Chapter 4. The Second Law of Thermodynamics

4.1 Reversible and Irreversible Processes

Learning Objectives

By the end of this section, you will be able to:

• Define reversible and irreversible processes
• State the second law of thermodynamics via an irreversible process

Consider an ideal gas that is held in half of a thermally insulated container by a wall in the middle of the container. The other half of the container is under vacuum with no molecules inside. Now, if we remove the wall in the middle quickly, the gas expands and fills up the entire container immediately, as shown in Figure 4.2.

Because half of the container is under vacuum before the gas expands there, we do not expect any work to be done by the system—that is, $W=0$—because no force from the vacuum is exerted on the gas during the expansion. If the container is thermally insulated from the rest of the environment, we do not expect any heat transfer to the system either, so $Q=0$. Then the first law of thermodynamics leads to the change of the internal energy of the system,

$\text{Δ}{E}_{\text{int}}=Q-W=0.$

For an ideal gas, if the internal energy doesn’t change, then the temperature stays the same. Thus, the equation of state of the ideal gas gives us the final pressure of the gas, $p=nRT\text{/}V={p}_{0}\text{/}2,$ where ${p}_{0}$ is the pressure of the gas before the expansion. The volume is doubled and the pressure is halved, but nothing else seems to have changed during the expansion.

All of this discussion is based on what we have learned so far and makes sense. Here is what puzzles us: Can all the molecules go backward to the original half of the container in some future time? Our intuition tells us that this is going to be very unlikely, even though nothing we have learned so far prevents such an event from happening, regardless of how small the probability is. What we are really asking is whether the expansion into the vacuum half of the container is reversible.

A reversible process is a process in which the system and environment can be restored to exactly the same initial states that they were in before the process occurred, if we go backward along the path of the process. The necessary condition for a reversible process is therefore the quasi-static requirement. Note that it is quite easy to restore a system to its original state; the hard part is to have its environment restored to its original state at the same time. For example, in the example of an ideal gas expanding into vacuum to twice its original volume, we can easily push it back with a piston and restore its temperature and pressure by removing some heat from the gas. The problem is that we cannot do it without changing something in its surroundings, such as dumping some heat there.

A reversible process is truly an ideal process that rarely happens. We can make certain processes close to reversible and therefore use the consequences of the corresponding reversible processes as a starting point or reference. In reality, almost all processes are irreversible, and some properties of the environment are altered when the properties of the system are restored. The expansion of an ideal gas, as we have just outlined, is irreversible because the process is not even quasi-static, that is, not in an equilibrium state at any moment of the expansion.

From the microscopic point of view, a particle described by Newton’s second law can go backward if we flip the direction of time. But this is not the case, in practical terms, in a macroscopic system with more than ${10}^{23}$ particles or molecules, where numerous collisions between these molecules tend to erase any trace of memory of the initial trajectory of each of the particles. For example, we can actually estimate the chance for all the particles in the expanded gas to go back to the original half of the container, but the current age of the universe is still not long enough for it to happen even once.

An irreversible process is what we encounter in reality almost all the time. The system and its environment cannot be restored to their original states at the same time. Because this is what happens in nature, it is also called a natural process. The sign of an irreversible process comes from the finite gradient between the states occurring in the actual process. For example, when heat flows from one object to another, there is a finite temperature difference (gradient) between the two objects. More importantly, at any given moment of the process, the system most likely is not at equilibrium or in a well-defined state. This phenomenon is called irreversibility.

Let us see another example of irreversibility in thermal processes. Consider two objects in thermal contact: one at temperature ${T}_{1}$ and the other at temperature ${T}_{2}>{T}_{1}$, as shown in Figure 4.3.

We know from common personal experience that heat flows from a hotter object to a colder one. For example, when we hold a few pieces of ice in our hands, we feel cold because heat has left our hands into the ice. The opposite is true when we hold one end of a metal rod while keeping the other end over a fire. Based on all of the experiments that have been done on spontaneous heat transfer, the following statement summarizes the governing principle:

Second Law of Thermodynamics (Clausius statement)

Heat never flows spontaneously from a colder object to a hotter object.

