The characteristic rotational temperature ( or ) is commonly used in

statistical thermodynamics In physics, statistical mechanics is a mathematical framework that applies Statistics, statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the ma ...

to simplify the expression of the

rotational partition function In chemistry, the rotational partition function relates the rotational degrees of freedom to the rotational part of the energy. Definition The total canonical partition function Z of a system of N identical, indistinguishable, noninteracting at ...

and the rotational contribution to molecular thermodynamic properties. It has units of

temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...

and is defined asP. Atkins and J. de Paula "Physical Chemistry", 9th edition (W.H. Freeman 2010), p.597 :

\theta_ = \frac = \frac,

where

\overline = B/hc

is the

rotational constant In rotordynamics, the rigid rotor is a mechanical model of rotating systems. An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space requires three angles, known as Euler angles. A special rig ...

, is a molecular

moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceler ...

, is the

Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...

, is the

speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...

, is the

reduced Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...

and is the

Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...

. The physical meaning of is as an estimate of the temperature at which

thermal energy The term "thermal energy" is used loosely in various contexts in physics and engineering. It can refer to several different well-defined physical concepts. These include the internal energy or enthalpy of a body of matter and radiation; heat, d ...

(of the order of ) is comparable to the spacing between rotational energy levels (of the order of ). At about this temperature the population of excited rotational levels becomes important. Some typical values are given in the table. In each case the value refers to the most common isotopic species.

References

See also