The Physics of Energy, page 33
(8.20)
becomes vanishingly small as , even when dQ is integrated to give a finite quantity of heat transfer. Heat transfer under quasi-equilibrium conditions can be reversed without additional energy input or entropy production by changing the sign of dT, in the limit where . Heat transfer under these conditions is called reversible heat transfer, for which
(8.21)
When other sources of entropy production are present, quasi-equilibrium processes are not necessarily reversible, though the converse – reversible processes are necessarily quasi-equilibrium – still holds. Consider, for example, slow compression of a gas by a piston that experiences friction with the cylinder in which it moves. If the compression proceeds slowly enough, the system can be kept arbitrarily close to thermodynamic equilibrium throughout the process. The process is not, however, reversible, since frictional losses cannot be reversed. To be reversible, the entropy of the system plus its environment must not increase, since the reverse process must not violate the 2 Law.
Like many other concepts in thermodynamics, quasi-equilibrium and reversibility are idealizations. In any real system, small deviations from thermal equilibrium, such as temperature gradients in the fluid, violate the quasi-equilibrium assumption and lead to generation of additional entropy. Thus, for a real system heat transfer is never really reversible in the sense defined above.
Equation (8.21) provides a convenient way to compute the entropy difference between two states of a thermodynamic system. If A and B are two equilibrium states and is a path from A to B in the space of thermodynamic states, along which the system remains close to equilibrium, then eq. (8.21) can be integrated along to obtain the entropy difference, ,
(8.22)
In §8.1 and in Figure 8.1 we examined the entropy increase when an ideal gas is allowed to expand freely from volume V to volume . A free expansion is irreversible and therefore eq. (8.22) cannot be used directly to compute the entropy change. As illustrated in Example 8.2, however, we can find a reversible path that leads to the same final state, and this enables us to compute the entropy change in a free expansion, and to verify the conclusion that the entropy increases by .
Example 8.2Entropy Change in the Free Expansion of an Ideal Gas, Revisited
Suppose, as illustrated in Figure 8.1, N molecules of an ideal gas, initially at temperature are held in avolume . By the ideal gas law, the initial pressure is . This equilibrium state is marked A in the plot of the pressure–volume (pV) plane at right.
Suddenly the gas is allowed to expand irreversibly until its volume has doubled to , and then the gas is allowed to settle once again into thermal equilibrium. No work was done and no heat was added to the gas, so its internal energy did not change. Since the internal energy of an ideal gas depends only on its temperature (eq. (5.6)), its temperature has not changed either: . The pressure has decreased to . This final equilibrium state is marked B in the figure. By how much has its entropy changed?
Note that the free expansion cannot be denoted by a path in the pV-plane because the system does not pass through a sequence of near-equilibrium states with well-defined temperature and pressure as it expands. Nevertheless, since entropy is a state function, the change in entropy of the gas does not depend upon the process used to get from A to B, so we may choose a reversible path from A to B and use eq. (8.22) to compute the entropy difference . In particular, we can start by adding heat reversibly to the gas at constant pressure, allowing it to expand from V to – the path marked AP in the figure. We can then remove heat from the gas at constant volume, allowing the pressure to drop from P to – the path marked PB in the figure. Along AP, , while along PB, , where are the (possibly temperature-dependent) heat capacities of the ideal gas at constant pressure and volume. Thus,
where we have used the fact that for an ideal gas (eq. (5.31)). Since the ideal gas law also fixes , we find
in agreement with the information-theoretic argument of §8.4.2.
In a reversible process, the total entropy of the system plus the environment does not change. As we find in the following section, maximizing efficiency of thermal energy conversion involves minimizing the production of additional entropy. Thus, reversible processes are often used as a starting point for constructing idealized thermodynamic cycles for converting thermal energy to mechanical energy. We return to this subject in §10.
8.5Limit to Efficiency
We have alluded repeatedly to the thermodynamic limit on the efficiency with which heat can be converted into work. We now have the tools to derive this limit. The clearest and most general statement of the efficiency limit applies to a cyclic process in which an apparatus takes in heat energy from a reservoir that is in thermal equilibrium at a high temperature , uses this heat energy to perform useful mechanical (or electromagnetic) work W, and then returns to its original state, ready to perform this task again. Such an apparatus is called a heat engine. The engine may also (indeed must) expel some heat to the environment, which we describe as another reservoir in thermal equilibrium at temperature . The reservoirs are assumed to be so large that adding or removing heat does not change their temperatures appreciably. §10 is devoted to the study of some specific types of heat engines. For the moment we leave the details unspecified and consider the engine schematically, as depicted in Figure 8.4.
Figure 8.4 A schematic depiction of a heat engine. The entropy added to the engine is greater than or equal to and the entropy dumped is less than .
