Principle of Equipartition of Energy -Dr S P Singh (Dept of Chemistry, A N College, Patna)
Historical Background 1843: The equipartition of kinetic energy was proposed by John James Waterston. 1845: more correctly proposed by John James Waterston. 1859: James Clerk Maxwell argued that the kinetic heat energy of a gas is equally divided between linear and rotational energy. ∑ Experimental observations of the specific heat capacities of gases also raised concerns about the validity of the equipartition theorem. ∑ Several explanations of equipartition's failure to account for molar heat capacities were proposed. 1876: Ludwig Boltzmann expanded this principle by showing that the average energy was divided equally among all the independent components of motion in a system. ∑ Boltzmann applied the equipartition theorem to provide a theoretical explanation of the Dulong-Petit Law for the specific heat capacities of solids. 1900: Lord Rayleigh instead put forward a more radical view that the equipartition theorem and the experimental assumption of thermal equilibrium were both correct; to reconcile them, he noted the need for a new principle that would provide an "escape from the destructive simplicity" of the equipartition theorem. 1906: Albert Einstein provided that escape, by showing that these anomalies in the specific heat were due to quantum effects, specifically the quantization of energy in the elastic modes of the solid. 1910: W H Nernst’s measurements of specific heats at low temperatures supported Einstein's theory, and led to the widespread acceptance of quantum theory among physicists.
Under the head we deal with the contributions of translational and vibrational motions to the energy and heat capacity of a molecule. Though the results are not correct except at sufficiently high temperatures, the procedure is nevertheless constructive.
Although the average value of the velocity component in any single direction is zero because equal no. of molecules have components +u and –u, the average value of kinetic energy associated with a particular component has always a + ve value for both +ve and – ve values of average velocities, kinetic energy either or equals . We now use the different steps of Maxwell’s distribution function to evaluate average kinetic (+ ) (− ) energy
1 1 ∴ ̅ = × 2 1 1 = ( ) ∙ ( ) ∙ ( ) × 2 1 1 = × × ( ) ( ) ( ) 2 1 = × × 2 × . 2 × . 2 × 2 and so on] [ , ∫ ( ) = 2 ∫ ( ) ̅ = × ∙ ∙ 8 × × × 1 1 = = 4 2 [ ] / / / / / ∴ ̅ = × × =
Similarly,= = × average= K.E. along Y and Z coordinates = and can be proved. ̅ = ̅ = ------(1) Total average K.E. (along all coordinates) ∴ ̅ = ̅ = ̅ = ------(2) where is Boltzmann Constant. ∴ 1 1 1 Expression (1) (for the translational motion) is the mathematical representation of law of 2 2 2 Equipartition ̅ = ̅ + ̅ + of ̅ energy,= which+ states+ as follows;=
“The total energy of a molecule for a dynamical system in thermal equilibrium is divided equally among the degrees of freedom.” Here ‘degree of freedom’ is meant by the possible modes or directions of motion. But for all sorts of motions the generalized statement is made as ‘Each kind of energy of a molecule that can be expressed in general form ax2, where x is a coordinate or momentum, that contributes an amount of to the average energy of one molecule.’ ax2 term is referred to as ‘square term’ or ‘quadratic term’. The translational (kinetic) energy of a molecule is equal to where x, y, z represent the components of its velocity in three directions at right angles. The momentum , for example, is equal to and so the energy is given by ( + + thus) consisting of 3 quadratic terms. By the principle of equipartition of energy, as proved above, it should be in agreement with equation, .( Translational+ + motion,)/2 thus of every 3 × molecule can be revealed into three possible directions as stated above. = = The rotational energy of a molecule assumed to be of constant dimensions is proportional to the square of the angular momentum, so that each type of the rotation is represented by one square term, and thus should contribute per mole to the energy content. A diatomic or any linear molecule, exhibits rotation about two axes perpendicular to the line joining the nuclii. The rotational energy of a diatomic or any linear molecule should thus be , i.e. T per mole. 2 × A non linear molecule on the other hand, can rotate about three axes, so that the rotation should contribute . i.e. per mole to the energy of the system. The equation for the energy of rotation has the form 3 × (for a linear molecule) ∈ .= + (for a non-linear molecule) where , , are angular velocities and , , are moments of inertia about the x, y and z ∈ .= + + axes respectively. In the case of a linear molecules . The energy of a harmonic oscillator is given by the= sum= of the two quadratic terms, one representing the K.E. and the other the P.E. of the vibrations. By the equipartition principle the energy is, thus, expected to be , i.e., T per mole for each possible mode of vibration. In general, a nonlinear molecule containing n atoms possess (3n – 6) moles of vibrations, so that the vibrational energy contribution should be (3n – 6) T per mole. Diatomic and any linear molecule, however have (3n – 5) vibrational modes and the vibrational energy is expected to be (3n – 5) T per mole. Vibrations comprise either stretching or bending of molecules. To sum up the total average energy per molecule should be (for linear molecule) (for non-linear molecule) ̅ = + + (3 − 5) If these ̅values= are +multiplied + (3 by −the 6 )Avogadro’s no. the average energies per mole is . So for mono atomic gas Similarly, for polyatomic gas = ( ̅ × = & × = ) (linear molecule) (non-linear molecule) = + + (3 − 5) If r and v denote the degree of freedom for rotation and vibration than the total molar energy, according =to principle + of equipartition + (3 − 6) of energy will be
1 = (3 + + ) × and hence the heat capacity at the constant volume, 2
1 = = (3 + + ) The heat capacity at constant pressure, 2
= (3 + + ) + = (5 + + ) Heat capacity ratio, ∴ = = Application
(i) Ideal gases provide an important application of the equipartition theorem Equipartition to derive the classical ideal gas law from Newtonian mechanics. The equipartition theorem provides a convenient way to derive the corresponding laws for an extreme relativisticideal gas. (ii) The equipartition theorem may also be used to derive the energy and pressure of "non-ideal gases" in which the particles also interact with one another through conservative forces whose potential U(r) depends only on the distance r between the particles. (iii) An anharmonic oscillator (in contrast to a simple harmonic oscillator) is one in which the potential energy is not quadratic in the extension q (measures the deviation of the system from equilibrium). Such oscillators provide a complementary point of view on the equipartition theorem. (iv) The equipartition theorem and the related virial theorem have long been used as a tool in astrophysics. (v) The equipartition theorem can be used to derive the Brownian motion of a particle from the Langevin equation. (vi) The same formulae may be applied to determining the conditions for star formation in giant molecular clouds..
Limitation
(i) The law of equipartition breaks down when the thermal energy kBT is significantly smaller than the spacing between energy levels. (ii) The law of equipartition holds only for ergodic systems in thermal equilibrium, which implies that all states with the same energy must be equally and likely to be populated.
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