Hotter than infinite temperatures.
Introduction
Explore the meaning of negative temperatures.
Material
3 poker chips
3 stairs
Assembly
Use a card to label the lower stair 0 energy units (the ground state) the middle stair 1, and the upper stair 2, these numbers are the energy levels of each stair.
Getting Started
One definition of temperature, T, is that 1/T =
dS/dU
where U is the internal energy in a system and
S is the entropy.
Approximate T = DU/DS
Where the symbol D
stands for change. So temperature is the ratio
of the change in internal energy to the change in entropy.
(This definition of temperature of a gas assumes we do no work on the
gas.)
For a history of different definitions of temperature see What is Temperature?
In systems like a box of ideal gas, when heat flows into the box, the internal energy of the box goes up and the entropy in the box goes up. So temperature of an ideal gas is always positive (the ratio of two positive numbers.)
Because the atoms are in a box they can only inhabit discrete energy levels (from quantum theory) the three stairs represent the three possible energy levels
Entropy is defined as a constant times the logarithm of the number of possible ways to create each state of the gas, W. So that S = k ln W. However, for simplicity, here we will use a simpler form of entropy and say it is S = log W where we use the base 10 logarithm.
We can make a simple model that uses this definition to produce negative temperatures.
To Do and Notice
Place all the poker chips on the lowest step, step
0.
The total energy of the system, U, is 4 x 0 = 0.
The entropy of the system, S, is proportional to the logarithm of the
number of ways, W,
of placing the chips all on one level. For simplicity let's choose
the proportionality constant to be 1 when we use the base 10 log.
There is only one way to do this. So the entropy is S = Log 1 =
0.
Move one chip to the higher level. This higher
level is an energy 1 unit above the lowest step.
The energy of the new system is 1.One chip is in the level with
energy 1.
The energy change between the two systems is
DU =
1.
The number of ways of arranging the chips is 4, and the entropy of
the new system is proportional to log 4. The change in entropy is
D S = log
4  log 1.
The temperature is proportional to D
U/DS
= 1/log 4 which is positive.
Move two chips to the higher level.
The energy of the new system is 2.
The change from the previous level is D
U = 1
There are 6 different ways to arrange 4 chips on 2 levels 2 on each
level.
The entropy of the new system is proportional to log 6, the change in
entropy is log 6  log 4.
The temperature is 1/(log 6  log 4) which is positive
Move three chips to the upper level
The energy is 3, D
U = 1
There are 4 ways to arrange chips on two levels so that there are 3
on one level and 1 on another.
The entropy is log 4, the change in entropy is log 4  log 6
The temperature is T = 1/(log 6  log 4)
This is a negative temperature.
The energy of the system went up, yet it became more
ordered.
What’s Going On?
A system with a finite number of energy levels, such as two, can have a lower entropy as the upper level becomes full, this is called a population inversion and is found in lasers, so negative temperatures are used to describe lasers.
The counting of states here assumed that the individual particles can be told apart. Slightly different counting applies if the particles are indistinguishable.
Going Further
Negative temperatures are hotter than infinite temperatures
Repeat the above activity with 3 poker chips.
Start with all the chips on the lowest level.
Move one chip to the higher level.
The temperature is T = D
U/DS
= 1/(log 3)
Move a second chip and the temperature is
T = 1/(log 3  log 3) = •
Move a third chip and the temperature is
T = 1/(log 1  log 3) = 1/log 3
So negative temperatures are on the other side of infinity from
positive numbers.
Negative temperatures are hotter than infinity.
Etc.
The temperature here is in a weird unit since we have chosen our definition of entropy to be S = log W, instead of S = k ln W, where ln is the natural logarithm and k is Boltzman's constant k = 1.38 x 10^{23} J/K.
The temperature in kelvins is proportional to the temperature we use here.
The definition of the temperature of a monatomic ideal gas used in the 19'th century that temperature is the
average
kinetic energy of
random
translational motion
per molecule
is identical to this new definition when the new temperature is applied to an ideal gas. However, the new definition is independent of the idealness of the gas used, and it also allows us to extend the definition of temperature into new realms.
Scientific Explorations with Linda Shore and Paul Doherty 

20 July 2001 