10-1. Explain how the second law of thermodynamics limits conversion of heat

Chapter 10 Thermodynamics With 1000 coins, the probability of obtaining all heads is so small as to be negligible. We could shake the tray for many years without seeing the ordered arrangement again. In summary, the following is to be noted from this illustration: The number of possible coin arrangements is large, and only one of them is the ordered arrangement; therefore, although any one of the coin arrangements—including the ordered one—is equally likely, the probability of returning to an ordered arrangement is small. As the number of coins in the ensemble increases, the probability of returning to an ordered arrangement decreases. In other words, if we disturb an ordered arrangement, it is likely to become disordered. This type of behavior is characteristic of all events that involve a collective behavior of many components.

The Second Law of Thermodynamics is a statement about the type of prob abilistic behavior illustrated by our coin experiment. One statement of the second law is: The direction of spontaneous change in a system is from an
arrangement of lesser probability to an arrangement of greater probability; that is, from order to disorder. This statement may seem to be so obvious as to be trivial, but, once the universal applicability of the second law is recognized, its implications are seen to be enormous. We can deduce from the second law the limitations on information transmission, the meaning of time sequence, and even the fate of the universe. These subjects, however, are beyond the scope of our discussion.

One important implication of the second law is the limitation on the con version of heat and internal energy to work. This restriction can be understood by examining the diﬀerence between heat and other forms of energy.

10.3Diﬀerence between Heat and Other Forms of
Energy

We deﬁned heat as energy being transferred from a hotter to a colder body. Yet when we examined the details of this energy transfer, we saw that it could be attributed to transfer of a speciﬁc type of energy such as kinetic, vibrational, electromagnetic, or any combination of these (see Chapter 9). For this reason, it may not seem obvious why the concept of heat is necessary. It is, in fact, possible to develop a theory of thermodynamics without using the concept of heat explicitly, but we would then have to deal with each type of energy transfer separately, and this would be diﬃcult and cumbersome. In many cases, energy is being transferred to or from a body by diﬀerent methods, and keeping track of each of these is often not possible and usually not necessary. No matter how energy enters the body, its eﬀect is the same. It raises the internal energy of the body. The concept of heat energy is, therefore, very useful.

The main feature that distinguishes heat from other forms of energy is the random nature of its manifestations. For example, when heat ﬂows via

Section 10.3 Diﬀerence between Heat and Other Forms of EnergyFIGURE 10.2 The motion of a piston.

conduction from one part of the material to another part, the ﬂow occurs through the sequential increase in the internal energy along the material. This internal energy is in the form of random chaotic motion of atoms. Similarly, when heat is transferred by radiation, the propagating waves travel in random directions. The radiation is emitted over a wide wavelength (color) range, and the phases of the wave along the wave front are random. By comparison, other forms of energy are more ordered. Chemical energy, for example, is present by virtue of speciﬁc arrangements of atoms in a molecule. Potential energy is due to the well-deﬁned position, or conﬁguration, of an object.

While one form of energy can be converted to another, heat energy, because of its random nature, cannot be completely converted to other forms of energy. We will use the behavior of a gas to illustrate our discussion. First, let us examine how heat is converted to work in a heat engine (for example, the steam engine). Consider a gas in a cylinder with a piston (see Fig. 10.2).

Heat ﬂows into the gas; this increases the kinetic energy of the gas molecules and, therefore, raises the internal energy of the gas. The molecules moving in the direction of the piston collide with the piston and exert a force on it. Under the inﬂuence of this force, the piston moves. In this way, heat is converted into work via internal energy.

The heat added to the gas causes the molecules in the cylinder to move in random directions, but only the molecules that move in the direction of the piston can exert a force on it. Therefore, the kinetic energy of only the molecules that move toward the piston can be converted into work. For the added heat to be completely converted into work, all the gas molecules would have to move in the direction of the piston motion. In a large ensemble of molecules, this is very unlikely.

The odds against the complete conversion of 1 cal of heat into work can be expressed in terms of a group of monkeys who are hitting typewriter keys at random and who by chance type out the complete works of Shakespeare without error. The probability that 1 cal of heat would be completely converted to work is about the same as the probability that the monkeys would type

Chapter 10 ThermodynamicsFIGURE 10.3 Conversion of heat to work.

Shakespeare’s works 15 quadrillion times in succession. (This example is taken from [11-2].)

The distinction between work and heat is this: In work, the energy is in an ordered motion; in heat, the energy is in random motion. Although some of the random thermal motion can be ordered again, the ordering of all the motion is very improbable. Because the probability of completely converting heat to work is vanishingly small, the Second Law of Thermodynamics states categorically that it is impossible.

Heat can be partially converted to work as it ﬂows from a region of higher temperature T1 to a region of lower temperature T2 (see Fig. 10.3). A quantitative treatment of thermodynamics shows (see, for example, [11-5]) that the maximum ratio of work to the input heat is

Work

1 − T2

(10.1)

Heat input

T1

Here the temperature is measured on the absolute scale.

From this equation, it is evident that heat can be completely converted into work only if the heat is rejected into a reservoir at absolute zero temperature.

