter originates from the third virial coefficient, which is the third term in a series expansion of the pressure of the colloidal dispersion in terms of the density.
The chapters take up only a bit over 200 pages, and the bulk of the volume is devoted to appendices of technical information on such matters as virial theory, the formation of commensurate planetary orbits, perturbations of the Oort Cloud, global warming, and physical constants and useful data.
Equation (94) in its meaning resembles the virial theorem for a stationary system of particles bound by its proper gravitational field -- in this system the absolute value of the total potential energy is approximately equal to double kinetic energy of all particles.
Bendict-Webb-Rubin (BWR) family includes complicated EOS and empirical extensions of the virial EOS (Riazi, 2003).
11]=second virial coefficient of the standard substance at column operating temperature [d[m.
We employ state-of-the-art pair and three-body potentials with path-integral Monte Carlo (PIMC) methods to calculate the third density virial coefficient C(T) for helium.
Vega: Bilinear virial identities and applications, to appear in Ann.
the calculation of the second and third molar virial coefficients [B.
For the first time, this study shows the X-ray emission and gas density and temperature out to - and even beyond - the virial radius, where the cluster continues to form.

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