THERMODYNAMICS APPLIED TO EARTH MATERIALS James Connolly, NW/E/78.1, 2-7804 [email protected] These notes and other materials for this course are available at: www.perplex.ethz.ch/thermo_course SOME PRELIMINARIES MAPLE TECHNICAL ISSUES Installing Maple - Expand the Mac (OSX), LINUX or PC (Windows) zip archive in www.perplex.ethz.ch\thermo_course\maple. Follow the instructions in the Maple installation text file! N.B. if you use Windows, then I recommend you install the 32 bit (x86) version regardless of whether you have 64 bit machine, the “Classic” Maple (cwmaple.exe) is only available in the x86 version. I have not checked myself, but I have been told that the “Classic” Maple is not available in the LINUX/OSX versions. Display settings – if you wish to keep the format of your scripts similar to those on the course web-page and you are not using “Classic” Maple, then: under Tools->Options->Display set Input display to Maple Notation and Typsetting level to Maple Standard, these minimize complications caused by invisible formatting characters; and under Tools->Options->Interface set Default format for new worksheets to Worksheet. Maple tutorial – this course does not require much skill with Maple and I will attempt to provide you with the essential syntax. However, if you want a better understanding of its syntax and capabilities, then the tutorial present in www.perplex.ethz.ch\thermo_course\maple may be useful. Classic vs Java scripts – the Maple scripts prepared for this course were prepared in the Classic Maple version (file type *.mws). These scripts can be used in the Java version of Maple, but if you do so please save them as Java version files (file type *.mw), Classic version files that have been manipulated in the Java version become illegible if saved as Classic version files. Running Classic Maple on Windows – The Classic version can be accessed from Start-> All programs -> Maple 18 -> Classic Worksheet Maple 18, or simply create a direct shortcut to the Classic version, which is usually at C:\Program Files\Maple 18\bin.win\cwmaple.exe. A word to the wise – Save your work frequently; do not rely on the auto-backup function, especially not if you are using the Java version of Maple. PROBLEM SETS The problem sets may be solved as a group effort. If you solve a problem set as a group, then please submit one copy of the solved problems for the entire group. In the first half of the semester, the problem sets are due within two weeks of their assignment. If you do a problem with Maple, then I am happy to accept an e-mail copy of your Maple script rather than a hand-written answer. However, be sure that the script can be executed sequentially, to verify this, type “restart;” or press the Maple menu restart button and then step through the script. Just as in a handwritten problem set, maple scripts should be documented by explanatory comments, preferably inserted as text (the “T” menu button). In particular, it is essential that the units be given for any dimensional results. Some firewalls do not allow Maple scripts as attachments; therefore, it is best to enclose the Maple script in a zip file. If necessary, then I will offer a tutorial for the more complicated problem sets, probably on Thursday’s, 14:15- 16:15. The penalty for solving problems in the tutorial is that the note for a perfect solution is 5. If you attend a tutorial, then please indicate this on your problem set. THE FIRST LECTURE AND THE CORRESPONDING PROBLEM SET To illustrate the purpose and goals of this course, in the first lecture I will begin at the end by considering how phase diagrams are constructed from thermodynamic models. There is no script for this lecture, but a short review of the basic concepts is in p 12-16 of www.perplex.ethz.ch\thermo_course\chapter_0\potenza.pdf. The problem set for this lecture consists of the four problems outlined in the maple scripts at ...\thermo_course\chapter_0. 1: THERMODYNAMIC PROCESSES AND follows that the integral of a state function along a LAWS closed path, i.e., a path that returns the system to its “Thermodynamics is the science of the impossible. initial state must be zero. Thus an alternative It enables you to tell with certainty what cannot statement of the first law is happen. Thermodynamics is noncommittal about ∫ d0U = . 0.2 the things that are possible. Thermodynamics is at But to define a closed path you must first define its best when nothing can happen, a condition state. Early thermodynamicists understood that a called equilibrium. The concept of equilibrium has system could only do work by changing its volume been fruitfully extended to reversible processes. or mass, therefore the properties V, M1, …, Mk are Here everything is impossible except one very state functions that define mechanical and chemical specific process and even this process is on the state. Furthermore, temperature (T) was assumed verge of being impossible.” to be a state function although the relationship – An anonymous, slightly inaccurate, wit. between temperature and heat was not entirely clear. Chemical thermodynamics is a theory developed to predict and understand the consequences of If a system loses heat, it can be considered to have processes. In principle it can be used to predict any done positive thermal work, accordingly Eq 0.1 can process, but it is mainly used to understand the be written more compactly processes of heat and mass transfer and isostatic k +2 dUW+= d0. dilation (i.e., a change of volume (V) in an ∑ i i=1 isotropic stress field). This restriction is implicit in Early thermodynamicists were concerned primarily the remainder of these notes. In this context, energy with the construction of steam engines, which is referred to as the internal energy (U) to convert heat to mechanical work; thus they had the emphasize that it is only the energy accessible prejudiced view that mechanical work was useful through the processes of interest. For a body of and that thermal work was useless except to the matter, i.e., a system, composed of k independently extent that it could be converted to useful work. variable kinds of mass (M1, …, Mk), there are thus While this attitude is no longer appropriate, it is k+2 independent processes. Intuitively one expects sometimes useful to isolate heat as a special kind of that any of these processes will change the energy work in order to understand the perspective of the of a system on which they operate. The first law of early thermodynamicists who were responsible for thermodynamics is a formal statement of this thermodynamic theory. intuition. Specifically, the first law states that energy of a system may only change if the system A MECHANICAL ANALOGY does work on its environment. Moreover, the first Although mechanics is more complex than law states that energy (U) is conservative, i.e., k +2 chemical thermodynamics in that it involves kinetic ddUQ−+∑ d Wi = 0 0.1 energy and vectors, it is familiar via elementary i=2 physics. Therefore it is helpful to introduce a where Q is the heat gained by the system and Wi is mechanical analogy to Eq 0.1, specifically the the work done by the system in the ith process, potential energy U of a stationary ball as a function where d is used to indicate an inexact (path of its horizontal position x along a frictionless 1- dependent) . The sign convention in Eq differential dimensional surface with height H(x) in a gravita- 0.1 is by no means universal; in particular, in many tional field (Fig 1.1). The potential energy of the textbooks W represents work done on the system in ball is proportional to the height H of the surface, which case the work and heat differentials have the i.e., U ∝ H(x) and by the first law (Eq 0.1) same sign. ddUW+= 0 0.3 the ball can do work if it lowers its potential energy Properties that are not path dependent are said to be by moving. In mechanics this work is state functions and the importance of the first law ddW= fx 0.4 is that it establishes energy as a state function. It where f is the force resisting the movement dx, thus 1 = − ddU fx 0.5 U In this particular case, the work differential happens to be exact because there is only one process (path) f=-∂ U/∂θx= possible. Eqs 0.3 and 0.4 can be considered, although it is not conventional, to define force as the negative spatial gradient in potential energy dU f ≡− . 0.6 x dx Fig 1.1 Energy of a stationary ball along a 1-d The ball will have no tendency to move if this surface, the force that would induce the ball to gradient is zero, thus the gradient can be viewed as roll is the negative of the gradient in the the potential for the only process (displacement of energy along the surface. A spatial gradient in energy is usually referred to as a potential. The the ball) possible in the system in the absence of potential is a measure of the amount of work external influence. For this reason spatial gradients that can be extracted from the ball by a displacement (dx ) in the along the surface. If in energy are often referred to as potentials for the system consists of only one ball it can be in work equilibrium only if the potential of the ball is dU zero.