Quasi-equilibrium and Reversible Processes
When a system is changed gradually, so that it remains infinitesimally close to thermodynamic equilibrium throughout the transformation, the process is called a quasi-equilibrium process. When heat dQ is transferred to a system in a quasi-equilibrium fashion and no other sources of entropy production are present, the process is reversible, and the entropy of the system increases by
Reversible processes do not increase the total entropy of the systems involved.
Additional entropy is generated when heat is transferred irreversibly.
Since both energy and entropy are state functions, and the process is cyclic, both the energy and entropy of the apparatus must be the same at the beginning and end of the cycle. First let us trace energy conservation. We suppose that the engine absorbs heat energy from , does an amount of work W, and expels heat energy to . Conservation of energy requires
(8.23)
Note the signs: means the engine does work, means the engine takes in heat from , and means the engine expels heat to .
Now consider the flow of entropy. If the transfer of heat from the high-temperature reservoir were reversible, then the entropy of the engine would increase by . In practice, the entropy of the engine increases by more than this, due for example to entropy creation in the friction of moving parts or free expansion of fluids. Thus, the entropy added to the engine when it absorbs heat from the reservoir at temperature can be written as
(8.24)
where (as required by the 2 Law).
Similarly, if heat were transferred reversibly into the low-temperature reservoir, then the entropy of the engine would decrease by , which is the amount by which the entropy of the low-temperature reservoir increases. In practice, the entropy of the engine does not decrease by this much because entropy is created by irreversibilities in the processes involved. Thus, the amount by which the engine's entropy drops when it expels heat at can be written as
(8.25)
where .
Since, by hypothesis, the engine is cyclic, the net change in its entropy must be zero, so . Combining eqs. (8.24) and (8.25) with this relation, and remembering that , gives
(8.26)
where the equality holds only in the limit that the engine operates reversibly throughout its cycle.
We define the efficiency of a heat engine to be the ratio of the work performed, W, to the input heat, ,
(8.27)
Combining eq. (8.23) (which follows from the 1st Law) and eq. (8.26) (which follows from the 2nd Law), we find
(8.28)
This thermodynamic limit on the efficiency of a heat engine is known as the Carnot limit, after Sadi Carnot, the nineteenth-century French physicist who discovered it. This law is of essential importance in the practical study of energy systems. The fact that the relations eqs. (8.24) and (8.25) are inequalities rather than equalities expresses the important physical reality that a heat engine cannot operate perfectly reversibly. Indeed, it is extremely difficult to prevent additional entropy production during an engine's cycle.
Note that we do not associate any entropy change in the engine with the work . Indeed, the essential difference between thermal energy and macroscopic mechanical or electromagnetic energy is the presence of entropy in the thermal case. A mechanical piston, for example, such as the one depicted in Figure 5.4, is idealized as a single macroscopic object with given position and velocity, and no entropy associated with ignorance of hidden degrees of freedom. In reality, some entropy will be created in the piston – through, for example, frictional heating. Energy lost in this fashion can be thought of as simply decreasing W, and hence the efficiency, further below the Carnot limit. Or, thinking of the whole process as part of the engine, this can be construed as an additional contribution to , with a corresponding increase in and .
The Carnot Limit
A heat engine that takes heat from a source at a high temperature , does work, expels heat to a sink at low temperature , and returns to its initial state, ready to begin the cycle again, has an efficiency η that cannot exceed the Carnot limit
The analysis leading to the Carnot limit has a number of other interesting implications. For example: no heat engine can transform heat into work without expelling heat to the environment; and no heat engine can remove heat from a low-temperature source, do work, and expel heat at a higher temperature. These constraints are so fundamental that in the early days of thermodynamics they served as statements of the 2 Law. The proofs of these assertions are left to the problems. It is important to note that the Carnot limit applies only to heat engines. It does not, for example, limit the efficiency with which electrical energy can be converted into useful work. The Carnot limit also does not say anything directly regarding the efficiency with which a non-cyclic process can convert heat into work; in later chapters, however, we develop the notion of exergy (§37, §36), which places a 2nd law bound on the energy that can be extracted in a situation where a substance is hotter or colder than the environment without reference to cyclic processes.
The practical consequences of the efficiency limit (8.28) are significant. Most energy used by humans either begins or passes an intermediate stage as thermal energy, and must be converted into mechanical energy to be used for purposes such as transport or electrical power. Thus, in systems ranging from car engines to nuclear power plants, much effort is made to maximize the efficiency of energy conversion from thermal to mechanical form (see Box 8.2). Treatment of these conversion systems forms a substantial component of the material in the remainder of this book. In §10, we describe a range of heat engines and the thermodynamic processes involved in their operation more explicitly.
Box 8.2 Heat Engine Efficiency: Ideal vs. Actual
Many primary energy resources provide thermal energy, either directly or indirectly. For use by mechanical or electrical systems, thermal energy must be transformed into another form using a heat engine or equivalent system. The Carnot limit gives an absolute upper bound on the fraction of thermal energy that can be extracted. For most real systems, the actual efficiency is significantly below the Carnot limit. Some of the systems studied later in the book are tabulated here with (very rough) estimates of the temperature of the thermal source, the Carnot efficiency, and approximate values of the actual efficiency for typical systems in the US using standard technology. In most cases the environment can be roughly approximated as 300 K.
