"About 14 billion years ago, the universe flared forth in the big bang, unfurling space, time, light, and matter in a singular, immense explosion. Less than one minute after this "Great Radiance," protons and neutrons emerged and combined to form nuclei of the simplest elements, mostly hydrogen and helium. About 300,000 years later, electrons combined with these nuclei for the first time, transforming matter into hydrogen and helium atoms." (Sampson)
0 - Immense explosion
1 min. - Protons and neutrons and combined to form nuclei of simplest elements
300,000 years - Electrons combined with nuclei to form atoms
Four Laws That Drive the Universe by Peter Atkins
1. THE ZEROTH LAW
The concept of temperature
"The zeroth law is an afterthought. Although it had long been known that such a law was essential to the logical structure of thermodynamics, it was not dignified with a name and number until early in the twentieth century. By then, the first and second laws had become so firmly established that there was no hope of going back and renumbering them. As will become apparent, each law provides an experimental foundation for the introduction of a thermodynamic property. The zeroth law establishes the meaning of what is perhaps the most familiar but is in fact the most enigmatic of these properties: temperature.
…slightly dusty definitions…
Thermo dynamics, like much of the rest of science, takes terms with an everyday meaning and sharpens them – some would say, hijacks them – so that they take on an exact an unambiguous meaning. We shall see that happening throughout this introduction to thermodynamics. It starts as soon as we enter its doors. The part of the universe that is at the centre of attention in thermodynamics is called the system. A system may be a block of iron, a beaker of water, an engine, a human body. It may even be a circumscribed part of each of those entities. The rest of the universe is called the surroundings. The surroundings are where we stand to make observations on the system and infer its properties. Quite often, the actual surroundings consist of a water bath maintained at constant temperature, but that is a more controllable approximation to the true surroundings, the rest of the world. The system and its surroundings jointly make up the universe. Whereas for us the universe is everything, for a less profligate thermodynamicist it might consist of a beaker of water (they system) immersed in a water bath (the surroundings).
system + surroundings = universe
A system is defined by its boundary. If matter can be added to or removed from the system, then it is said to be open. A bucket, or more refinedly an open flask, is an example, because we can just shovel in material. A system with a boundary that is impervious to matter is called closed. A sealed bottle is a closed system. A system with a boundary that is impervious to everything in the sense that the system remains unchanged regardless of anything that happens in the surroundings is called isolated. A stoppered vacuum flask of hot coffee is a good approximation to an isolated system.
Open system
Closed system
Isolated system
The properties of a system depend on the prevailing conditions. For instance, the pressure of a gas depends on the volume it occupies, and we can observe the effect of changing that volume if the system has flexible walls. “Flexible walls” is best thought of as meaning that the boundary of the system is rigid everywhere except for a patch – a piston – that can move in and out. Think of a bicycle pump with your finger sealing the orifice.
Properties are divided into two classes. An extensive property depends on the quantity of matter in the system – its extent. The mass of a system is an extensive property; so is its volume. Thus, 2 kg of iron occupies twice the volume of 1 kg of iron. An intensive property is independent of the amount of matter present. The temperature (whatever that is) and the density are examples. The temperature of water drawn from a thoroughly stirred hot tank is the same regardless of the size of the sample. The density of iron is 8.9 g cm^-3 regardless of whether we have a 1 kg block or a 2 kg block. We shall meet many examples of both kinds of property as we unfold thermodynamics and it is helpful to keep the distinction in mind.
Depends on the amount of matter – extensive property
Mass and volume
Independent of the amount of matter – intensive property
Temperature and density
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…the enigma of temperature and the zeroth law…
So much for these slightly dusty definitions. Now we shall use a piston – a movable patch in the boundary of a system – to introduce one important concept that will then be the basis for introducing the enigma of temperature and the zeroth law itself.
