The Nature of temperature: Lab January 16, 2007
A: The same number of molecules were on both sides of the chambers. When the door was opened,
the number increased on the lower temperature side of the chamber: I believe because the higher
temperature particles were moving faster and expanded more quickly to the other side of the
chamber.
B: Fewer particles at a certain temperature means that the temperature of the particles will change
the greatest. Temperature is quantity dependent.
C: When there are a fewer number of particles and the temperature difference is smaller than other
simulations, the time to reach equilibrium is greater.
Critical Thinking Question
1.
Suppose we place a glass of hot water in a cool room. What happens to the temperature of the water? What happens, if
anything, to the temperature of its surroundings?
1. When a glass of hot water is placed in a cooler room, the temperature of the glass of hot water
will decrease in temperature significantly. The temperature of the surroundings will increase
slightly, but most likely not within levels of detection.
Simulation II Identification of Temperature
(a) Does the kinetic energy and potential energy change as the system evolves? Why?
When the two solids are not in thermal contact with each other, then the kinetic energy and the potential
energy seems to remain constant. The Kinetic energy seems to change more than the potential energy. The
systems are isolated so there is nothing that interferes or leaves the substances.
What is the mean potential and kinetic energy of each system?
Potential for RED: -3.05 Kinetic: 0.9
Potential for GREEN: -4.4 Kinetic: 0.14
(b) Remove the barrier (select contact) and let the two systems interact with one another. Monitor the kinetic and
potential energy of each system. After equilibrium has been established between the two systems, compare the
average kinetic and potential energies to their values before the two systems came into contact. What quantity is
exchanged between the two systems? (The volume of each system is fixed.)
The average Potential is -3.7 and the kinetic is 0.16, which is an increase in kinetic with both systems greater
than the initial kinetic energy. Kinetic energy is exchanged between the two systems because it is what
changes.
(c) We are looking for a quantity that is the same in both systems after equilibrium has been established. Are the
average kinetic and potential energies the same? The average kinetic energy is the same for both substances.
Consider what would happen if you doubled the number of particles N and/or the system size L for each system?
Would the temperature change? When N is increased, the temperature increases. When the size is increased,
the temperature is decreased.
Does it make more sense to compare the average kinetic and potential energies or the kinetic and potential energies
per particle? What do you conclude about the possible identification of the temperature? The average Potential
energy and the average kinetic energy is the same as per molecule. But the Kinetic energy is better
measurement of temperature.
We know what happens when we place two bodies at different temperatures in thermal contact with one another – energy is
transferred from the hotter to the colder body until thermal equilibrium is reached and the two bodies have the same
temperature. Although the total energy would be conserved in any process, the distribution of energy changes in an
irreversible manner, i.e., hot objects always get colder. We will now consider another simple model that illustrates how
statistical concepts can help us understand this directional transfer of energy and the microscopic nature of thermal
equilibrium.
Model 1: Einstein Solid
Consider a system with n particles. Assume that the energy, i, of each particle, i, is given by integer values i = 0, 1, 2, 3…
n
and that the total energy of the system is E 
 . How many ways can E units of energy be distributed among n different
i
i1
particles? Suppose we use a purple dot to represent each unit of energy and vertical lines to differentiate the n particles.
Figure (1) shows two example configurations of E = 6 units of energy distributed among n = 4 particles,

Figure 1: Two configurations for which six units of energy are distributed among four particles. Each configuration is represented two different ways, (a) an
energy diagram where the yellow dots represent particles with energy, , and (b) a linear distribution of two equally probable item types, purple dots
representing energy units and lines distinguishing particles.
Note: this can be viewed as a distribution of two types of items, dots representing energy units and lines distinguishing
particles.
Critical Thinking Questions
2.
Illustrate two other configurations similar to Figure (1) representing unique distributions of E = 6 units of energy among
n = 4 particles.
Determine the total number of possible configurations for which E = 6 units of energy (dots) can be distributed among n
= 4 particles. (Hint: You should utilize one of the relationships learned previously to calculate the multiplicity.)
The type of counting used for this model, is often referred to as Bose-Einstein statistics. Show that the general
relationship used to calculate the number of configurations for this model is
3.
4.
