Virial Theorem

The virial theorem explains how kinetic energy and potential energy are related. Suppose the potential energy is proportional to the nth power of position (r); in that case, the average kinetic energy is equal to (n/2) times potential energy at that position, according to the virial theorem.

If, U 𝝰 r2

Then, K= n2(U)

If the motions are not random or isotropic, the virial theorem format must be altered, but the theorem is still applicable. But the virial theorem doesn’t apply to particles that are not bonded to a system. In this article, we provide you with virial theorem notes. 

The Formula for Virial Theorem

In both classical and quantum physics, the virial theorem is a crucial theorem pertaining to a system of moving particles. The virial theorem is useful when dealing with a large number of particles, and it is especially important in the case of central-force motion. The formula for virial theorem can be given as:

(T) = –12k=1N(Fk . rk)

Where, 

(T) = total kinetic energy

N = number of particles

Fk = force on the kth particle

rk = position

The virial theorem has been expanded in a number of ways, and the most notable is the tensor form. As a result, n times the average total potential energy VTOT = twice the average total kinetic energy T. VTOT  represents  the total potential energy  of the  system or the sum of the potential energy of all pairs of particles in the system.

Virial theorem also deals with average properties and has applications to statistical mechanics. It can be used to make a crude estimate of the mass of a cluster of galaxies. 

Overview of Virial Theorem

 According to the lecture “On a mechanical theorem applicable to heat,” the system’s mean vis viva equals its virial, or the average kinetic energy equals the average potential energy. The virial theorem can be easily deduced from Lagrange’s identity as applied to classical gravitational dynamics, which was first published in 1772 in Lagrange’s “Essay on the Problem of Three Bodies.”

However, because statistical dynamics had not yet united the distinct disciplines of thermodynamics and classical dynamics at the time of creation, the interpretations that led to the construction of the equations were highly different.

James Clerk Maxwell, Subrahmanyan Chandrasekhar and many others used, popularised, generalised, and developed the theorem later. 

Richard Bader demonstrated that a complete system’s charge distribution can be partitioned into kinetic and potential energies that obey the virial theorem. 

Virial Theorem for Dark Matter

Fritz Zwicky first used the virial theorem to prove the presence of unseen matter, now known as dark matter. The virial theorem connects a self-gravitating body’s total kinetic energy, T, due to the motions of its constituent parts to the body’s gravitational potential energy, U.

0 = 2T + U

We get the following results by rearranging the above equation and making some simple assumptions about dark matter.

T=(Mv2/2) and U = ( GM2/R)

for galaxies:

M= v2R/G

Where M is the galaxy’s total mass, v is the galaxy’s mean velocity (combined rotation and velocity dispersion), G is Newton’s gravitational constant, and R is the galaxy’s effective radius (size). This equation is crucial because it connects two observable aspects of galaxies (velocity dispersion and effective half-light radius) to a basic yet unobservable property — galaxy mass.

As a result, the virial theorem is at the heart of a number of galaxy physical relations. One technique astronomers use to discover the presence of dark matter in galaxies and clusters of galaxies is to compare mass estimations based on the virial theorem against mass estimates based on galaxies’ luminosities.

The Generalised Version of Virial Theorem

The idea that electrostatic forces determine the stability of a system of charged particles and potential energy changes drive the stabilisation of molecules in a system has been known for ages. Although the virial theorem does not hold in these systems, covalent bonds do form, and the wave mechanical bonding analysis produces similar results to that of the Coulomb potentials.

The main driving force is, once again, electron delocalization, which reduces the interatomic kinetic energy component. The role of the virial theorem in the context of covalent binding is discussed in depth.

Conclusion

The history of the virial theorem and the overview of the virial theorem is quite vast and modified in various ways that conclusively formulated the standard virial theorem. The virial theorem and the overall collected data were used to offer an “empirical” study of the covalent bond in H2. This study demonstrates that the involvement of electron kinetic energy in the creation of covalent bonds defies current thinking in two seemingly contradictory ways. You can find useful information and important definitions to make your own set of virial theorem notes.