Suppose we have two closed systems, each with a piston on one side and pinned into place to make a rigid container. The two pistons are connected with a rigid rod so that as one moves out the other moves in. We release the pins on the piston. If the piston on the left drives the piston on the right into the system, we can infer that the pressure on the left was higher than that on the right, even though we have not made a direct measure of the two pressures. If the piston on the right won the battle, then we should infer that the pressure on the right was higher than that on the left. If nothing had happened when we released the pins, we should infer that the pressures of the two systems were the same, whatever they might be. The technical expression for the condition arising from the equality of pressures is mechanical equilibrium. Thermodynamicists get very excited, or at least get very interested, when nothing happens, and this condition of equilibrium will grow in importance as we go through the laws. We need one more aspect of mechanical equilibrium: it will seem trivial at this point, but establishes the analogy that will enable us to introduce that concept of temperature. Suppose the two systems, which we shall call A and B, are in mechanical equilibrium when they are brought together and the pins are released. That is, they have the same pressure. Now suppose we break that link and establish a link between system A and a third system, C, equipped with a piston. Suppose we observe no change: we infer that the systems A and C are in mechanical equilibrium and we can go on to say that they have the same pressure. Now suppose be break that link and put system C in mechanical contact with system B. Even without doing the experiment, we know what will happen: nothing. Because systems A and B have the same pressure, and A and C have the same pressure, we can be confident that systems C and B have the same pressure, and that pressure is a universal indicator of mechanical equilibrium.
Now we move from mechanics to thermodynamics and the world of the zeroth law. Suppose that system A has rigid walls made of metal and system B likewise. When we put the two systems in contact, they might undergo some kind of physical change. For instance, their pressures might change or we could see a change in colour through a peephole. In everyday language we would say that ‘heat has flowed from one system to the other’ and their properties have changes accordingly. Don’t imagine, though, that we know what heat is yet: that mystery is an aspect of the first law, and we aren’t even at the zeroth law yet.
It may be the case that no change occurs when the two systems are in contact even though they are made of metal. In that case we say that the two systems are in thermal equilibrium. Now consider three systems, just as we did when talking about mechanical equilibrium. It is found that if A is put in contact with B and found to be in thermal equilibrium, and B is put in contact with C and found to be in thermal equilibrium, the when C is put in contact with A, it is always found that the two are in thermal equilibrium. This rather trite observation is the essential content of the zeroth law of thermodynamics:
if A is in thermal equilibrium with B, and B is in thermal equilibrium with C, then C will be in thermal equilibrium with A.
The zeroth law implies that just as the pressure is a physical property that enables us to anticipate when systems will be in mechanical equilibrium when brought together regardless of their composition and size, then there exists a property that enables us to anticipate when two systems will be in thermal equilibrium regardless of their composition and size: we call this universal property the temperature. We can now summarize the statement about the mutual thermal equilibrium of the three systems simply by saying that they all have the same temperature.
We are not yet claiming that we know what temperature is, all we are doing is recognizing that the zeroth law implies the existence of a criterion of thermal equilibrium: if the temperatures of two systems are the same, then they will be in thermal equilibrium when put in contact through conducting walls and an observer of the two systems will have the excitement of noting that nothing changes.
We can now introduce two more contributions to the vocabulary of thermodynamics. Rigid walls that permit changes of state when closed systems are brought into contact – that is, in the language of Chapter 2, permit the conduction of heat – are called diathermic (from the Greek words for ‘through’ and ‘warm’). Typically, diathermic walls are made of metal, but any conducting material would do. Saucepans are diathermic vessels. If no change occurs, then either the temperatures are the same or – if we know that they are different – then the walls are classified as adiabatic (‘impassable’). We can anticipate that walls are adiabatic if they are thermally insulated, such as in a vacuum flask or if the system is embedded in foamed polystyrene.
The zeroth law is the basis of the existence of a thermometer, a device for measuring temperature. A thermometer is just a special case of the system B that we talked about earlier. It is a system with a property that might change when put in contact with a system with diathermic walls. A typical thermometer makes use of the thermal properties of material. Thus, if we have a system B (‘the thermometer ‘) and put it in thermal contact with A and find that the thermometer does not change, and then we put the thermometer in contact with C and find that it still doesn’t change, then we can report that A and C are at the same temperature.
