Entropy And Probability

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Entropy tells it where to go

A. Introduction

The concept of entropy was originally developed from the study of possible limitations on the transformation of energy from one form to another. It is therefore particularly pertinent to concerns about energy shortages in modern industrial societies. Although it was once thought that boundless resources of energy are available for human use, some people fear that we may be running out of energy and that there is a great need for energy conservation. (This is not the same energy conservation expressed in the first law of thermodynamics.) Energy shortages are better described as shortages of available and useful energy, where the meaning of useful is at least partially determined by possible economic, political, and environmental consequences of the conversion of energy to a "useful" form.

The availability of energy of a system is related to entropy. The concept of entropy, however, is much more consequential and fundamental than its relationship to energy transformations indicates. It is particularly significant for chemistry and chemical processes, and is an important part of the studies of scientists and engineers. For example, the entropy of a system determines whether it will exist in a particular solid, liquid, or gas phase, and how difficult it is to change from one phase to another. The microscopic interpretation of the entropy concept leads to ideas of order-disorder, organization and disorganization, irreversibility, and probabilities within the kinetic-molecular model of matter. These ideas have been extended and broadened to become significant parts of information theory and communication theory and have been applied to living systems as well. They have also been applied by analogy to political and economic systems.

This chapter is primarily concerned with the development of the entropy concept and its applications to energy transformations, reversibility and irreversibility, and the overall direction of processes in physical systems. One of the major consequences of the entropy concept is the recognition of heat as a ' 'degraded'' form of energy. When energy is in the form of heat, a certain fraction of it is unavailable for use, and this fact must be taken into account in the design of energy transformation systems.

1. Energy as a Parameter of a System

The law of conservation of energy states that although energy may be changed from one form to another, the total amount of energy in an isolated system can be neither increased nor decreased, but rather must remain unchanged. As already stated in Chapter 4, if a system is not isolated, the energy conservation law is formulated in a different, but equivalent, manner. The state of a system can be characterized or described, partially at least, in terms of its total energy content. The state of the system can be further described in terms of other properties, such as its temperature, its size or volume, its mass, its internal pressure, its electrical condition, and so on. These properties, all of which can be measured or calculated in some way, are called parameters of the system. If the state of a system changes, at least some of the parameters of the system must change. In particular, it is possible that its total energy content may change.

The law of conservation of energy states that the change in energy content of the system must be equal to the amount of energy added to the system during the change minus any energy removed from the system during the change. In other words, the conservation law states that it is possible to do bookkeeping on the energy content of a system—any increases or decreases in the energy content must be accounted for in terms of energy (in any form) added to or taken away from the system. The law of conservation of energy used in this manner is called the first law of thermodynamics.

The concept of energy and its conservation, although making it possible to set up an equation, sets no limits on the quantities entering into the equation. It gives no idea about how much energy can be added or taken away or what rules, if any, should govern the transformation of energy from one form to another. Nor does the energy concept state what fraction of the energy of a system should be in the form of kinetic energy or heat or potential energy or electrical energy, and so on. In fact, it is not even possible to say how much energy a system has in an absolute sense because motion and position are relative. Therefore, kinetic energy and potential energy are relative, and it is only changes in energy that are significant, as discussed in the previous chapter.

2. Entropy as a System Parameter

There are rules that govern energy transformations. The concept of entropy, and related ideas, deals with these rules. In fact, the word entropy was coined by the German physicist Rudolf J. E. Clausius in 1865 (although he developed the concept in 1854) from Greek words meaning transformation content. Entropy, like energy, is also a parameter that can be used to describe the state of a system. Although the energy concept deals with an abstract intangible "substance" that exists only in relative amounts and is recognized only by its changes, by contrast the entropy concept deals with how much energy is subject to change or transformation from one form to another.

Historically, the entropy concept developed from the study of "heat" and temperature and the recognition that heat is a rather special form of energy that is not completely convertible to other forms. Other forms of energy, however, are completely convertible to heat. Moreover, it seems that in all transformations of energy from one form to another, some energy must be transformed into heat.