W
(E  n 1)!
.
E!(n 1)!
(1)
Create a plot of number of configurations, W, as a function of the total energy, E = 0…6, for a four-particle system. How
does multiplicity depend on energy?

90
80
Configurations
70
60
50
40
30
20
10
0
0
1
2
3
4
Energy
5
6
7
Model 2: Insulating Einstein Solids
The evaluation of the number of ways you can distribute energy among a collection of particles or systems is important in
physical calculations since any collection will tend toward the configuration of maximum multiplicity (maximum entropy).
We will now consider a model system that can exchange energy with another system. Consider an isolated system of N = 4
particles that is divided into two subsystems surrounded by insulating, rigid, impermeable outer walls and separated by a
similar partition (see Figure 2).
Figure 2: Two subsystems each with two particles separated by an insulating, rigid, and impermeable wall.
Subsystem A consists of two particles with total energy EA = 5; subsystem B consists of two particles, with total energy EB =
1. The partition is an internal constraint that prevents the transfer of energy from one subsystem to another and in this case
keeps EA = 5 and EB = 1. (The internal constraint also keeps the volume and number of particles in each subsystem fixed.)
Critical Thinking Questions
7. Identify the multiplicity, WA, number of configurations for subsystem A.
8. Identify the multiplicity, WB, number of configurations for subsystem B.
9. Identify the total multiplicity for the composite system, i.e., subsystem A and subsystem B before the removal of the
internal constraint, (i.e., subsystems are separated by an insulating boundary).
Model 3: Conducting Einstein Solids
We now consider a simple example of a thermal interaction. Suppose that the insulating, rigid, impermeable partition
separating subsystems A and B is changed to a conducting, rigid, impermeable partition. The partition maintains the volumes
VA and VB, and hence the single particle energy eigenvalues are not changed. Because the partition is impermeable, the
particles cannot penetrate the partition and go from one subsystem to the other. However, energy can be transferred from one
subsystem to the other, subject only to the constraint that the total energy of subsystems A and B is constant.
Figure 3: Two subsystems each with two particles separated by a conducting, rigid, impermeable wall.
Critical Thinking Questions
10. Identify the total multiplicity for the composite system, i.e., subsystem A and subsystem B after the removal of the
internal constraint, (i.e. subsystems are separated by a conducting boundary).
11. Complete the following table detailing various quantities for all possible values of EA for the composite system of
Model 3.
EA
6
WA(EA)
WB(EB)
WAWB
PA(EA)
5
4
3
2
1
0
12. What is the probability of subsystem A gaining energy when the internal constraint is removed?
13. What is the probability that the energy remains unchanged when the internal constraint is removed?
14. What is the probability that energy is transferred from the “hotter” to the “colder” subsystem when the internal
constraint is removed?
15. What is the driving force for the transfer of energy between subsystems?
16. Determine the average energy, <EA>, of the composite system.
17. Create a plot of P(EA) vs. EA . For this model what is the relationship between the most probable energy, Emp, and
the average energy, <EA>, of the composite system?
Simulation III: Einstein Solid (http://stp.clarku.edu/simulations/einsteinsolid)
(a) Enter the data for Model 3 and confirm that the probability distribution, number of initial configurations, WA = A,
most probable energy, Emp, the average energy, <EA>, and probability that energy flows from A to B is consistent
with your previous answers.
(b) We will now use the applet to investigate a variety of different systems in order to understand the qualitative
behavior of this model as function of the parameters, EA,, EB, NA, and NB. Note: When investigating the effects of
various quantities, you need to be careful not to change too many variables at once. Be sure you understand the
effects of one variable change, before you start to consider the effects of a second variable. The following table has
been constructed as a guide to help you systematically consider a number of different interesting relationships that
can be understood from this simple model. Feel free to try additional combinations until you are satisfied that you
can accurately predict the results of this model.
EA
EB
NA
NB
6
2
2
2
6
2
20
20
6
2
200
200
12
2
2
2
12
2
20
20
12
2
200
200
60
2
2
2
60
2
20
20
60
2
200
200
60
2
30
10
Emp
<EA>
PE (AB)
60
2
35
5
60
2
39
1
31
31
20
20
31
31
30
10