There are several scales of temperature, and how they are established is fundamentally the domain of the second law (see Chapter 3). However, it would be too cumbersome to avoid referring to these scales until then, though formally that could be done, and everyone is aware of the Celsius (centigrade) and Fahrenheit scales. The Swedish astronomer Anders Celsius (1701-1744) after whom the former is named devised a scale on which water froze at 100^o and boiled at 0^o, the opposite of the current version of his scale (0^oC and 100^oC, respectively). The German instrument maker Daniel Fahrenheit (1686-1736) was the first to use mercury in a thermometer: he set 0^o at the lowest temperature he could reach with a mixture of salt, ice, and water, and for 100^o he chose his body temperature, a readily transportable but unreliable standard. On this scale water freezes at 32^oF and boils at 212^oF.
The temporary advantage of Fahrenheit’s scale was that with the primitive technology of the time, negative values were rarely needed. As we shall see, however, there is an absolute zero of temperature, a zero that cannot be passed and where negative temperatures have no meaning except in a certain formal sense, not one that depends on the technology of the time (se Chapter 5). It is therefore natural to measure temperatures by setting 0 at this lowest attainable zero and to refer to such absolute temperatures as the thermodynamic temperature. Thermodynamic temperatures are denoted T, and whenever that symbol is used in this book, it means the absolute temperature with T = 0 corresponding to the lowest possible temperature. The most common scale of thermodynamic temperatures is the Kelvin scale, which uses degrees (‘kelvins’, K) of the same size as the Celsius scale. On this scale, water freezes at 273 K (that is, at 273 Celsius-sized degrees above absolute zero; the degree sign is not used on the Kelvin scale) and boils at 373 K. Put another way, the absolute zero of temperature lies at -273^oC. Very occasionally you will come across the Rankine scale, in which absolute temperatures are expressed using degrees of the same size as Fahrenheit’s.
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In each of the first three chapters I shall introduce a property from the point of view of an external observer. Then I shall enrich our understanding by showing how that property is illuminated by thinking about what is going on inside the system. Speaking about the ‘inside’ of a system, its structure in terms of atoms and molecules, is alien to classical thermodynamics, but it adds deep insight, and science is all about insight.
Classical thermodynamics is the part of thermodynamics that emerged during the nineteenth century before everyone was fully convinced about the reality of atoms, and concerns relationships between bulk properties. You can do classical thermodynamics even if you don’t believe in atoms. Towards the end of the nineteenth century, when most scientists accepted that atoms were real and not just an accounting device, there emerged the version of thermodynamics called statistical thermodynamics, which sought to account for the bulk properties of matter in terms of its constituent atoms. The ‘statistical’ part of the name comes from the fact that in the discussion of bulk properties we don’t need to think about the behavior of individual atoms but we do need to think about the average behaviour of myriad atoms. For instance, the pressure exerted by a gas arises from the impact of its molecules on the walls of the container; but to understand and calculate that pressure, we don’t need to calculate the contribution of every single molecule: we can just look at the average of the storm of molecules on the walls. In short, whereas dynamics deals with the behaviour of individual bodies, thermodynamics deals with the average behaviour of vast numbers of them.
The central concept of statistical thermodynamics as far as we are concerned in this chapter is an expression derived by Ludwig Boltzmann (1844-1906) towards the end of the nineteenth century. That was not long before he committed suicide, partly because he found intolerable the opposition to his ideas from colleagues who were not convinced about the reality of atoms. Just as the zeroth law introduces the concept of temperature from the viewpoint of bulk properties, so the expression that Boltzmann derived introduces it from the viewpoint of atoms, and illuminates its meaning.
To understand the nature of Boltzmann’s expression, we need to know that an atom can exist with only certain energies.
Less than one minute after this "Great Radiance," protons and neutrons emerged and combined to form nuclei of the simplest elements, mostly hydrogen and helium. About 300,000 years later, electrons combined with these nuclei for the first time, transforming matter into hydrogen and helium atoms.
