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A Survey of Basic Thermodynamics

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A Survey of Basic Thermodynamics A SURVEY OF BASIC THERMODYNAMICS W. F. Vinen School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, UK Abstract A brief summary is given of the thermodynamics that underlies the use of su- perconducting and cryogenic techniques in particle accelerators and detectors. It is intended to remind participants at the School of basic principles, and it is not an introduction to the subject. A rigorous treatment is not possible within the limited space available. 1. INTRODUCTION: MACROSTATES AND MICROSTATES Thermodynamics is concerned with macroscopic systems, with the concept of temperature, and with the way in which the behaviour of such systems is constrained by very general principles, called the Laws of Thermodynamics, which originate from a recognition that the random motion of particles in the system is governed by general statistical principles. It describes macroscopic systems by macroscopic parameters, such as pressure, p, volume, V , magnetization, M~ , applied magnetic field, H~ , and total internal energy, E, all of which have obvious meanings, and it introduces new and more subtle macroscopic parameters, such the temperature, T , and entropy, S. The constraints are expressed in terms of allowed relationships between these parameters. The so-called extensive quantities, V; M~ ; E and S, are proportional to the mass of material present; we shall generally assume that each relates to one mole. The various macroscopic parameters define a macrostate of the system. We can also think about a microscopic state, or microstate, of the system, defined, for example, by specifying the energy or position of each individual atom or molecule. It is crucial to understand that a very large number of different microstates correspond to the same macrostate. For example, the same total internal energy, E, can come about from a very large number of different ways of distributing this energy among the individual atoms or molecules. We denote this large number by W . Other things being equal, the probability of the system being found in one microstate is the same as that of finding it in another microstate. Thus the probability that the system is in a given macrostate must be proportional to the corresponding number, W , of microstates. The number of microstates, W , is often called the statistical weight. 2. INTERNAL ENERGY AND THE FIRST LAW OF THERMODYNAMICS In essence the First Law states that, if the macroscopic parameters such as pressure, volume, magnetiza- tion and applied magnetic field are specified, then so also is the internal energy, E. For a non-magnetic fluid system, the important parameters are usually p and V . Then the First Law states that E is a properly defined function of p and V : E = E(p; V ). If you take a fluid system through some cycle and return it to its original p and V , its internal energy returns to its original value. We say that E is a function of state. Obvious? No, it depends on the fact that the individual molecules are very quickly randomized among all the accessible microstates, the accessible microstates being those that are consistent with the given p and V . The previous history is irrelevant. 3. ENTROPY AND THE SECOND LAW OF THERMODYNAMICS The number of microstates, W , is a very useful parameter because it relates to the probability of finding a given macrostate. But W is usually an inconveniently large number, so it is better to work in terms of ln W . We define the entropy as S = kB ln W , where kB is a constant, the value of which will be fixed later. Note that if we have two systems, A and B, the number of microstates of the combined system is WAWB, so that the combined entropy is additive; it is similarly proportional to the mass of the system. The Second Law of Thermodynamics states in part that S is a function of state: S = S(p; V ). Again, not obvious, but again depends on the fact that a system quickly randomizes over the accessible microstates (consistent with the given p and V ), independently of the previous history. 4. THE PRINCIPLE OF INCREASE IN ENTROPY This Principle completes our statement of the Second Law. Suppose that a system is isolated from its surroundings, so that any change in the system can have no effect on the surroundings and vice versa. Then, if the system can be imagined to exist in two macroscopic states, corresponding to two different values of W , it is more likely to be found in the one with the larger W . In practice, one value of W is usually enormously larger than the other, so that the system will almost certainly be found in the one with the larger W . In other words, in the one with the larger entropy. For example, consider a gas in a box. Suppose that at some initial time all the gas were in the left hand side of the box, corresponding to W = W1. If the gas were allowed to fill the whole box, W = W2. It turns out that W2 >>>> W1. So the gas must expand to fill the whole box, in the sense that it must have only a negligible probability of being found in one side of the box only. In this sense entropy tends to a maximum value. Note that the principle that a system tends to maximize its entropy applies only to an isolated system. Otherwise one would need to worry about changes in W or S for the surroundings. Often systems are not isolated, and therefore in due course we shall look for principles that are more convenient to use than the Principle of Increase in Entropy. 5. THE KELVIN TEMPERATURE We define the Kelvin temperature by the relation: T = (@E=@S)V . Why? And does this definition accord with our everyday experience? Suppose that we have two systems, A and B, between which energy can flow, while each is maintained at constant volume (this flow of energy is actually a flow of “heat”). The two systems are not isolated from each other, and therefore we can apply the principle of increase in entropy only to the combined system. If we transfer energy δE from one system to the other we have @S @S δS = A δE − B δE: (1) @E @E A V B V (Remember that S is additive.) Therefore maximizing the entropy means that TA = TB. Therefore equilibrium requires equality of Kelvin temperature, in accord with our experience. (Note that we have in fact proved that the concept of temperature is meaningful; this is related to what is sometimes called the Zeroth Law of Thermodynamics.) If initially TA > TB, (heat) energy must flow from A to B (hot to cold), if S is to increase, which again accords with our expectations. 6. THE BOLTZMANN DISTRIBUTION, AND THE MEANING OF T = 0 Imagine a volume of gas, at constant volume, V , and constant internal energy, E0. Focus on one particu- lar molecule of the gas. If its energy is , the rest of the gas has energy E0 − . Therefore the probability that the molecule has this energy is the probability that the rest of the system has energy E0 − . We might try writing for this probability @W W (E0 − ) = W (E0) − ; (2) @E but W varies too rapidly with E to allow us to use this formula (Taylor expansion). Therefore work with ln W , instead of W , and hence write @ ln W (E0 − ) = ln W (E0) − ln W = ln W (E0) − : (3) @E k T =0 B Therefore the probability that a molecule has energy is proportional to exp (−/kBT ). Note the mean- ing of T = 0. At this temperature no molecule can be in an excited state, and therefore the whole gas must be in its state of minimum energy. 7. THE EQUATION OF STATE, AND THE RELATIONSHIP BETWEEN THE KELVIN TEM- PERATURE AND THE IDEAL GAS TEMPERATURE The relation between p, V and T is called the equation of state. Given the Boltzmann distribution we can calculate the equation of state for an ideal gas. We find pV = NAkBT , where NA is Avogadro’s number. Therefore the Kelvin temperature is the same as the ideal gas temperature if we take kB = R=NA, where −23 −1 R is the gas constant. kB is then called Boltzmann's constant (1:38 × 10 JK ). 8. WORK AND HEAT Energy can be added to a system in the form of either work or heat. For example, work is done on a gas if it is compressed in a cylinder fitted with a piston. Heat can be added to the system by thermal conduction through the walls of the containing vessel. The total increase in energy is the sum of the heat added and the work done. dE = dq + dw: (4) Note the convention of signs: energy added to a system is positive. q and w are not functions of state (you cannot associate a particular amount of heat with a particular state), and therefore dq and dw are not proper differentials (hence the bars over the ds). 9. REVERSIBLE PROCESSES If the piston compresses the gas slowly and if there is no friction, the process is said to be reversible, because a small change in conditions (force on the piston) can reverse the direction of the process. The work done can then be written dwrev = −pdV: (5) If the compression is not slow the gas swirls around and the pressure is not defined. If the gas expands into a vacuum through a valve (an extremely irreversible process), no work is done. Heat can also flow reversibly, if the temperature difference is very small.
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