Ephemera (i.e. will be taken down)
NOTE: This document is the provenance of the other review materials in this series. The form has not been optimized, and it will be taken offline as soon as the content finds a new home. I include this ephemera because it contains valuable information, regardless of its level of completeness. MTP 12/11/10

Equations of the Week: Translation into English
Scientists, engineers, et cetera speak in the Equation language, while most Harvard students speak in the English language. By linking mathematic and scientific jargon and concepts to your existing everyday observations from cooking food, we have tried to teach you how to read, speak, and understand this Equation language. Hopefully we have been more or less successful. Below I aim to solidify the viscous fluid of information that has made its way into your mind—I attempt to “cross-link” the myriad strands of new concepts floating in your head into a solid foundation for you to view the world.
Herein I explain a few concepts about the equations of the week with the goal of elucidating the core content the course.
Week 5
In Week 5, we continued to view equations as a balance. The idea is in any physical system, there are competing forces acting on the system elements. This week the forces were:
+ the electrostatic forces between different regions of a single polymer chain; between different regions of different polymer chains; and between regions of polymer chains and charged particles dissolved in water—acids, bases, and salt solutions. Electrostatic forces occur between electric charges: Like charges are pushed (or forced) away from each other; opposite charges are pulled (forced) towards each other.

+ the hydrophobic forces between regions of polymer chains and a solution of water. Hydrophobic molecules are push (forced) away from water molecules.

+ the entropic (or thermal) forces between a heat source and a solution of polymer chains. Thermal energy or “heat is seen as a form of energy arising from the random motion of the molecules of bodies.” One method of transferring this random motion of molecules (vibrations or thermal energy) is through direct contact. That is, the molecules of a hotter material will be moving around faster than the molecules of a colder matter. This faster motion (higher velocity) has its own forces associated with it and thus has its own energy associated with it (kinetic energy). You can find the sum of this kinetic energy for all of the individual molecules of a material. The sum of kinetic energy for a set of vibrating molecules at any given temperature is defined as the thermal energy at the temperature. Saying it differently, molecules are always moving around; at any given temperature they have an average speed at which they travel. The relationship linking this speed to a quantity we can measure—the temperature—is an energy balance! The temperature allows us to calculate an average of the velocity for a collection of particles. For us here, the energy associated with this speed is more useful than the molecular speed itself. This is because we are balancing the energies aldfjal;sf a;sdfjkSo we balance The thermal energy can be found by for an Temperature is a direct measure of heat (thermal energy)Heat makes things vibrate, pushing (forcing) them to move around randomly, providing more states that can be explored and increasing the entropy.

* You may be slightly confused at this point, because in this course, we didn't call these forces—we called them energies—but since force and energy are intimately related (in a way that we didn't explain) and most people understand the concept of force better than energy, I thought it would be helpful to begin this elucidation using a discussion of the competition between forces.
To illustrate this competition between forces Professor Brenner beautifully used the tug-of-war
tug of war picture
analogy. In the sense that different parts of a system are pushing and pulling on each other until some part yields, there is quite literally a war going on between the forces! When some force “wins”, interesting things happen. That is to say, in physical systems, when there is a change in the energy balance—a force “wins”'—we typically see dramatic changes. This week, when the electrostatic and entropic forces beat the hydrophobic forces we saw milk turn to cheese! To say it differently, when the sum of the entropic (or thermal) energy and the electrostatic energy was greater than the hydrophobic energy in our system of polymers in solution (milk) we saw the protein chains denature and then coagulate (curddle).
      Now let's say this in math. To translate the above concepts into the Equation language we start by defining some vocabulary:
+ hydrophobic energy is defined as UHydrophobicity
+ electrostatic energy is defined as UElectrostatics
+ entropic (or thermal) energy is defined as UEntropic or UThermal
, or NkbT
UHydrophobicity = NkbT + UElectrostatics
.