Although objects can be cooled to within a very small fraction of absolute zero, absolute zero cannot be attained. Therefore, heat cannot be completely converted into work.

10.4

Thermodynamics of Living Systems

It is obvious that animals need food to live, but the reason for this is less obvious. The idea that animals need energy because they consume energy is, strictly speaking, incorrect. We know from the First Law of Thermodynamics that energy is conserved. The body does not consume energy, it changes it from one form to another. In fact, the ﬁrst law could lead us to the erroneous conclusion that animals should be able to function without a source of external energy. The body takes in energy that is in the chemical bonds of the food molecules and converts it to heat. If the weight and the temperature of the body remain constant and if the body performs no external work, the energy input to the body equals exactly the heat energy leaving the body. We may suppose that if the heat outﬂow could be stopped—by good insulation, for example—the body could survive without food. As we know, this supposition is wrong. The need for energy is made apparent by examining the functioning of the body in the light of the Second Law of Thermodynamics.

The body is a highly ordered system. A single protein molecule in the body may consist of a million atoms bound together in an ordered sequence.

Cells are more complex still. Their specialized functions within the body depend on a speciﬁc structure and location. We know from the Second Law of Thermodynamics that such a highly ordered system, left to itself, tends to become disordered, and once it is disordered, it ceases to function. Work must be done on the system continuously to prevent it from falling apart. For example, the blood circulating in veins and arteries is subject to friction, which changes kinetic energy to heat and slows the ﬂow of blood. If a force were not applied to the blood, it would stop ﬂowing in a few seconds. The concentration of minerals inside a cell diﬀers from that in the surrounding environment.

This represents an ordered arrangement. The natural tendency is toward an equalization with the environment. Work must be done to prevent the contents of the cell from leaking out. Finally, cells that die must be replaced, and if the animal is growing, new tissue must be manufactured. For such replacement and growth, new proteins and other cell constituents must be put together from smaller, relatively more random subcomponents. Thus, the process of life consists of building and maintaining ordered structures. In the face of the natural tendency toward disorder, this activity requires work. The situation is somewhat analogous to a pillar made of small, slippery, uneven blocks that tend to slide out of the structure. The pillar remains standing only if blocks are continuously pushed back.

The work necessary to maintain the ordered structures in the body is obtained from the chemical energy in food. Except for the energy utilized in external work done by the muscles, all the energy provided by food is ultimately converted into heat by friction and other dissipative processes in the body. Once the temperature of the body is at the desired level, all the heat generated by the body must leave through the various cooling mechanisms of the body (see Chapter 11). The heat must be dissipated because, unlike heat engines (such as the turbine or the steam engine), the body does not have the ability to obtain work from heat energy. The body can obtain work only

Chapter 10 Thermodynamics from chemical energy. Even if the body did have mechanisms for using heat to perform work, the amount of work it could obtain in this way would be small. Once again, the second law sets the limit. The temperature diﬀerences in the body are small—not more than about 7 C◦ between the interior and the exterior. With the interior temperature T1 at 310 K (37◦C) and the exterior temperature T1 at 303 K, the eﬃciency of heat conversion to work would be (from Eq. 10.1) at most only about 2%.

Of all the various forms of energy, the body can utilize only the chemical binding energy of the molecules which constitute food. The body does not have a mechanism to convert the other forms of energy into work. A person could bask in the sun indeﬁnitely, receiving large quantities of radiant energy, and yet die of starvation. Plants, on the other hand, are able to utilize radiant energy. As animals use chemical energy, so plants utilize solar radiation to provide the energy for the ordering processes necessary for life.

The organic materials produced in the life cycle of plants provide food energy for herbivorous animals, which in turn are food for the carnivorous animals that eat them. The sun is, thus, the ultimate source of energy for life on Earth.

Since living systems create order out of relative disorder (for example, by synthesizing large complex molecules out of randomly arranged subunits), it may appear at ﬁrst glance that they violate the Second Law of Thermodynamics, but this is not the case. To ascertain that the second law is valid, we must examine the whole process of life, which includes not only the living unit but also the energy that it consumes and the by-products that it rejects. To begin with, the food that is consumed by an animal contains a considerable degree of order. The atoms in the food molecules are not randomly arranged but are ordered in speciﬁc patterns. When the chemical energy in the molecular bindings of the food is released, the ordered structures are broken down. The eliminated waste products are considerably more disordered than the food taken in. The ordered chemical energy is converted by the body into disordered heat energy.

The amount of disorder in a system can be expressed quantitatively by means of a concept called entropy. Calculations show that, in all cases, the increase in the entropy (disorder) in the surroundings produced by the living system is always greater than the decrease in entropy (i.e., ordering) obtained in the living system itself. The total process of life, therefore, obeys the second law. Thus, living systems are perturbations in the ﬂow toward disorder. They keep themselves ordered for a while at the expense of the environment. This is a diﬃcult task requiring the use of the most complex mechanisms found in nature. When these mechanisms fail, as they eventually must, the order falls apart, and the organism dies.