3. Statistical Nature of Entropy

Entropy (and temperature also) is ultimately explained in terms of statistical concepts because of the atomic nature of matter. (In fact, the full impact and power of the energy concept itself is not appreciated without statistical considerations and the resulting connection of energy with temperature and entropy.) The idea of entropy can be generalized, through its connection with statistics and probability, to be a useful means of describing relative amounts of organization (order versus disorder) and thus becomes useful for studying various states of matter (gas, liquid, solid, plasma, liquid crystal, etc.).

B. Heat and Temperature

1. Distinction between Heat and Temperature

To understand the relationship between the concepts of energy and entropy, it is first necessary to make a clear distinction between heat and temperature. One useful way to make this distinction is to use the caloric theory of heat. (Although the caloric theory is erroneous, the analogy it makes between heat and a fluid helps to introduce some otherwise very abstract ideas.) The relationship between heat flow into an object and its temperature is expressed by the heat capacity of the object. The heat capacity of an object or system is defined as the amount of heat it takes to raise the temperature of the object by one degree (see Chapter 4, Section Bl).

Temperature is described in terms of intensity or concentration of internal molecular energy, that is, the internal molecular energy per unit amount of the substance. For example, a thimble filled with boiling hot water can have a higher temperature than a bathtub full of lukewarm water, yet it takes much less heat energy to raise the water in the thimble to its high temperature than the bathtub water to its lower temperature. The distinction between temperature and heat is explained in terms of the kinetic-molecular theory of matter, which describes temperature as related to the average energy of microscopic modes of disorganized motion per molecule of the water, and heat in terms of changes in the total energy of microscopic modes of disorganized motion of the molecules. The water molecules in the thimble have more energy on the average than the water molecules in the bathtub, but the thimble needed far less heat than the bathtub because there are far fewer molecules in the thimble than in the bathtub.

2. Properties of Temperature

Although temperature can be "explained" in terms of concentration of molecular energy, the usefulness of the temperature concept actually depends on two other fundamental aspects of heat. These aspects are recognized in common everyday experience: thermal equilibrium and heat flow.

2.1 Temperature and Thermal Equilibrium Thermometers are used to measure the temperature of an object or a system. Almost all thermometers depend on the existence of thermal equilibrium. This means that if two objects or systems are allowed to interact with each other, they will eventually come to have the same temperature. Depending on the circumstances, this may happen very quickly or very slowly, but eventually the temperature of one or both of the objects will change in such a way that they will both have the same final temperature.

For example, the measurement of the temperature of a feverish hospital patient depends on the interaction between one object, the patient, and a second object, the thermometer. The nurse puts the thermometer into the mouth of the patient and waits for several minutes. While the interaction is going on, the state of the thermometer is changing. For this example, we assume that the thermometer is the mercury in glass type. The length of the mercury column inside the thermometer is a parameter of the thermometer, and it increases as the temperature of the thermometer increases. When the temperature of the thermometer becomes the same as that of the patient, the length of the mercury no longer changes, and therefore the thermometer is in thermal equilibrium both internally and with the patient. This means two things: (1) the thermometer will not change any further, and (2) the temperature of the thermometer is the same as the temperature of the patient. Only after equilibrium is reached may the nurse remove the thermometer from the patient's mouth and read the scale to determine the temperature of the patient.

2.2 Temperature and Equations of State Actually the thermometer does not measure the temperature directly, but rather some other parameter is measured. In the case of the mercury in glass thermometer, the parameter measured is the length of the mercury in the glass tube. In other types of thermometers, other parameters are measured. In resistance thermometers, the electrical resistance of the thermometer is the parameter that is measured. In thermoelectric thermometers, the thermoelectric voltage is the parameter measured, whereas in a common oven thermometer, the shape of a coil to which a pointer is attached is the parameter that changes. Almost any kind of a system can be used as a thermometer. In all cases, the parameter that is actually measured must be related to the temperature parameter in a definite mathematical manner. This mathematical relationship is called an equation of state of the system. (In this case, the thermometer is the system.) This simply means that if some parameters of a system change, then the other parameters must also change, and the changes can be calculated using the equation of state. The dial reading or scale reading of the thermometer represents the results of solving the equation of state for the temperature. The equation of state of the system being used as a thermometer determines the temperature scale of the thermometer.

