Peoples Physics Book Version-3

http://www.ck12.org Chapter 19. Thermodynamics and Heat Engines

The left side of this is identical to the left side of equation [3], whereas the only variable on the right side is
temperature. By setting the left sides equal, we find:

2

3

n(KE)avg=nkT

or

T=

2

3 k

(KE)avg

Therefore, according to the kinetic theory of an monoatomic ideal gas, the quantity we called temperature is
— up to a constant coefficient — a direct measure of the average kinetic energy of the atoms in the gas. This
definition of temperature is much more specific and it is based essentially on Newtonian mechanics.

Temperature, Again

Now that we have defined temperature for a monoatomic gas, a relevant question is: can we extend this definition to
other substances? It turns out that yes, we can, but with a significant caveat. In fact, according to classical kinetic
theory, temperature is always proportional to the average kinetic energy of molecules in a substance. The constant
of proportionality, however, is not always the same.

Consider: the only way to increase the kinetic energies of the atoms in a monoatomic gas is to increase their
translational velocities. Accordingly, we assumed above that the kinetic energies of such atoms are stored equally in
the three components (x,y,andz) of their velocities.

On the other hand, other gases — called diatomic — consist of two atoms held by a bond. This bond can be modeled
as a spring, and the two atoms and bond together as a harmonic oscillator. Now, a single molecule’s kinetic energy
can be increasedeither by increasing its speed, by making it vibrate in simple harmonic motion, or by making
it rotate around its center of mass. This difference is understood in physics through the concept ofdegrees of
freedom: each degree of freedom for a molecule or particle corresponds to a possibility of increasing its kinetic
energy independently of the kinetic energy in other degrees.

It might seem to you that monatomic gases should have one degree of freedom: their velocity. They have three
because their velocity can be altered in one of three mutually perpendicular directions without changing the kinetic
energy in other two — just like a centripetal force does not change the kinetic energy of an object, since it is always
perpendicular to its velocity. These are called translational degrees of freedom.

Diatomic gas molecules, on the other hand have more: the three translational explained above still exist, but there
are now also vibrational and rotational degrees of freedom. Monatomic and diatomic degrees of freedom can be
illustrated like this: