3.1 Energy minimum principle

Transcription

1 Ashley Robison My Preferences Site Tools FAQ Sign Out If you like us, please share us on social media. The latest UCD Hyperlibrary newsletter is now complete, check it out. ChemWiki BioWiki GeoWiki StatWiki PhysWiki MathWiki SolarWiki Periodic Table of the Elements Reference Tables Physical Constants Units & Conversions Lab Techniques ChemWiki: The Dynamic Chemistry E-textbook > Physical Chemistry > Thermodynamics > Advanced Thermodynamics > 3. Basic properties of U, S, and their differentials 3. Basic properties of U, S, and their differentials 3.1 Energy minimum principle can be written as, is a vect of all independent internal extensive variable (e.g. all but one U k, and all other X i ). Because S is monotonic in U and continuous, we can invert to. This relation is fully equivalent to the fundamental relation. Because of the shape of, as shown in the figure, maximizing the entropy at constant U is equivalent to minimizing the energy at constant S. This is the familiar version from mechanics, system properties are usually fmulated in terms of energies, instead of entropies. Postulate 2': Minimizing Energy The internal energy of a composite system at constant is minimized at equilibrium. 1/8

2 3.2 Intensive parameters: Temperature Wking f now with f a simple system, we can write I think was increct in the source and I fixed it... with appropriate constraints n each derivative by postulate three. By construction, and the are intensive variables. F example, and We consider in detail the properties of the energy derivative, and then briefly by analogy other intensive variables. Let all the (such as,,, etc.) equal to zero: no mechanical macroscopic variables are being altered except f energy. It then follows that because. Therefe f small (quasistatic) charges in heat, the change in system entropy is linearly proptional to the heat increment. Thus as we add energy to the uncontrollable degrees of freedom of our system, entropy increases, in accd with the notion that entropy is disder. Furtherme, we can rewrite this as When is larger, the entropy increases less f a given heat input. What is this quantity? Consider a closed composite system of two subsystems and separated by a diathermal wall. A diathermal wall allows only heat flow, so again. At equilibrium, accding to P2 But f a closed system by P1, from which follows that At equilibrium, f any variation of, which can only be true if Thus, is the quantity that is equalized between two subsystems when heat is allowed to flow between them. This is the most straightfward definition of temperature: the thing that becomes equal when heat stops flowing from one place to another. We can thus identify the intensive variable as the temperature of the system. Temperature is always guaranteed to be positive by P3 because entropy is a monotonically increasing function of energy. 2/8

3 Finally, if, we can rewrite the third postulate as me commonly known as the third law of thermodynamics. As all the energy is removed from a system by lowering its temperature, the system becomes completely dered. It is wth noting that there are systems (glasses), reaching this limit takes an indinate amount of time. A very general principle of quantum mechanics guarantees that the third law holds even in those cases, if we can actually get the system to equilibrium: a codinate spin Hamiltonian always has a single groundstate of symmetry. This is the state any system reaches as. In practice, this state may just not be reachable even approximately in glasses, and heuristic replacements of the third law have been developed f this case, which is really a non-equilibrium case. To summarize always by postulate P2 by P2 f a quasistatic process when no wk is done always by postulate P3 f two systems in thermal equilibrium always by P3, difficult to reach even approximately in some cases Thus and have all the intuitive characteristics of temperature and disder, and we can take them as representing temperature and disder. The latter can be justified even me deeply by making use of statistical mechanics in later chapters, the second postulate follows from microscopic properties of the system. A note on units: must have units of energy. It would be convenient to let have units of energy (as an energy per unit size of the system ) and to let be unitless, but f histical reasons, has arbitrary units of Kelvin and S has units of Joules/Kelvin to compensate. 3.3 Other extensive-intensive Variable Pairs The me complex a composite system becomes, the me extensive variables it requires beyond example:, leading to additional intensive variables. F Pressure (volume) leads to an energy change The intensive derivative is called the pressure of the system. has units of Joules, so must have units of Joules/m 3 N/m 2. Thus certainly has the units we nmally associate with pressure, fce per unit area. Usually because squeezing a system increases its energy. Thus is generally a positive quantity, again in accd with our intuition. Note however that there is no postulate that says must be positive. In fact, we can bring systems to negative pressure by pulling on the system, putting tension on it. Is is in fact pressure? It is easy to see that it is, by applying the minimum energy principle to a diathermal flexible wall, in analogy to what was done f temperature above: by Postulate 1 by Postulate 2' In the third line, we assume a closed system and reversible process, so and. When the energy has reached equilibrium, the equation must hold f any small perturbation of the entropy volume of subsystem 1, which can only be satisfied if (again), and. 3/8

