6.7: Heat Engines and the Second Law of Thermodynamics

Learning Objectives

• Describe the processes of a simple heat engine.
• Explain the relationship between work done by a gas and change in its volume.

Figure \(\PageIndex{1}\): Beginning with the Industrial Revolution, humans have harnessed power through the use of the first law of thermodynamics, before we even understood it completely. This photo, of a steam engine at the Turbinia Works, dates from 1911, a mere 61 years after the first explicit statement of the first law of thermodynamics by Rudolph Clausius. (credit: public domain; author unknown)

The field of thermodynamics was largely driven by the demands of the Industrial Revolution, and the interest in having mechanical devices to help do work on a very large scale. It was through this process that the Second Law of Thermodynamics was first discovered. It was not until much later that it was discovered how this law applies to all sorts of processes, even those undertaken by living creatures. In this section we will first discuss on a very broad level the types of devices which were initially used to understand thermodynamics, and then in each of the subsections we will examine the framings of the second law of thermodynamics that arose from these experiments.

The scientists of the Industrial Revolution were focused on finding ways in which to use heat transfer to do work for us. The devices they used for this investigation are called heat engines. Car engines and steam turbines that generate electricity are examples of heat engines. Figure \(\PageIndex{2}\) shows schematically how the first law of thermodynamics applies to the typical heat engine.

Figure \(\PageIndex{2}\): Schematic representation of a heat engine, governed, of course, by the first law of thermodynamics (and other laws of thermodynamics we will discuss later).

Figure \(\PageIndex{3}\): (a) Heat transfer to the gas in a cylinder increases the internal energy of the gas, creating higher pressure and temperature. (b) The force exerted on the movable cylinder does work as the gas expands. Gas pressure and temperature decrease when it expands, indicating that the gas’s internal energy has been decreased by doing work. (c) Heat transfer to the environment further reduces pressure in the gas so that the piston can be more easily returned to its starting position.

The illustrations above show one of the ways in which heat transfer does work. Fuel combustion produces heat transfer to a gas in a cylinder, increasing the pressure of the gas and thereby the force it exerts on a movable piston. The gas does work on the outside world, as this force moves the piston through some distance. Heat transfer to the gas cylinder results in work being done. To repeat this process, the piston needs to be returned to its starting point. Heat transfer now occurs from the gas to the surroundings so that its pressure decreases, and a force is exerted by the surroundings to push the piston back through some distance. Variations of this process are employed daily in hundreds of millions of heat engines. Here, we consider some of the thermodynamic processes on which heat engines are based.

Work Done by a Gas

As an example, we will examine a process by which a gas does work on a piston at constant pressure. There are other possible ways in which a gas can do work on a piston, but this is the simplest one to illustrate the point. Since the pressure is constant, the force exerted is constant and the work done is given as shown in the following derivation.

\[P \Delta V. \nonumber \]

Figure \(\PageIndex{4}\): An isobaric expansion of a gas requires heat transfer to keep the pressure constant. Since pressure is constant, the work done is \(P \Delta V\).

Recall from mechanics that work done by a force \(F\) on an object undergoing displacement \(d\) is

\[W=F d. \nonumber \]

See the symbols as shown in Figure \(\PageIndex{4}\). Now force is pressure times area (\(F=P A\)), and so

\[W=\text { PAd }. \nonumber \]

Because the volume of a cylinder is its cross-sectional area \(A\) times its length \(d\), we see that \(A d=\Delta V\), the change in volume; thus,

\[W=P \Delta V \text { (isobaric process) }. \nonumber \]

Note that if \(\Delta V\) is positive, then \(W\) is positive, meaning that positive work is done by the gas on the outside world.

(Note that the pressure involved in this work that we have called \(P\) is the pressure of the gas inside the tank. If we call the pressure outside the tank \(P_{\text {ext }}\), an expanding gas would be working against the external pressure; the work done would therefore be \(W=-P_{\text {ext }} \Delta V\) (isobaric process). There are some—especially chemists—who use this definition of work, and not the definition based on internal pressure, as the basis of the First Law of Thermodynamics. This definition reverses the sign conventions for work, and results in a statement of the first law that becomes \(\Delta U=Q+W\). In this textbook, we will use the physics convention of using work done by the system on the surrounding, not the other way around.)

This is the key lesson from the above derivation: a gas expanding under pressure does work on its surrounding, and unless additional energy is added through heat transfer, the internal energy of the gas decreases. We will examine the experimental results that come about as a consequence of this fact later.

Thermodynamic Processes

We introduced an example process above in discussing work done by a gas. This process is an example of a thermodynamic process. A thermodynamic process describes a change that happens to a gas, which results in change in its pressure (\(P\)), volume (\(V\)), and/or temperature (\(T\)). There are several named thermodynamic processes: isobaric, isochoric, isothermal, and adiabatic. (It turns out the one we considered was isobaric.) These processes are given special names because they occur under some restrictions, which gives them their special properties. The details of these processes are beyond the scope of this course, but they have been well studied and lead to some very interesting results which help us to understand the Second Law of Thermodynamics.

Figure \(\PageIndex{5}\): This thermodynamic process is called the Otto Cycle, and is an example of a cyclic process. In this cycle, a mixture of air and fuel enters the chamber and is ignited. The chemical reaction that occurs creates a large pressure which does work in part (c). However, the work done is not equal to all of the energy from the chemical reaction, as some of the energy is released as a hot exhaust gas. The internal combustion engines of gasoline powered vehicles are an example of the Otto Cycle.

Reversible Processes

Some of these thermodynamic processes are reversible in principle. 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. Real macroscopic processes are never exactly reversible. In the previous example, our system is a gas (like that in Figure \(\PageIndex{4}\)), and its environment is the piston, cylinder, and the rest of the universe. If there are any energy-dissipating mechanisms, such as friction or turbulence, then heat transfer to the environment occurs for either direction of the piston. So although the gas inside the piston might be returned to its original state, the environment will not—it will have been heated. Reversibility requires the direction of heat transfer to reverse for the reverse path. Since dissipative mechanisms cannot be completely eliminated, real processes cannot be reversible.

There must be reasons that real macroscopic processes cannot be reversible. We can imagine them going in reverse. For example, heat transfer occurs spontaneously from hot to cold and never spontaneously the reverse. Yet it would not violate the first law of thermodynamics for this to happen. In fact, all spontaneous processes, such as bubbles bursting, never go in reverse. There is a second thermodynamic law that forbids them from going in reverse. When we study this law, we will learn something about nature and also find that such a law limits the efficiency of heat engines. We will find that heat engines with the greatest possible theoretical efficiency would have to use reversible processes, and even they cannot convert all heat transfer into doing work.

Section Summary

• One of the important implications of the first law of thermodynamics is that machines can be harnessed to do work that humans previously did by hand or by external energy supplies such as running water or the heat of the Sun. A machine that uses heat transfer to do work is known as a heat engine.
• There are several simple processes, used by heat engines, that flow from the first law of thermodynamics, and they differ from one another based on how they affect pressure, volume, temperature, and heat transfer.
• If the work done is performed on the outside environment, work (\(W\)) will be a positive value. If the work done is done to the heat engine system, work (\(W\)) will be a negative value.
• Some thermodynamic processes are reversible in theory; that is, both the thermodynamic system and the environment can be returned to their initial states. However, because of loss of energy owing to the second law of thermodynamics, complete reversibility does not work in practice.

Glossary

heat engine

a device that converts heat transfer into work

thermodynamic process

describes a change to a gas, which results in changes to its pressure, volume, and/or temperature

reversible process

both the system and its environment can return to exactly the states they were in by following the reverse path.