  # Entropy Activity

-Introduction

-The reaction coordinate

-The energy landscape

-Population distributions

-Motion at constant temperature, and exchange of energy with the heat bath

-Thermally activated processes

-Energy and temperature determine the populations

-Entropy and free energy

-Mathematical derivation of free energy

 < Previous | Next >

## Population distributions

In the previous simulations, you determined the strength of kick needed to knock the box over or stand it up. Molecules are always being kicked by their surroundings, and the strength of these kicks is related to the temperature of the system. For molecules in solution, the kicks come from the surrounding solvent molecules. However, the details of the surroundings are not essential to making predictions of how the system will behave. In fact, we can just view the surroundings as a "heat bath" that exchanges energy (heat) with the system through random kicks.

For our box, a good analogy of a heat bath is to consider placing the box on a shaking platform. The platform kicks the box randomly, with the strength of the kicks being set by the temperature of the heat bath. Before putting boxes on a shaking platform, let's consider placing balls on the platform. The height to which a ball rises after being kicked by the platform is an indication of how hard it was kicked. The average height of the balls is then an indication of the average strength of the kicks.

Virtual Activity

- Population distributions (Java Required) [Opens in a separate window. Please read through all content in the lower window, using the scroll bar.]

In the above activity, the ratio between the number of particles in state 2 and the number in state 1 is equal to: Where P2 and P1 are the "populations" of state 2 and state 1, i.e. the average number of particles in each of these states.

Note that P2/P1 depends only on ΔE and RT. ΔE is the difference in energy between the two states, and RT is the thermal energy (i.e. a measure of the average kinetic energy of the particles, or in our example, the average height of the bouncing balls). To predict P2/P1, we don't need to know the detailed pathway between the states, we just need to know the difference in state energies and the temperature. (The pathway can change how long it takes to reach thermal equilibrium, i.e. how long it takes the populations of the two platforms in our simulation to reach steady state.)

If you watch the simulation for a while, you'll notice that every so often a ball gets a very strong kick from the platform. Although these strong kicks are rare, they are important since they give the system enough energy to make it to a high-energy platform. Even small increases in temperature can substantially increase the number of times a system gets a strong enough kick to reach a highly activated state, which would be represented here as a high energy platform. Try changing the temperature and noting how often a ball gets a strong kick.

 < Previous | Next >