This statement turns out to be one of several different ways of stating the second law of thermodynamics. The form of this statement is credited to German physicist Rudolf Clausius (1822−1888) and is referred to as the Clausius statement of the second law of thermodynamics. The word “spontaneously” here means no other effort has been made by a third party, or one that is neither the hotter nor colder object. We will introduce some other major statements of the second law and show that they imply each other. In fact, all the different statements of the second law of thermodynamics can be shown to be equivalent, and all lead to the irreversibility of spontaneous heat flow between macroscopic objects of a very large number of molecules or particles.

Both isothermal and adiabatic processes sketched on a pV graph (discussed in The First Law of Thermodynamics) are reversible in principle because the system is always at an equilibrium state at any point of the processes and can go forward or backward along the given curves. Other idealized processes can be represented by pV curves; Table 4.1 summarizes the most common reversible processes.

Summary of Simple Thermodynamic Processes
Process Constant Quantity and Resulting Fact
Isobaric Constant pressure $W=p\text{Δ}V$
Isochoric Constant volume $W=0$
Isothermal Constant temperature $\text{Δ}T=0$
Adiabatic No heat transfer $Q=0$

Summary

• A reversible process is one in which both the system and its environment can return to exactly the states they were in by following the reverse path.
• An irreversible process is one in which the system and its environment cannot return together to exactly the states that they were in.
• The irreversibility of any natural process results from the second law of thermodynamics.

Conceptual Questions

State an example of a process that occurs in nature that is as close to reversible as it can be.

Show Solution

Some possible solutions are frictionless movement; restrained compression or expansion; energy transfer as heat due to infinitesimal temperature nonuniformity; electric current flow through a zero resistance; restrained chemical reaction; and mixing of two samples of the same substance at the same state.

Problems

A tank contains 111.0 g chlorine gas $\left({\text{Cl}}_{2}\right),$ which is at temperature $82.0\phantom{\rule{0.2em}{0ex}}\text{°C}$ and absolute pressure $5.70\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{5}\phantom{\rule{0.2em}{0ex}}\text{Pa}.$ The temperature of the air outside the tank is $20.0\phantom{\rule{0.2em}{0ex}}\text{°C}$. The molar mass of ${\text{Cl}}_{2}$ is 70.9 g/mol. (a) What is the volume of the tank? (b) What is the internal energy of the gas? (c) What is the work done by the gas if the temperature and pressure inside the tank drop to $31.0\phantom{\rule{0.2em}{0ex}}\text{°C}$ and $3.80\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{5}\phantom{\rule{0.2em}{0ex}}\text{Pa}$, respectively, due to a leak?

A mole of ideal monatomic gas at $0\phantom{\rule{0.2em}{0ex}}\text{°C}$ and 1.00 atm is warmed up to expand isobarically to triple its volume. How much heat is transferred during the process?

Show Solution

$11.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}\phantom{\rule{0.2em}{0ex}}\text{J}$

A mole of an ideal gas at pressure 4.00 atm and temperature 298 K expands isothermally to double its volume. What is the work done by the gas?

After a free expansion to quadruple its volume, a mole of ideal diatomic gas is compressed back to its original volume isobarically and then cooled down to its original temperature. What is the minimum heat removed from the gas in the final step to restoring its state?

Show Solution

$4.5\phantom{\rule{0.2em}{0ex}}p{V}_{0}$

Glossary

Clausius statement of the second law of thermodynamics
heat never flows spontaneously from a colder object to a hotter object
irreversibility
phenomenon associated with a natural process
irreversible process
process in which neither the system nor its environment can be restored to their original states at the same time
reversible process
process in which both the system and the external environment theoretically can be returned to their original states