Virtuallabs
From Ccliwiki
States, Energy, Degeneracy, Entropy, and Free Energy
Lessons and Applets
Site Content
This set of lessons and applets aims to enable first-year undergraduate students to understand a number of difficult concepts. As each concept is introduced, an applet, accompanied by a series of exercises involving the applet, will be provided to aid the students' understanding of the concepts. Through the lessons and applets, students will become familiar with the following concepts:
- states
- energies of states
- energy barriers between states
- degeneracies of states
- entropy
- free energy
Site Layout
This website is set up with a navigation bar on the left and a content window on the right. In the navigation bar, students may select a particular lesson and access the introduction to the lesson. Then, by clicking on the applet link in the navigation bar, the student can access the applet corresponding to the lesson. Clicking on the applet link also creates a small window at the bottom of the screen which gives instructions on what to do with the applet. Both the applet and this instruction window are visible simultaneously, for maximum convenience.
Pretest
Before taking the virtual labs, please take a pretest at http://www.surveymonkey.com/s.aspx?sm=sBAsGK48duvD1RK1SCMUBA_3d_3d.
BRIEF SURVEY: After completing all the virtual labs, please take a survey at http://www.surveymonkey.com/s.aspx?sm=2pHB3ueOdo32KJVoVSefKQ_3d_3d to give us feedback on the Virtual Labs.
Thank you. We value your input.
Contents |
Lesson 1
States of Physical Systems
At any given time a physical system at equilibrium can be said to occupy one of many possible states. There are three types of equilibrium states that a system can occupy:
- Unstable: If a system in an unstable state is perturbed in any way, it will exit this state and after a period of transience, will settle in a stable or metastable state. Because the slightest perturbation will cause the system to leave an unstable state, systems are generally never observed in unstable states.
- Metastable: If a system in a metastable state undergoes a small perturbation, it will remain in this state; however, if the perturbtation is large enough, it will exit this state and, after a period of transience, settle in a different metastable or stable state.
- Stable: This state is the most energetically favored of all possible states. Like metastable states, small perturbations will leave the system unaffected, but large enough perturbations will cause the system to transition to a metastable state. However, the perturbation required to move a system from the stable state to a metastable state is larger than the perturbation required to move a system from a metastable state to the stable state.
In the applet with this lesson, you will use the real-world example of a cardboard box to explore the concept of states. You will be asked to determine which configuration of the box corresponds to the stable, metastable, and unstable states. You will also have an opportunity to observe that perturbations are necessary to cause the system to transition from one state to another.
View Virtual Lab 1
Lesson 2
States and their Energies
Introducing the Energy Landscape
One of the important aspects of a state is the fact that a system in a particular state has a well-defined energy. In the case of the cardboard box, we can calculate the box's potential energy based on the height of its center of mass, represented in the applet by the intersection of the two diagonal lines. Furthermore, we can calculate the height of the center of mass, not only when the box is at equilibrium, but also as it is transitioning from one state to another. If we plot the height of the center of mass versus the angle one side of the box is making with the ground, we can create what is known as an "energy landscape". The energy landscape gives us a visual representation of the relative energies of the 3 equilibrium states that the box may occupy, in addition to the energies of the transient states in between the equilibrium states.
In this applet, you will have the opportunity to repeat the activities in lesson 1, except this time, you will be shown the energy landscape for this system and you will be able to observe where the system is on the energy landscape as the system undergoes a transition from one state to another.
View Virtual Lab 2
Lesson 3
Energy Landscape part 2
We saw in the previous lesson that each state that the system could occupy had a well-defined energy and that plotting that energy as a function of some kind of reaction coordinate (in this case, angle) resulted in what is known as an energy landscape. Do you think the energy landscape is dependent upon the specifics of the system?
In this exercise you will have the opportunity to explore how the energy landscape changes when you change the shape of the box.
