Jump to ContentJump to Main Navigation
Bonding, Structure and Solid-State Chemistry$

Mark Ladd

Print publication date: 2016

Print ISBN-13: 9780198729945

Published to Oxford Scholarship Online: May 2016

DOI: 10.1093/acprof:oso/9780198729945.001.0001

Show Summary Details
Page of

PRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2018. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a monograph in OSO for personal use (for details see www.oxfordscholarship.com/page/privacy-policy). Subscriber: null; date: 16 January 2019

(p.397) A5 Mean Classical Thermal Energy and the Equipartition Theorem

(p.397) A5 Mean Classical Thermal Energy and the Equipartition Theorem

Source:
Bonding, Structure and Solid-State Chemistry
Author(s):

Mark Ladd

Publisher:
Oxford University Press

A5.1 Mean kinetic energy

Consider a system of particles, each of mass m but with differing values of speed v; the kinetic energy of each particle is mv2/2, which may be resolved along mutually perpendicular x, y and z axes. The mean value <X> of any distribution of the form ϕ(X) is given by:

(A5.1)
<X>=Xϕ(X)dXϕ(X)dX,

where the integration is taken over the range of the variable. Assuming that the energies of the particles follow a Boltzmann distribution, then from Appendix A4, the mean value for the kinetic energy uK for a single particle is:

(A5.2)
<uK>=<mv2/2>=(mv2/2)exp(mv2/2kT)dvxdvydvzexp(mv2/2kT)dvxdvydvz

Since positive and negative directions of v are equally probable, introducing spherical coordinates from Appendix A6 and replacing r of that discussion by v, Eq. (A5.2) may be now written as:

(A5.3)
<uK>=20(mv2/2)exp(mv2/2kT)v2dv0πsinθdθ02πdϕ20exp(mv2/2kT)v2dv0πsinθdθ02πdϕ

which simplifies to:

(A5.4)
<uK>=(m/2)0v4exp(mv2/2kT)dv0v2exp(mv2/2kT)dv

(p.398) Making the substitution t=pv2,wherep=m/2kT, and following the argument of Example A7.1 in

Appendix A7, Eq. (A5.4) becomes:

(A5.5)
<uK>=(m/4)p5/2Γ(5/2)(1/2)p3/2Γ(3/2)=(m/2)p1(3/2)Γ(3/2)(3/2)Γ(1/2)=(m/2)p1(3/2)2Γ(1/2)(3/2)Γ(1/2)=3m/4p=(3/2)kT

and for the system of n particles, the average kinetic energy <UK> is given by:

(A5.7)
<UK>=32nkT

A5.2 Equipartition theorem

A molecule possesses energy in terms of translation, rotation and vibration. The equipartition theorem, which derives from the Boltzmann distribution (Appendix A4), states that for a system of particles at thermal equilibrium under a temperature T, the mean value of each quadratic contribution to the energy of the system is kT/2. A quadratic contribution is one that can be expressed as the square of a property such as position or velocity.

Translational energy is kinetic and involves three quadratic terms:

(A5.7)
UK=(1/2)mvx2+(1/2)mvy2+(1/2)mvz2

with each quadratic term in this expression equal to kT/2. Hence, the total classical kinetic energy is (3/2)kT; it is a mean value, since all directions of translation are equally probable.

A rigid rotor has two perpendicular axes about which it can rotate, each leading to a quadratic kinetic energy contribution; rotations about the axis of the rotor have no effect on the energy. The key term in a rotor is the moment of inertia; it has the form mr2 in the simplest case and constitutes a quadratic term. In accordance with the equipartition theorem, each such term contributes kT/2 to the energy; thus, the total mean rotational energy is kT for two quadratic terms. The vibrational energy of an atomic oscillator involves quadratic terms in both kinetic and potential energy; hence, there are again two quadratic terms and a total mean vibrational energy of kT.

A monatomic species such as argon possesses only energy of translation of total mean value (3/2)RT per mole. A diatomic molecule like oxygen may rotate about two directions perpendicular to each other and to the molecular axis, with each rotation contributing to the energy of the molecule. Rotation about the molecular axis does not contribute to the energy of the molecule which has the total mean value of (5/2)RT per mole, that is, (3/2)RT for translation plus RT for rotation.