Exploration 20.4: Equipartition Theorem
Please wait for the animation to
The kinetic energy of a particle can be due to motion in the x,
y, and z directions, as well as to rotations. The equiparition of energy
theorem says that the kinetic energy of an atom or particle is, on average,
equally distributed between the different modes (different degrees of freedom) available. In a monatomic gas, an individual atom has three degrees of freedom
because it can move in the x, y and z directions. The energy per particle has
an average value of (f/2)kBT, where f is the number of
degrees of freedom, kB is the Boltzmann constant, and T is the
- In this animation of a monatomic gas in a box, why do the
particles only have 2 degrees
of freedom? The table shows the total kinetic energy of all particles in
the box, as well as the average kinetic energies of particles in the
box (the animation averages over a 10-s period, so you need to wait
10 s to read the averages).
- Record the total energy.
- What is
the energy per particle?
- If the energy is given in joules/kB, what
is the temperature inside the box?
Try this animation of a diatomic gas with 20
particles. Notice that the graph
shows the total kinetic energy of the diatomic particles and the kinetic energies of
translation (motion in x and y directions) and rotation.
- Why is the translational kinetic energy, on average, about two times the rotational kinetic
energy? (The animation averages over a 10-s interval, so you need to wait
for the animation to run for at least 10 s to read the average values of
- From the total energy, what is the energy per particle?
- If the energy is given in joules/kB, what is the temperature in the
box? (Remember that <energy>/particle = (f/2)kBT and in
this case, f = 3 (Why?).)
Now, try a mixture of 20 monatomic particles and 20
- Why is the temperature of the gas in the box a single value (not one value for atoms and
another for molecules)? Hint: Think about the air surrounding you at
essentially a constant temperature, unless the heater or air conditioner just
turned on and made one section of the air a different temperature. Air
is made up of monatomic particles (helium) and diatomic particles (water, oxygen,
- After waiting at least 10 s, compare the average values of the
kinetic energies. What
value is the average monatomic kinetic energy close to?
- Why should those two values (the two averages that you found in part (i)), averaged over a long period of time, be equal
to each other and greater than the
rotational kinetic energy of the diatomic particles?
- Explain why the total energy
should be equal to (2/2)20kBT + (3/2)20kBT.
- From the total
energy (given in joules/kB), what is the temperature?
- In this animation, if a mixture has 15 atoms, how many diatomic particles
should it have so that the average kinetic energies of both particles are the
same? Try setting the number of monatomic particles and diatomic
particles to check your
Exploration authored by Anne J. Cox.
Script authored by Wolfgang Christian modified by Anne J. Cox.
Applet authored by Ernesto Martin and modified by Wolfgang Christian.
© 2004 by Prentice-Hall, Inc. A Pearson Company