4 Thus, P is the quantity which is the same in two subsystems when they are connected by a flexible wall. This is the most straightfward definition of pressure: the thing that is equalized between two systems when the volume can change to whatever it wants. is a pressure, not just in units, but agrees with our intuitive notion of what a pressure should be. Surface Area (area in surface system) has units of (N/m) and is therefe the surface tension. Magnetization (magnetization) is the externally applied magnetic field. Mole Number (mole number) is the chemical potential equalized when particles are allowed to flow. Length (length) is the linear tension fce. In general Many me conjugate pairs of extensive and intensive variables are possible, but this gives the general picture. F an arbitrary variation in have we is the vect of all intensive variables except temperature. Often, we will use as an example, when dealing with a simple 3-dimensional 1-component system. 3.4 First der homogeneity Consider f a closed system. Because is extensive,. This agrees with the intuitive notion that 2 identical disdered systems amount to twice as much disder as a single one. Similarly,. Differentiating both sides with respect to yields When, this yields 4/8

5 Thus the energy has a surprisingly simple fm: it is simply a bilinear function of the intensive and extensive parameters; it is known as the Euler fm. The fmula f energy looks like the fmula f with the removed. F example, f a simple onecomponent system. Solving f yields an analogous fmula in the entropy representation, f example The entropy is also a simple bilinear function of its intensive and extensive parameters. 3.5 Gibbs-Duhem relation The differential of combined with first der homogeneity requires that not all intensive parameters be independent. F a completely arbitrary variation of, But we know from earlier that Using this Gibbs-Duhem relation, one intensive parameter can be expressed in terms of the others. F example, consider a simple multicomponent system: One chemical potential change can be expressed in terms of pressure, temperature, and the other chemical potentials. In general, an -component simple 3-D system has only degrees of freedom. This will be useful f multi-phase systems. F example, let two phases of the same substance be at equilibrium, and particle flow is allowed from one phase to another. Then ( particles would flow to the phase of lower chemical potential accding to 3.), and to remain at equilibrium when the chemical potential changes,. Combining the Gibbs- Duhem relations f each phase, and Thus letting traces out the, \(P\) conditions the two phases are at equilibrium. This is known as the Clausius equation. 3.6 Equations of State and the Fundamental Relation Often we do not know the fundamental equation instead we know equation involving intensive variables, known as equations 5/8

6 of state. F example, Similarly, the derivative with respect to any other yields the cresponding equation of state. These are called equations of state in nmal fm, and express one intensive variable in terms of all the extensive variables. There are as many equations of state as there are extensive variables f the system (e.g. f a simple -component system). Note that an equation of state does not contain the same amount of infmation as the iginal fundamental relation; it can be integrated up to a constant that depends on all extensive variables except the one involved in the derivative, but that part of ( ), if we derive equations of state from cannot simply be left out. If all the equations of state in nmal fm are known, we can reconstruct the fundamental relation by using the Euler fm from 4 this is also solvable f because of P3. If they are not known in nmal fm, we may also be able to obtain the fundamental relation by integrating a differential fm, such as If needed, we can compute one intensive variable from the Gibbs-Duhem relation, so we need one less equation of state (only f a simple - component system) to evaluate the fundamental relation. Finally, equations of state may also be substituted into one another, yielding equations that depend on me than one intensive variable. These are also referred to as equations of state, but they are not in nmal fm. Let us consider two examples of how to determine a fundamental relation. We start with the fundamental relation f a rubber band, we can write down reasonable guesses f both equations of state needed. We need equations of state so and can be eliminated to yield : a) ; is the relaxed length of the rubber band, and we are treating it like a linear spring once stretched. An unusual feature is that increases with. At higher T polymer chains wrinkle into me random coils, causing shrinkage, and increasing the tension f the same length. b), as long as depends only linearly on only. The reason is that so can be any single-valued function of as long as it is independent of ; f simplicity we pick, as f an ideal gas. We can now insert the two equations of state into the differential fm, and integrate it The constant can be determined by invoking the third law. However, note that this can lead to singularities if the equations of state themselves are not crect at low temperature, as is the case in this example. Meover, note that most be intensive, and must be extensive so that \(S\) is extensive. From the fundamental relation we can calculate any desired properties of the rubber band. Alternatively, we could try to obtain the fundamental relation in terms of, but then we would need and \(F(S,L)\) instead of and, which were not available. Similarly, to plug into, we would need and ; we have the fmer, but not the latter: the equation of state f is in terms of another intensive variables, and not in the basic fm required f the Euler fm. Note that plugging into to get a will not help either because this does not yield an equation of state in nmal fm as it would have been obtained by taking the derivative of. As another example, consider the fundamental relation f an ideal monatomic gas. In this case, we will derive one of the equations of state from the others, get all three equations of state in nmal fm, befe inserting all three to obtain the fundamental relation. The gas has 1 component, so we need equations of state to get started: 6/8

7 Here the two well-known equations of state f an ideal gas are written in terms of intensive variables and. Again the first equation depends on two intensive variables and is not in standard fm. We can bring both equations into standard fm as follows: We now need as the third equation of state. Proceeding with the Gibbs-Duhem relation, We must eliminate since we fmulated and as a function of, not. Using the bilinear fm of, Next we eliminate and by using equations 1 and 2 : We then divide by u on both sides, rearrange, and integrate: This is the third equation of state, f the chemical potential. We now have all intensive parameters as nmal fm equations of state, to construct the fundamental relations. (Of course, the homogeneous first der property means that to get and, we just multiply by.) Doing, f example, Note that this equation of state violates Postulate 3: so ; but as, it approaches. Thus, either,, both must be high-temperature approximations that fail as. At low, excluded volume effects, particle interaction, and quantum effects come into play. The ideal gas equation would have to be replaced by a me accurate equation, such as the van der Waals equation to satisfy the third law closer to. In that sense, thermodynamics can point out to us when approximate equations of state break down. 3.7 Stability and Second Derivatives The first derivatives (intensive parameters) are very useful because they crespond to quantities that are equalized among equilibrated subsystems. However, the first der relationship, although necessary by Postulate 2 at equilibrium, is not sufficient. The extremum in must be a 7/8

8 maximum: accding to Postulate 1: Extrema with are also possible (minima, saddles, degenerate points). However, thermodynamics cannot make statements about such points without some further assumptions that go beyond the postulates. This suggests that the study of second derivative will be fruitful, to ensure that one is wking near a stable equilibrium point. Three of these second derivatives encountered later are F a simple system, only three second derivatives are linearly independent if we exclude ones based on energy have only three second derivatives,. The reason is that the terms in the is not a perfect differential. Rather than picking those three, we will usually wk with the first independent set, cresponding to quantities with me obvious physical interpretations to chemists wking at constant pressure and temperature. We consider the cresponding fundamental relations in the next chapter. Contributs Prof. Martin Gruebele (University of Illinois, Urbana Champaign) Copyright 2015 Chemwiki Powered by MindTouch Unless otherwise noted, content in the UC Davis ChemWiki is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License. Permissions beyond the scope of this license may be available at copyright@ucdavis.edu. Questions and concerns can be directed toward Prof. Delmar Larsen (dlarsen@ucdavis.edu), Founder and Direct. Terms of Use 8/8