This is the domain of quantum mechanics, but we do not need any of that subject’s details, only that single conclusion. At a given temperature – in the bulk sense – a collection of atoms consists of some in their lowest energy state (their ‘ground state’), some in the next higher energy state, and so on, with populations that diminish in progressively higher energy states. When the populations of the states have settled down into their ‘equilibrium’ populations, and although atoms continue to jump between energy levels there is no net change in the populations, it turns out that these populations can be calculated from a knowledge of the energies of the states and a single parameter, B (beta).
Populations of the various energy states can be calculated from a knowledge of the energies of the states and beta.
Another way of thinking about the problem is to think of a series of shelves fixed at different heights on a wall, the shelves representing the allowed energy states and their heights the allowed energies. The nature of these energies is immaterial: they may correspond, for instance, to the translational, rotational, or vibrational motion of molecules. Then we think of tossing balls (representing the molecules) at the shelves and noting where they land. It turns out that the most probable distribution of populations (the numbers of balls that land on each shelf) for a large number of throws, subject to the requirement that the total energy has a particular value, can be expressed in terms of that single parameter B.
The precise form of the distribution of the molecules over their allowed states, or the balls over the shelves, is called the Boltzmann distribution. This distribution is so important that it is important to see its form. To simplify matters, we shall express it in terms of the ratio of the population of a state of energy E to the population of the lowest state, energy 0:
Population of state of energy E
---------------------------------------- = e^-BE
Population of state of energy 0
We see that for states of progressively higher energy, the populations decrease exponentially: there are fewer balls on the high shelves than on the lower shelves. We also see that as the parameter B increases, then the relative population of a state of given energy decreases and the balls sink down on to the lower shelves. They retain their exponential distribution, with progressively fewer balls in the upper levels, but the populations die away more quickly with increasing energy.
When the Boltzmann distribution is used to calculate the properties of a collection of molecules, such as the pressure of a gaseous sample, it turns out that it can be identified with the reciprocal of the (absolute) temperature. Specifically, B = 1/kT, where k is a fundamental constant called Boltzmann’s constant. To bring B into line with the Kelvin temperature scale, k has the value 1.38 X 10^-23 joules per kelvin. 1 The point to remember is that, because B is proportional to 1/T, as the temperature goes up, B goes down, and vice versa.
1. Energy is reported in joules (J): 1J = 1 kg m^2 s^-2. We could think of 1 J as the energy of a 2kg ball travelling at 1 m s^-1. Each pulse of the human heart expends an energy of about 1 J.
There are several points worth making here. First, the huge importance of the Boltzmann distribution is that it reveals the molecular significance of temperature: temperature is the parameter that tells us the most probable distribution of populations of molecules over the available states of a system at equilibrium. When the temperature is high (B low), many states have significant populations; when the temperature is low (B high), only the states close to the lowest state have significant populations. (Figure 4). Regardless of the actual values of the populations, they invariably follow an exponential distribution of the kind given by the Boltzmann expression. In terms of our balls-on-shelves analogy, low temperatures (high B) corresponds to our throwing the balls weakly at the shelves so that only the lowest are occupied. High temperatures (low B) corresponds to our throwing the balls vigorously at the shelves, so that even high shelves are populated significantly. Temperature, then, is just a parameter that summarizes the relative populations of energy levels in a system at equilibrium.
Caption for Figure 4.The Boltzmann distribution is an exponentially decaying function of the energy. As the temperature is increased, the populations migrate from lower energy levels to higher energy levels. At absolute zero, only the lowest state is occupied; at infinite temperature, all states are equally populated.
The second point is that B is a more natural parameter for expressing temperature than T itself. Thus, whereas later we shall see that absolute zero of temperature (T = 0) is unattainable in a finite number of steps, which may be puzzling, it is far less surprising that an infinite value of B (the value of B when T = o) is unattainable in a finite number of steps. However, although B is the more natural way of expressing temperatures, it is ill-suited to everyday use. Thus water freezes at 0^oC (273 K), corresponding to B = 2.65 X 10^20 J^-1, and boils at 100^oC (373 K), corresponding to B = 1.94 X 10^20 J^-1. These are not values that spring readily off the tongue. Nor are the values of B that typify a cool day (10^oC, corresponding to 2.56 X 10^20 j^-1) and a warmer one (20^oC, corresponding to 2.47 X 10^20 J^-1).