View Virtual Lab 3
Lesson 4
Introduction to Energy Distributions
In the previous 3 lessons, you were responsible for choosing the magnitude of the kicks that the box received and you used information from the energy landscape to determine the kick magnitude necessary to cause state transitions. While it's interesting to explore the different states that a box can take and the kicks necessary to cause a state transition, you may have felt that this all has more to do with physics than chemistry.
As it turns out, kicks similar to those you applied to the box in the previous lessons are happening all around you right now - very rapidly on a tiny scale. Atoms in a liquid or solid are always vibrating and their constant vibration is responsible for the "kicks" happening all around you. Most of the time, these kicks don't cause any change to the thing being kicked. However, occasionally, if the kick magnitude is large enough, the kick can cause a physical or chemical change. A simple example of a physical change would be water molecules leaving a liquid phase and entering a gas phase after being kicked by another energetic water molecule or, perhaps, the rapidly vibrating surface of a hot pan. An example of a chemical change might be a diatomic molecule dissociating into two separate atoms after the interatomic bond is broken by an energetic kick from the molecule's surroundings.
As stated above, atoms are always vibrating. The vibration of atoms is linked to temperature. In fact, temperature is defined in terms of the kinetic energy of a collection of atoms. The larger the vibrations, the greater the kinetic energy, the higher the temperature. So, it shouldn't be surprising to find out that in a system that is at a high temperature, kicks of a large magnitude are more likely to be observed than in a system at a low temperature. An expression that allows us to calculate the probability of observing a kick of a given magnitude for all possible magnitude values is called a distribution, and is generally a function of temperature. If you have taken a statistics class or studied statistics in a math class, you may already be familiar with the concept of distributions. One commonly discussed distribution is the uniform distribution, in which the probability of observing a particular value is uniform, or constant, over a specific domain and zero outside of that domain. Another commonly used distribution is the normal, or Gaussian, distribution, in which the probability of observing a particular value is highest at the distribution's mean and decays to almost zero by two standard deviations away from the mean.
In this lesson, the box that we have examined in prior lessons will be given kicks just like before, except that, this time, the energy of the kick is chosen randomly from a distribution, rather than by you. In addition, the box will be given kicks over and over, just as an object in contact with a constantly vibrating surface would experience. You will have the opportunity to see how the distribution used in assigning the kick energies to the box varies with temperature.
View Virtual Lab 4
Lesson 5
A New Model System
In the previous examples, we used a cardboard box to illustrate states, state energy, and the energy landscape. However, looking at other models can help to give us a more complete picture of some of these concepts.
In this lesson, we look at a model that depicts balls bouncing on raised platforms. The platforms can be thought of as a quantized version of the energy landscape presented earlier. The balls are initially placed in the system above the platforms and fall downwards, due to gravity. Whenever a ball lands on a platform, it is given a kick, similar to the kicks we gave the cardboard boxes in previous examples. The kick causes the ball to fly back up into the air for some period of time, after which it falls back onto a platform and is given another kick. Like the previous example, the distribution from which the kick magnitudes are chosen depends on temperature, with larger kicks being more likely at high temperatures. The balls' motion is uncoupled and they do not interact with one another.
It may be helpful to imagine that the balls are light-weight beads and that the platforms have speakers directly under them. The sound being played through the speaker produces mechanical vibrations on the surfaces of the platforms, which give the beads a kick.
The interesting thing about this model is that it gives us a natural way to look at the occupancies of states. We can simply observe how many beads are sitting on one platform and that tells us how many beads are in that state. Using this model, we will explore how occupancy varies with the relative heights of the platforms and the height of a thin "energy barrier" platform in between wider platforms. In subsequent lessons, we will use this model to help us better understand the box model.
View Virtual Lab 5
Lesson 6
Energy and temperature determine the populations
View Virtual Lab 6
Lesson 7
Entropy and free energy
View Virtual Lab 7
Post-test
After taking the virtual labs, please take a post-test at http://www.surveymonkey.com/s.aspx?sm=d_2fbnOBnOGfdofMPYxNofHw_3d_3d
