Message-Id: <v01540b11b12df1d0b31e@[192.174.2.173]>
Date: Thu, 12 Mar 1998 12:49:16 -0800
To: pinhole@exploratorium.edu
From: pauld@exploratorium.edu (Paul Doherty)
Subject: temperature
Hi Paul,
I know temperature to be an measure of the average kinetic energy of the
particles that make up a substance. With a monatomic gas, piece of
cake--translational KE. How about a di/polyatomic gas? How do all modes of
motion (translational, rotational, vibrational) get in the act? What is
temperature a measure of when it comes to solids and liquids? And how about
the cosmic microwave background temperature of 2.7 Kelvins: is there stuff
out there that is at the same temperature and therefore at thermal
equilibrium? Is there any matter in space colder than 2.7 K ? Should I ask
about negative Kelvins now?
Jay Goldberg
Good questions Jay
As you indicate, for an ideal gas, temperature is proportional to:
The average
random
kinetic energy of translational motion
per molecule.
Every word is important here:
You must average which means that you need two and usually many more
molecules to have a temperature.
Random means that you subtract the center of mass motion of all the molecules.
(This way the temperature doesn't change when you run past a box of
molecules therby increasing the apparent speed of all of them from your
point of view. By using only the random motions you ignore the common
motion of all the molecules due to your changing point of view.)
kinetic energy (so we will be able to convert the temperature into joules
per molecule)
of translational motion so rotation of the molecules and vibrations don't count.
Per molecule this is the important difference that makes temperature
different from energy.
Rotational and vibrational energy doesn't count toward temperature!
However, when the molecules collide (we don't have a perfect ideal gas)
energy is shared between rotation, translation and vibration. So that each
independent motion gets the same energy.
for example, each molecule gets an energy of translational motion in the x
direction of, U, such that U = 1/2kT
Where k is the conversion factor into joules and is known as boltzmann's
constant 1.38 x 10^-23 J/K
T is the temperature in kelvins.
There is also 1/2 kT in the y direction and also in the z direction.
When we add the energies in all 3 directions we find that the total
translational energy of the molecule in 3-D is U = 3/2 kT.
By the way, since KE = 1/2 mv^2
where m is the mass of the molecule in kg
and v is its speed
then
1/2 mv^2 = 3/2 kT
v = (3kT/m)^0.5
and the component of the molecular speed is proportional to the square root
of the temperature and inversely proportional to the molecular mass.
(I'm simplifying here v is really the root-mean-square speed)
A diatomic molecule can vibrate as well, and 1/2 kT of energy goes into the
kinetic energy of vibration. An additional 1/2 kT goes into the potential
energy of vibration.
It can also rotate, and 1/2 kT of kinetic energy of rotation goes into each
of the two independent ways it can rotate.
If the molecule is stretched along the z axis it can rotate about the x and
about the y axis.
The kinetic energy in vibration, and rotation, and the potential energy do
not count toward the definition of temperature.
When the gas condenses into a liquid or a solid translational motion is
slowly constrained and turns into vibration, so that we cannot use the
ideal gas definition any more. We define the temperature of a liquid or
solid by putting it into contact with an ideal gas, knowing that two
objects in contact will come to the same temperature. Then we use the
definition of temperature for the ideal gas to find the temperature of the
solid.
Paul D