Avogadro’s law / Equipartition Theorem / Degrees of Freedom
In this section,
Feynman examines three concepts: Avogadro’s law, the equipartition theorem, and
degrees of freedom. A more appropriate title might be “Theoretical Basis
of the Ideal Gas Law” or “Avogadro’s Law and Equipartition Theorem,” as these concepts
provide a statistical foundation for understanding the ideal gas law. The
concept of degrees of freedom was delivered at the beginning of next lecture.
1. Avogadro’s law:
“Furthermore, at the same temperature and pressure
and volume, the number of atoms is determined; it too is a
universal constant! So equal volumes of different gases, at the same
pressure and temperature, have the same number of molecules, because of
Newton’s laws. That is an amazing conclusion! (Feynman et al, 1963, p. 39-10).”
The ideal gas law
can be viewed as a synthesis of Newton’s laws of motion, Avogadro’s law, and
the equipartition theorem:
(1) Newton’s
laws provide the mechanical framework, describing gas molecules travel in
straight lines (First Law), exchange momentum during collisions (Second and
Third Laws), and exert pressure through impacts against container walls.
(2) Avogadro’s
law establishes that equal volumes of gases at the same temperature and
pressure contain equal numbers of molecules, allowing pressure to scale with
the number of particles (or moles), independent of chemical identity.
(3) The equipartition theorem connects temperature to the average kinetic energy per molecule, giving a statistical definition of temperature grounded in molecular motion.
Together, these principles provide the microscopic basis of
the ideal gas law: pressure arises from Newtonian momentum transfer and
temperature from the equipartition of energy across degrees of freedom. The
accuracy of the ideal gas law reflects not only the predictive power of
Newtonian mechanics, but also the energy distribution suggested by the
equipartition theorem and the statistical regularity expressed in Avogadro’s
law.
While
Avogadro’s law states that the volume of an ideal gas is directly proportional
to the number of moles at constant temperature and pressure, this relationship
is an idealization that real gases only approximately follow. In practice,
gases exhibit intermolecular attractions and occupy finite volumes—factors
accounted for in the van der Waals equation by the gas-specific constants a
(which corrects for attractive forces) and b (which corrects for
molecular size). The Redlich–Kwong equation offers a further refinement
by more accurately fitting experimental data across a broader range of
conditions. These corrections indicate that molar volume and particle count can
vary with the physical properties of the real gas, thereby violating the
assumptions of ideal gas. Therefore, Avogadro’s law does not strictly hold for
real gases, particularly at high pressures or low temperatures where non-ideal
effects become significant.
In the audio recording [48:50], Feynman remarks, “It is one of those
famous gas laws whose names I do not remember—Boyle’s, Charles’s, and everybody
else wrapped into one.” He could have referred to Gay-Lussac’s law, which he did not name
explicitly. In 1834, Benoît Paul Émile Clapeyron unified Boyle’s law, Charles’s
law, Avogadro’s law, and Gay-Lussac’s law into what is now known as the ideal
gas law.
2.
Equipartition Theorem:
“The equilibrium
conditions are the same. No matter where the piston is, its speed of motion
must be such that it passes energy to the molecules in just the right way. So
it makes no difference about the spring. The speed at which
the piston has to move, on the average, is the same. So our theorem,
that the mean value of the kinetic energy in one direction is ½kT, is
true whether there are forces present or not (Feynman et al., 1963, p.
39-10).”
Feynman’s
statement—“the mean value of the kinetic energy in one direction is ½kT, is
true whether there are forces present or not”—captures the essence of the
equipartition theorem but it risks being misleading. The theorem assumes that the
system is in thermal equilibrium, a condition that only holds when all forces
involved are conservative. Dissipative forces, such as friction or viscous
drag, violate this premise by continuously extracting energy from the system, thereby
disrupting equilibrium. For example, in a gas subject to viscous damping, kinetic
energy is gradually transformed into thermal energy, causing local temperature fluctuations.
A more accurate formulation would state: “The mean kinetic energy associated with each degree of freedom is ½kT, provided the
system is in thermal equilibrium and all forces are conservative.” This caveat is
crucial for applying the theorem to real-world systems, where energy losses are
not negligible.
“Incidentally, we have also
proved at the same time that the average kinetic energy of the internal motions
of the diatomic molecule, disregarding the bodily motion of the CM,
is (3/2)kT! For, the total kinetic energy of the parts of the molecule is
½mAvA2 + ½mBvB2,
whose average is (3/2)kT+(3/2)kT, or 3kT. The kinetic energy of the
center-of-mass motion is (3/2)kT, so the average kinetic energy of the
rotational and vibratory motions of the two atoms inside the molecule is the
difference, (3/2)kT (Feynman et al., 1963, p. 39-11).”
Feynman’s derivation of the average kinetic energy of a diatomic molecule as (3/2)kT, obtained by subtracting the center-of-mass motion, offers valuable insight—but requires qualifications. The equipartition theorem is not a universal law, but a classical approximation that holds only under specific physical conditions, such as quantum effects are negligible. For example, Einstein’s (1907) model showed that vibrational degrees of freedom freeze out at low temperature when kT << ℏω. At room temperature, diatomic molecules exhibit energy contributions from translational motion (3/2)kT and rotational motion (kT), while vibrational modes contribute minimally and become significant only at higher temperatures (See figure below). This underscores the importance of specifying the conditions under which the theorem applies—its dependence on temperature and molecular structure. Strictly speaking, the equipartition theorem is derived using Hamiltonian mechanics, applied within the framework of statistical ensembles that describe how systems behave on average.
 |
| (Source: Young, Freedman, & Ford, 2012) |
3. Degrees of freedom:
“These “independent
directions of motion” are sometimes called the degrees of freedom of
the system. The number of degrees of freedom of a molecule composed
of r atoms is 3r, since each atom needs three coordinates to
define its position (Feynman et al., 1963, p. 39-12).”
The degrees of
freedom (DOF) in the equipartition theorem
represent independent ways a system can store energy. Each DOF acts
like a storage bin, holding an average of ½kT of
energy. Translational DOFs correspond to motion along the three spatial
directions, while rotational DOFs depend on its molecular structure. For a
rotational DOF to contribute, the molecule must have a nonzero moment of
inertia about the corresponding axis. Diatomic molecules like N₂ or O₂ can
rotate about two axes perpendicular to the bond axis, but rotation around the
bond axis contributes negligibly because the moment of inertia is extremely
small. In contrast, monatomic gases have no internal structure—effectively all
mass is concentrated at a point—so any rotation is effectively like a
relabeling of coordinates (without change in rotational kinetic energy). In
general, molecules possess 0 (monatomic), 2 (diatomic), or 3
(polyatomic) rotational degrees of freedom; in certain special systems
like adsorbed large molecules, rotational DOF can be effectively reduced
to just one due to constrained motion.
“Our theorem, applied to
the r-atom molecule, says that the molecule will have, on the average,
3rkT/2 joules of kinetic energy, of which 3/2kT is kinetic
energy of the center-of-mass motion of the entire molecule, and the rest, 3/2(r−1)kT,
is internal vibrational and rotational kinetic energy (Feynman et al., 1963, p.
39-12).”
The application of
the equipartition theorem to an r-atom molecule, as outlined by Feynman in one
sentence, can be clarified as follows. According to this theorem, a molecule
composed of r atoms in thermal equilibrium has an average kinetic energy
of (3r/2)kT, corresponding to three translational degrees of freedom per atom.
Of this total, (3/2)kT is related to the translational motion of the molecule’s
center of mass. The remaining (3/2)(r−1)kT is associated with internal
motions—rotations and vibrations involving the relative movement of atoms
within the molecule. However, this classical model is only an approximation. It
does not account for the quantum nature of molecular motion, especially the
quantization of vibrational and rotational energy levels, which become
significant at low temperatures or for light molecules. While some degrees of
freedom may be thermally inaccessible, vibrational motion can contribute both
kinetic and potential energy to the system.
Note: In Einige allgemeine Sätze über Wärmegleichgewicht, Boltzmann
(1871) writes:
“If intramolecular motions are present, and if f is the number of
degrees of freedom of a molecule, then the corresponding fraction of the
average kinetic energy......”
Review questions:
1. Do you agree
with Feynman that equal volumes of different
gases, at the same pressure and temperature, have the same number of molecules,
because of Newton’s laws?
2. Would you state the equipartition theorem as “the mean value of the
kinetic energy in one direction is ½kT, is true whether there are
forces present or not”?
3. How would you
explain the degrees of freedom using an analogy?
The Moral of the
Lesson (In Feynman’s Spirit): Your body is like a molecule—its
energy depends on how freely it can move! A hunched posture isn’t just about
looks; it’s like freezing out degrees of freedom. Just as a
rigid diatomic molecule at low temperature can’t access all its rotational and vibrational modes, a hunched spine limits your mechanical range, restricting movement and
reducing kinetic energy output. The result? Fewer calories burned, a slower
metabolism, higher blood sugar, and—like a system trapped in a low-temperature
state—a body stuck in a suboptimal equilibrium. But when you stand
tall, align your spine, and move fluidly, you're not just defying
gravity—you're activating dormant degrees of freedom and reclaiming your
full thermodynamic potential.
In short, the best
posture is changing posture by increasing your body’s degrees of freedom.
References:
Boltzmann,
L. (1871) Some General Statements on Thermal Equilibrium. Wiener Berichte,
63, 679-711.
Einstein,
A. (1907). "Die Plancksche Theorie der Strahlung und die Theorie
der spezifischen Wärme" (Planck's Theory of Radiation and the
Theory of Specific Heat). Annalen der Physik, 22(1),
180-190.
Feynman,
R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on
Physics, Vol I: Mainly
mechanics, radiation, and heat. Reading, MA: Addison-Wesley.
Young, H. D., Freedman, R. A., & Ford, A. L. (2012). Sears and Zemansky's University physics with Modern Physics, 13th ed. Pearson Higher Education.
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