Throughout the rest of the book we consider many different types of thermal energy conversion, and explore the reasons why their actual efficiency falls short of the Carnot limit.
One clear implication of the Carnot limit is that the greatest thermodynamic efficiency can be achieved when thermal energy is extracted from a source at a temperature that is as high as possible compared to the temperature of the environment into which entropy must be dumped. Note, however, that for the distinct processes within a thermodynamic cycle, the maximization of efficiency for the cycle as a whole requires that thermal energy transfer from one part of the system to another must occur across temperature differentials that are as small as possible to avoid production of extra entropy.
8.6The Boltzmann Distribution
Now that we have a rigorous definition of temperature, we can give a more precise description of a system in thermal equilibrium at a fixed temperature. While a system in thermodynamic equilibrium at fixed energy is equally likely to be in each of its available microstates, a system that is in thermodynamic equilibrium itself and in thermal equilibrium with an environment at fixed temperature T is characterized by a more complicated probability distribution known as the Boltzmann distribution. This probability distribution, often referred to as a thermal distribution, has a universal form in which the probability of a given microstate depends only on the energy E of the state and the temperature T. In this section we derive this distribution. One application of this result, which we use in the following section, is to a system of many particles, such as in an ideal gas. For such a system, each particle can be thought of as being in thermal equilibrium with the rest of the system and has a probability distribution on states given by the Boltzmann distribution, even when the total energy of the gas itself is fixed.
8.6.1 Derivation of the Boltzmann Distribution
Once again, as in §8.4.3 where we gave a precise definition of temperature, we consider two systems that are allowed to come into thermal equilibrium with one another at temperature T. This time, however, as depicted in Figure 8.5, we imagine that one system, the thermal reservoir , is very large – effectively infinite – and that the other system is much smaller (and could consist even of only a single particle, in which case Figure 8.5 exaggerates the size of dramatically).
Figure 8.5 A large thermal reservoir, , and a small system in thermal contact, but otherwise isolated from the environment.
Together, and have a fixed total energy . Energy can move back and forth between and , so the energy apportioned to states of is not fixed. The question we wish to address is then: What is the probability distribution on states of when is in thermal equilibrium with at temperature T?
Since the combined system is in thermodynamic equilibrium and has fixed total energy, all states of the combined system with energy U are equally probable. Suppose is in a state s with energy E that occurs with probability p. This leaves energy available to . According to eq. (8.11), the number of microstates of with energy is
(8.29)
Since each combined state of is equally probable, the probability of the given state s is proportional to the number of compatible microstates of
(8.30)
where C is a constant. Note that this probability only depends on the energy E of the state s, so we can simply write it as .
To explicitly compute , we differentiate with respect to E
(8.31)
This simple differential equation for the probability has the solution
(8.32)
where Z is a constant. Since is a probability, it must sum to one when all possible states of are considered. This fixes Z to be
(8.33)
where j is an index over all possible states of the system . Z, which is a function of the temperature, is known as the partition function and is an important characteristic of systems in thermal equilibrium. The probability distribution (8.32) with this overall normalization is the Boltzmann distribution and the factor , known as the Boltzmann factor, suppresses the probability of finding states with energy much greater than . The Boltzmann distribution gives the probability of finding the system in a given state with energy E when is in thermal equilibrium at temperature T. Note that is not necessarily the probability of finding the system with energy E, since there may be (and often are) many states with energy E.
The Boltzmann Distribution
If a system is in equilibrium with a (much larger) thermal reservoir at temperature T, we say that is in thermal equilibrium at temperature T. The probability of finding in a state with energy is then
where Z is known as the partition function.
8.6.2 Ensemble Averages
We have constructed several types of ensembles of states with specific probability distributions. Isolated systems of definite energy U that have reached thermodynamic equilibrium form the microcanonical ensemble. The system has microstates, each of which occurs with equal probability . On the other hand, systems in thermal equilibrium with a heat reservoir at temperature T form a different ensemble known as the canonical ensemble. This ensemble has a probability distribution given by the Boltzmann distribution (8.32).
Often, when we observe a system, we would like to know its average properties. For example, although the precise value of the energy for a system in thermal equilibrium at temperature T changes over time as small fluctuations move energy between the system and the reservoir, we may want to know the average value of this energy. For a system in thermal equilibrium at temperature T, what we mean by the “average” of a quantity is the average over many measurements of that quantity when we pick elements at random from the canonical ensemble. For a system in thermal equilibrium at fixed energy, we take the average over the microcanonical ensemble. For an ergodic system, this is the same as the time-averaged value of the quantity.