Every system has its own equation of state. For example, helium gas at moderate pressures and not too low temperatures has the following equation of state relating three of its parameters: PV = RT, where P stands for pressure, V for volume, T for temperature (using the thermodynamic or absolute temperature scale, which will be defined below), and R is a proportionality constant. This equation of state is often called the general gas law or the perfect gas law and is a mathematical relationship among the three parameters mentioned. At all times, no matter what state the helium gas is in, the three parameters—temperature, pressure, and volume—must satisfy the equation. (It is also possible to determine the energy content, U, and the entropy, S, of the system once the equation of state is known.) The equation of state of the system can be drawn as a graph of two of the parameters if the other one is held constant. Figure 5.1 shows graphs of pressure versus volume for helium gas at several different temperatures.

The equation of state and resulting graphs for helium are relatively simple because helium is a gas at all but the lowest temperatures. The equation of state and the resulting graphs of pressure versus volume for water are much more

Volume

Figure 5.1. Equation of state for helium gas. Each curve is an "isotherm" showing how pressure and volume can change while the temperature remains constant. The curves 1, 2, 3, 4 are for successively higher temperatures.

Volume

Figure 5.1. Equation of state for helium gas. Each curve is an "isotherm" showing how pressure and volume can change while the temperature remains constant. The curves 1, 2, 3, 4 are for successively higher temperatures.

complex because water can be solid, liquid, or gas, depending on the temperature and pressure. Some graphs for water are shown in Fig. 5.2.

Both helium and water could be used as thermometers, and in fact, helium is sometimes used as a thermometer for scientific measurements at very low tem-

Figure 5.2. Equation of state for water. Each curve is an "isotherm" showing how pressure and volume can change while temperature remains constant. The curves 1, 2, 3, 4 are for successively higher temperatures. The flat parts of curves 1 and 2 cover a range of volumes for which part of the water is liquid and part gas. Water is completely liquid to the left of the flat parts, and completely a gas to the right. Curve 3 is at a temperature (374° Celsius) for which it is impossible to tell the difference between liquid and gas, whereas at all higher temperatures water can be only a gas.

Figure 5.2. Equation of state for water. Each curve is an "isotherm" showing how pressure and volume can change while temperature remains constant. The curves 1, 2, 3, 4 are for successively higher temperatures. The flat parts of curves 1 and 2 cover a range of volumes for which part of the water is liquid and part gas. Water is completely liquid to the left of the flat parts, and completely a gas to the right. Curve 3 is at a temperature (374° Celsius) for which it is impossible to tell the difference between liquid and gas, whereas at all higher temperatures water can be only a gas.

peratures; except for such special purposes, however, helium is not a very practical thermometric material. Water is useful in establishing certain points on temperature scales, but otherwise is not a practical thermometric substance. When the barometric pressure is 760 mm (30 inches), water freezes at 0° Celsius (32° Fahrenheit), and boils at 100° Celsius (212° Fahrenheit). Temperature scales are discussed further in the following optional section and in Section F below.

Optional Section 5.1 Temperature Scales

The same concept of thermal equilibrium is used to define temperature scales in a very precise and reproducible manner; this is called "fixed point" definition of temperature scales. For example, the Celsius and Fahrenheit scales were at one time defined using the freezing points and boiling points of water. On the Celsius (sometimes called Centigrade) scale, the freezing point of water is by definition 0° and the boiling point of water by definition 100°. The water must be pure, and the measurement must be made at standard atmospheric pressure which is 760 mm or equivalently 14.5 lb/sq in. Thus for the common mercury in glass thermometer immersed in a mixture of ice and water, the equilibrium position of the "top" of the mercury column was marked on the glass tube (see Fig. 5.0.1) as 0°. Then with the thermometer immersed in boiling water, the mercury expanded along the length of the glass tube to a new equilibrium position, which was marked on the glass. Then the part of the

Ice-water mixture

Boiling water

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