Modern Thermodynamics

- Chapter 1

35

(vy, vy+d vy) and (vz, vz+d vz).

As shown in the Fig. 1.9, each point in the velocity space corresponds to a velocity vector; P(v)dvx dvy

dvz is the probability that the velocity of a molecule lies within an elemental volume dvx, dvy and dvz at

the point (vx vy vz). P(v) is called the probability density in the velocity space.

The mathematical form of P(v) was obtained by James Clerk Maxwell; the concept was later

generalized by Ludwig Boltzmann (1844-1906) to the probability distribution of the total energy, E, of

the molecule. According to the principle discovered by Boltzmann, when a system reaches

thermodynamic equilibrium, the probability that a molecule is in a state with energy E is proportional to

Exp(–E/kBT). If ρ(E) is the number of different states in which the molecule has energy E:

P(E) ∝ ρ(E)e−E / k

B

T (1.6.9)

The quantity ρ(E) is called the density of states. Relation (1.6.9), called the Boltzmann principle, is

one of the fundamental principles of physics. Using this principle, equilibrium thermodynamic properties

of a substance can be derived from molecular energies E -- a subject called Statistical Thermodynamics

presented in Chapter 17. In this introductory section, however, we will only study some elementary

consequences of this principle.

The energy of a molecule E = Etrans + Erot + Evib + Eint + …. in which Etrans is the kinetic energy of

translational motion of the whole molecule, Erot is the energy of rotational motion, Evib is the energy of

vibrational motion, Eint is the energy of the molecule's interaction with other molecules and fields such as

electric, magnetic or gravitational fields, and so on. According to the Boltzmann principle, the

probability that a molecule will have an translational kinetic energy Etrans is proportional to Exp(-

Etrans/kBT) (the probabilities associated with other forms of energy are factors that multiply this term).

Since the kinetic energy due to translational motion of the molecule is mv2/2, we can write the probability

as a function of the velocity v by which we mean probability that a molecule’s velocity is in an elemental

cube in velocity space, as shown in the Fig. 1.9. For a continuous variable such as velocity, we must

define a probability density P(v) so that the probability that a molecule's velocity is in an elemental cube

of volume dvx dvy dvz located at the tip of the velocity vector v is P(v)dvx dvy dvz. According to the

Boltzmann principle, this probability is: