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3.21 Spring 2002: Lecture 01

Thermodynamics and Kinetics


$\displaystyle dU = \delta W + \delta Q$ (01-1)

For the limiting case of a reversible process,

$\displaystyle dS = \frac{\delta Q}{T}$ (01-2)


$\displaystyle S(U) = k \log \omega_U$ (01-3)

Rigorous and mostly complete foundation of equilibrium of materials.

Two fundamental results.

  1. 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.

  2. If a system is in equilibrium with reservoir that maintains a constant potential (e.g. $ P$ and $ T$), then there exists a free energy function for that system (e.g., $ G(P,T)$) that is minimized at equilibrium. Therefore, a necessary condition for equilibrium is that every small variation in $ G$ must be non-negative:1 $ (\delta G)_{P=\mbox{const.}, T=\mbox{const.}} \geq 0$.

Figure 1-1: Representation of all possible values of the molar free energy function that is minimized at equilibrium.

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

Figure: If the two curves on the left represent the lowest possible values of an appropriate thermodynamic function function that is minimized at equilibrium (such as $ \overline{G}(P,T)$ at a fixed value of $ P$, $ T$) and if each phase constrained to have a homogeneous composition $ X$, then these curves are the equilibrium values of the appropriate free energy as a function of $ X$ (i.e., $ \overline{G}(P,T,X)$).

If the constraint that the system has a homogeneous composition is removed, then if any combination of compositions $ X_i$--distributed among all possible phases in such a way that the average composition is $ \ensuremath{{X}_\circ}$--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


A field associates a physical quantity with a position, $ \ensuremath{\vec{x}}= (x,y,z)$ at a time, $ t$.2A field may also be a function of time: $ f(\ensuremath{\vec{x}}, t)$ where $ f$ 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 $ c({\ensuremath{\vec{x}}})$ such as the one illustrated in Figure 1-3.

$\displaystyle c(\vec{x} + \vec{v}t) = c(\vec{x}) + \ensuremath{\nabla}c \cdot \...
...} \ensuremath{\left.\mbox{\rule{0pt}{16pt}}\right\vert}_{t=0} \Delta t + \ldots$ (01-4)

The instantaneous rate of change of $ c$ with respect to $ t$ is therefore:

$\displaystyle \ensuremath{\frac{d {c}}{d {t}}}= \ensuremath{\nabla}c \cdot v$ (01-5)

The gradient is parallel to the direction of steepest ascent.

This can be generalized even further by considering a time-dependent field $ c({\ensuremath{\vec{x}}},t)$, the instantaneous rate of change of $ c$ with velocity $ \vec{c}(\vec{x})$ is

$\displaystyle \ensuremath{\frac{d {c}}{d {t}}}= \ensuremath{\nabla}c \cdot v + \ensuremath{\frac{\partial{c}}{\partial{t}}}$ (01-6)

Continuum Limits Nature is fundamentally discrete, how is that we can discuss things like derivates?

Figure 1-4: Infinitesimal volume with dimensions $ dx$, $ dy$, and $ dz$ located at $ \ensuremath{\vec{x}}$.

Figure: Behavior of the concentration at a point $ c(\Delta V(\vec{x}))$ as the volume shrinks towards the point $ \vec{x}$.

$\displaystyle c(\vec{x}) = \lim{\Delta V \to 0} \frac{ \int_{\Delta V} \xi( \vec{x})( \delta(\vec{x_1} + \vec{x_2} + \ldots + \vec{x_N}) dV} {\delta V}$ (01-7)

Figure: The function $ \xi(\vec{x})$ performs a weighted sampling of points near $ \ensuremath{\vec{x}}= 0$.

Let $ \vec{\Delta A}$ be an oriented patch of area, $ \vec{\Delta A} = \hat{n} \Delta A = (A_x , A_y , A_z)$. If $ \dot{M_i}$ is the rate at which $ i$ flows through a unit area, it follows that

Figure: The vector $ \vec{\Delta A}$.

$\displaystyle \dot{M_i}(\vec{\Delta A}) \propto \vert \vec{\Delta A} \vert$ (01-8)

The proportionality factor must be a vector field:

$\displaystyle \dot{M_i}(\vec{\Delta A}) = \ensuremath{\vec{J}}_i \vec{\Delta A}$ (01-9)

This defines the local flux as the continuum limit of:

$\displaystyle \frac{\dot{M_i}(\vec{\Delta A})}{\Delta A} = \vec{J_i}(\vec{x}) \cdot \hat{n}$ (01-10)

The rate at which $ i$ accumulates in a volume $ \Delta V = dx dy dz$ (with outward oriented normals) during time interval $ \Delta t$ is:

$\displaystyle \Delta M_i = (i$    Flowing in $\displaystyle ) - (i$    Flowing out $\displaystyle ) + ($Rate of Production of$\displaystyle i)$ (01-11)

\begin{displaymath}\begin{array}{ll} \delta M_{i} = & - \vec{J}(x + dx/2, 0, 0) ...
...a t \\  & + \dot{\rho}_i(\vec{x}) \Delta t \Delta V \end{array}\end{displaymath} (01-12)

where $ \dot{\rho}_i(\vec{x})$ is the density of the rate of production of $ i$ in $ \Delta V$.

Expanding to first order in $ dx, dy , dz$, subtracting, and using the continuum limit,

$\displaystyle \frac{\partial c}{\partial t} = -\nabla \cdot \vec{J} + \dot{\rho}_m$ (01-13)

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 production

Note that Eq. 1-13 could have been derived directly from:

\begin{displaymath}\begin{array}{ll} \dot{M_i} &= -\int_{\mathcal{B}(\Delta V)} ...
... \cdot \ensuremath{\vec{J}}_i + \dot{\rho_i}) dV\\  \end{array}\end{displaymath} (01-14)

where $ \mathcal{B}(\Delta V)$ is the oriented surface around $ \Delta V$ and the

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W. Craig Carter 2002-02-06