The third point is that the existence and value of the fundamental constant k is simply a consequence of our insisting on using a conventional scale of temperature rather than the truly fundamental scale based on B. The Fahrenheit, Celsius, and Kelvin scales are misguided: the reciprocal of temperature, essentially B, is more meaningful, more natural, as a measure of temperature. There is not hope, though, that it will ever be accepted, for history and the potency of simple numbers, like 0 and 100, and even 32 and 212, are too deeply embedded in our culture, and just too convenient for everyday use.
Although Boltzmann’s constant k is commonly listed as a fundamental constant, it is actually only a recovery from a historical mistake. If Ludwig Boltzmann had done his work before Fahrenheit and Celsius had done theirs, then it would have been seen that B was the natural measure of temperature, and we might have become used to expressing temperatures in the units of inverse joules with warmer systems at low values of B and cooler systems at high values. However, conventions had become established, with warmer systems at higher temperatures than cooler systems, and k was introduced, through kB = 1/T, to align the natural scale of temperature based on B to the conventional and deeply ingrained one based on T. thus, Boltzmann’s constant is nothing but a conversion factor between a well-established conventional scale and the one that, with hindsight, society might have adopted. Had it adopted B as its measure of temperature, Boltzmann’s constant would not have been necessary.
We shall end this section on a more positive note. We have established that the temperature, and specifically B, is a parameter that expresses the equilibrium distribution of the molecules of a system over their available energy states. One of the easiest systems to imagine in this connection is a perfect (or ‘ideal’) gas, in which we imagine the molecules as forming a chaotic swarm, some moving fast, others slow, travelling in straight lines until one molecule collides with another, rebounding in a different direction and with a different speed, and striking the walls in a storm if impacts and thereby giving rise to what we interpret as pressure. A gas is a chaotic assembly of molecules (indeed, the words ‘gas’ and ‘chaos’ stem from the same root), chaotic in spatial distribution and chaotic in the distribution of molecular speeds. Each speed corresponds to a certain kinetic energy, and so the Boltzmann distribution can be used to express, through the distribution of molecules over their possible translational energy states, their distribution of speeds, and to relate that distribution of speeds to the temperature. The resulting expression is called the Maxwell-Boltzmann distribution of speeds, for James Clerk Maxwell (1831-1879) first derived it in a slightly different way. When the calculation is carried through, it turns out that the average speed of the molecules increases as the square root of the absolute temperature. The average speed of molecules in the air on a warm day (25^oC, 298 K) is greater by 4 per cent than their average speed on a cold day (0^oc, 273 K). Thus, we can think of temperature as an indication of the average speeds of molecules in a gas, with high temperatures corresponding to high average speeds and low temperatures to lower average speeds (Figure 5).
Caption for Figure 5. The Maxwell-Boltzmann distribution of molecular speeds for molecules of various mass and at different temperatures. Note that light molecules have higher average speeds than heavy molecules. The distribution has consequences for the composition of planetary atmospheres, as light molecules (such as hydrogen and helium) may be able to escape into space.
A word or two of summary might be appropriate at this point. From the outside, from the viewpoint of an observer stationed, as always, in the surroundings, temperature is a property that reveals whether, when closed systems are in contact through diathermic boundaries, they will be in thermal equilibrium – their temperatures are the same – or whether there will be a consequent change of state – their temperatures are different – that will continue until the temperatures have equalized. From the inside, from the viewpoint of a microscopically eagle-eyed observer within the system, one able to discern the distribution of molecules over the available energy levels, the temperature is the single parameter that expresses those populations. As the temperature is increased, that observer will see the population extending up to higher energy states, and as it is lowered, the populations relax back to the states of lower energy. At any temperature, the relative populations of a state varies exponentially with the energy of the state. That states of higher energy are progressively populated as the temperature is raised means that more and more molecules are moving (including rotating and vibrating) more vigorously, or the atoms trapped at their locations in a solid are vibrating more vigorously about their average positions. Turmoil and temperature go hand in hand.
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