**Class Structure**(Described in Syllabus) Lectures will be mostly chalk on the board, supplemented by computer simulations and graphics. This is a 15 unit class. You are expected to spend 5 (waking) hours in lectures. The other ten hours should be occupied by reading the text or any supplementary materials, reviewing or recopying your lecture notes, and doing your homework. If you begin spending**significantly more**than 15 hours a week on this class, please confer with your classmates. The groups are designed to reduce the amount of time spent on homework--not to increase it. If the the class is generally spending more than 15 hours, let us know so we can adjust.**Grading****2 exams 25%/exam**- Two exams will be in-class during the lecture period.
**Homework 25%**- Class groups will be organized. Each group will turn in a joint homework and receive a single grade that is recorded for each member of the group. Everyone in the group receives the same grade.
**Class Participation 25%**- You will receive credit for asking good questions or making good
comments during lectures and recitations--
*if you state your name.*You will not receive credit for ``showing off'': a good question or a comment is one that benefits most of your classmates. There will be random questions to students during lectures. You should be prepared to answer simple questions succinctly or say that you honestly don't know-and call on a friend.

**Lecture Notes**will often be found on the web. You can use them any way you wish. The lecture notes will supplement (and sometimes correct) the textbook.**The textbook (KPIM)**for this course is being printed at CopyTech. You should be able to buy a copy next week at CopyTech for a very modest price. You are expected to read the book and understand the concepts that are covered in the lecture. Because the book is still in revision, comments/corrections on the book in the form of writing will be very welcome and will be regarded as class participation.**Office hours**can be found on the website.

__Thermodynamics and Kinetics__

**Joule/Kelvin**-
(01-1)

**Carnot/Clausius**- For the limiting case of a reversible process,
(01-2)

**Boltzmann**-
(01-3)

**Gibbs**- Rigorous and mostly complete foundation of
equilibrium of materials.
Two fundamental results.

*If an extensive quantity can be exchanged between two bodies, then a necessary condition for equilibrium is that the associated potential, which is an intensive quantity, must have the same value in each body.*

*If a system is in equilibrium with reservoir that maintains a constant potential (e.g. and ), then there exists a free energy function for that system (e.g., ) that is minimized at equilibrium. Therefore, a necessary condition for equilibrium is that every small variation in must be non-negative:*^{1}.

The study of materials processes is a combination of Thermodynamics and Kinetics. Kinetics relies on thermodynamics as a rigorous foundation. Kinetics is less rigorous and more approximate; but, perhaps applicable to more real systems.

Materials Processes | |

Relatively few variables | Complex interations involving a space and time continuum of variables. |

Precise stratements about equilibrium states or quasistatic processes | Approximate statements or models for complex evolving systems |

Powerful concepts and foundation, but limited to ideal systems | Approximate but predicitive statements for a wide range of phenomenae |

__Extending Thermodynamics to Kinetics__

If the constraint that the system has a homogeneous composition is removed, then if any combination of compositions --distributed among all possible phases in such a way that the average composition is --has a lower free energy than any homogeneous system, then the equilibrium free energy curve is the convex hull from below of all the homogenous free energy curves (i.e., the single-phase compositions of the homogeneous molar free energy curves plus the common tangent).

The values of `comparison' free energies of systems constructed from linear combinations of homogeneous equilibrium molar free energies is also bound from above--a set that is considerably smaller than non-equilibrium values of that function which is minimized during an approach to equilibrium.

__Mathematical Background__

- Fields
A field associates a physical quantity with a position, at a time, .

^{2}A field may also be a function of time: where is the physical quantity that depends on location and time.**Scalar Fields**-

**Vector Fields**-

**Tensor Fields**-

**Figure 1-3:**Two representations of a scalar field in two-dimensions are illustrated in the left and middle figures.Every sufficiently smooth scalar field has a natural vector field associated with it: the gradient field.

Consider a stationary scalar field such as the one illustrated in Figure 1-3.

The instantaneous rate of change of with respect to is therefore:

The gradient is parallel to the direction of steepest ascent.

This can be generalized even further by considering a time-dependent field , the instantaneous rate of change of with velocity is

- $&bull#bullet;$
- Continuum Limits
Nature is fundamentally discrete, how is that we can discuss things
like derivates?

- Fluxes
- Let
be an oriented patch of area,
.
If is the rate at which flows through
a unit area, it follows that
(01-8)

The proportionality factor must be a vector field:(01-9)

This defines the local flux as the continuum limit of:(01-10)

- Accumulation
- The rate at which accumulates in a volume
(with outward oriented normals)
during time interval is:
Flowing in Flowing out Rate of Production of (01-11)

(01-12)

where is the density of the rate of production of in .Expanding to first order in , subtracting, and using the continuum limit,

*The rate of accumulation of the density of an extensive quantity is***minus**the divergence of the flux of that quantity plus the rate of productionNote that Eq. 1-13 could have been derived directly from:

(01-14)

where is the oriented surface around and the