Learning Objectives
By the end of this section, you will be able to do the following:
- Define entropy
- Calculate the increase of entropy in a system with reversible and irreversible processes
- Explain the expected fate of the universe in entropic terms
- Calculate the increasing disorder of a system
The information presented in this section supports the following AP® learning objectives and science practices:
- 7.B.2.1: The student is able to connect qualitatively the second law of thermodynamics in terms of the state function called entropy and how it (entropy) behaves in reversible and irreversible processes. (S.P. 7.1)
There is yet another way of expressing the second law of thermodynamics. This version relates to a concept called entropy. By examining it, we shall see that the directions associated with the second law—heat transfer from hot to cold, for example—are related to the tendency in nature for systems to become disordered and for less energy to be available for use as work. The entropy of a system can in fact be shown to be a measure of its disorder and of the unavailability of energy to do work.
Making Connections: Entropy, Energy, and Work
Recall that the simple definition of energy is the ability to do work. Entropy is a measure of how much energy is not available to do work. Although all forms of energy are interconvertible, and all can be used to do work, it is not always possible, even in principle, to convert the entire available energy into work. That unavailable energy is of interest in thermodynamics, because the field of thermodynamics arose from efforts to convert heat to work.
We can see how entropy is defined by recalling our discussion of the Carnot engine. We noted that for a Carnot cycle, and hence for any reversible processes, . Rearranging terms yields
for any reversible process. and are absolute values of the heat transfer at temperatures and , respectively. This ratio of is defined to be the change in entropy for a reversible process,
where is the heat transfer, which is positive for heat transfer into and negative for heat transfer out of, and is the absolute temperature at which the reversible process takes place. The SI unit for entropy is joules per kelvin (J/K). If temperature changes during the process, then it is usually a good approximation (for small changes in temperature) to take to be the average temperature, avoiding the need to use integral calculus to find .
The definition of is strictly valid only for reversible processes, such as used in a Carnot engine. However, we can find precisely even for real, irreversible processes. The reason is that the entropy of a system, like internal energy , depends only on the state of the system and not how it reached that condition. Entropy is a property of state. Thus the change in entropy of a system between state 1 and state 2 is the same no matter how the change occurs. We just need to find or imagine a reversible process that takes us from state 1 to state 2 and calculate for that process. That will be the change in entropy for any process going from state 1 to state 2. (See Figure 15.34.)
Now let us take a look at the change in entropy of a Carnot engine and its heat reservoirs for one full cycle. The hot reservoir has a loss of entropy , because heat transfer occurs out of it. Remember that when heat transfers out, then has a negative sign. The cold reservoir has a gain of entropy , because heat transfer occurs into it. (We assume the reservoirs are sufficiently large that their temperatures are constant.) So the total change in entropy is
Thus, since we know that for a Carnot engine,
This result, which has general validity, means that the total change in entropy for a system in any reversible process is zero.
The entropy of various parts of the system may change, but the total change is zero. Furthermore, the system does not affect the entropy of its surroundings, since heat transfer between them does not occur. Thus the reversible process changes neither the total entropy of the system nor the entropy of its surroundings. Sometimes this is stated as follows: Reversible processes do not affect the total entropy of the universe. Real processes are not reversible, though, and they do change total entropy. We can, however, use hypothetical reversible processes to determine the value of entropy in real, irreversible processes. The following example illustrates this point.
Example 15.6 Entropy Increases in an Irreversible (Real) Process
Spontaneous heat transfer from hot to cold is an irreversible process. Calculate the total change in entropy if 4000 J of heat transfer occurs from a hot reservoir at to a cold reservoir at , assuming there is no temperature change in either reservoir. (See Figure 15.35.)
Strategy
How can we calculate the change in entropy for an irreversible process when is valid only for reversible processes? Remember that the total change in entropy of the hot and cold reservoirs will be the same whether a reversible or irreversible process is involved in heat transfer from hot to cold. So we can calculate the change in entropy of the hot reservoir for a hypothetical reversible process in which 4,000 J of heat transfer occurs from it; then we do the same for a hypothetical reversible process in which 4,000 J of heat transfer occurs to the cold reservoir. This produces the same changes in the hot and cold reservoirs that would occur if the heat transfer were allowed to occur irreversibly between them, and so it also produces the same changes in entropy.
Solution
We now calculate the two changes in entropy using . First, for the heat transfer from the hot reservoir,
And for the cold reservoir,
Thus, the total is
Discussion
There is an increase in entropy for the system of two heat reservoirs undergoing this irreversible heat transfer. We will see that this means there is a loss of ability to do work with this transferred energy. Entropy has increased, and energy has become unavailable to do work.
It is reasonable that entropy increases for heat transfer from hot to cold. Since the change in entropy is , there is a larger change at lower temperatures. The decrease in entropy of the hot object is therefore less than the increase in entropy of the cold object, producing an overall increase, just as in the previous example. This result is very general:
There is an increase in entropy for any system undergoing an irreversible process.
With respect to entropy, there are only two possibilities: entropy is constant for a reversible process, and it increases for an irreversible process. There is a fourth version of the second law of thermodynamics stated in terms of entropy:
The total entropy of a system either increases or remains constant in any process; it never decreases.
For example, heat transfer cannot occur spontaneously from cold to hot, because entropy would decrease.
Entropy is very different from energy. Entropy is not conserved but increases in all real processes. Reversible processes (such as in Carnot engines) are the processes in which the most heat transfer to work takes place and are also the ones that keep entropy constant. Thus we are led to make a connection between entropy and the availability of energy to do work.