The Second Law of Thermodynamics

Any form of energy can be converted into any other form, and energy can also be transferred from one object or system to another. However, certain kinds of energy transfers only go in one direction. For instance, if you drop an ice cube into a cup of hot coffee, heat (thermal energy) will flow from the coffee into the ice, and the ice cube will melt. The reverse of that process never happens. Imagine taking a video of the melting ice cube, then playing the video backwards: the ice cube begins as a tiny speck in the steaming cup of coffee, and gradually grows larger. That sort of un-melting process never happens in real life. The second law of thermodynamics provides an account of which sorts of physical processes are allowed to happen in reverse, and which are not.

Early Formulations of the Second Law

The earliest formulation of the second law was proposed by German physicist Rudolf Clausius (1822-1888), who expressed the idea as follows:

“Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.”Rudolf Clausius, “On a Modified Form of the Second Fundamental Theorem in the Mechanical Theory of Heat,” The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science series 4 volume 12 (1856): 86. Available online, here.

In other words, heat doesn’t flow out of something colder and into something hotter unless some other physical process is at work. Heat doesn’t flow spontaneously (“on its own,” so to speak) from colder things to hotter things. Heat can flow spontaneously from hot to cold, but not vice versa.

Of course, it is possible for heat to flow from colder things to hotter things when other processes are involved. For example, an air conditioner extracts heat from the cooler air inside a building and transfers that heat to the warmer air outside. But the flow of heat from colder air to warmer air is not the only process involved. In order for the air conditioner to work, another flow of energy is required—namely, the energy that powers the air conditioner via its electrical power cord.

How Air Conditioners and Refrigerators Work

In order to transfer heat out of something that is already colder than its environment, air conditioners and refrigerators use a mechanism called a heat pump. Here’s how it works:

  1. First, a piston compresses some fluid (gas or liquid) inside a cylinder. The energy used to move the piston is converted partly to elastic potential energy, but some of the energy is converted to heat as the moving piston speeds up the motions of the fluid’s molecules. So, the fluid gets hotter when it is compressed.
  2. The hot, pressurized fluid is then circulated through a system of tubes that allow heat to flow from the hot fluid into the outside air.
  3. After the pressurized fluid cools off to a lukewarm temperature (the same temperature as the outside air), it is returned to the cylinder and allowed to expand, pushing the piston back to its original position. Since the fluid exerts a force that moves the piston, work is done on the piston. The energy used to do this work comes partly from the elastic potential energy of the compressed fluid, but some of the energy comes from the motions of the fluid’s molecules—i.e., from heat. As fast-moving (high-temperature) molecules bounce against the receding piston, some of their kinetic energy is transferred to the piston, and as a result they move more slowly afterwards. In this way, some of the lukewarm fluid’s heat energy is converted to kinetic energy in the piston, so the fluid gets colder as it expands.
  4. The cold, uncompressed fluid is circulated through a system of tubes that cool the air inside a refrigerator or building, by allowing heat from the inside air to flow into the colder fluid.
  5. The fluid is then returned to the cylinder, and the whole process is repeated.
photograph of William Thomson (Lord Kelvin)
William Thomson
(Lord Kelvin)
image source (public domain)
1824 - 1907

Another early version of the second law was developed by Scottish physicist William Thomson, better known by his honorary title Lord Kelvin. He recognized that in order for heat to be used as a source of energy for doing work, there must be something colder for the heat to flow into. In his words:

“It is impossible … to derive mechanical effect [i.e. work] from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.”William Thomson, “On the Dynamical Theory of Heat, with numerical results deduced from Mr Joule’s equivalent of a Thermal Unit, and M. Regnault’s Observations on Steam,” Transactions of the Royal Society of Edinburgh volume 20 (1853), 265. Available online, here.

This version of the second law has important practical implications. Even though heat is a form of energy (and energy is the ability to do work), heat won’t do any work—it won’t exert a force that moves something—unless it is flowing from something hotter into something colder. And even then, not all of the heat can be used to do work, as we’ll see in what follows.

A device that uses heat energy to do work is called a heat engine. Car engines are examples of heat engines: heat from the burning gasoline is used to exert force that moves the car. In order to function, a heat engine must transfer thermal energy from something hot (called the engine’s high-temperature reservoir) into something cold (called its low-temperature reservoir). For example, a car engine transfers heat from the burning gasoline into the much cooler atmosphere. The engine won’t work if the air around the car is the same temperature as the burning fuel. (If it ever does get that hot outside, of course, engine failures will be the least of our problems!)

Since a heat engine transfers thermal energy from one reservoir into another, not all of the energy from the high-temperature reservoir can be used to do work. Some of that heat must be transferred into the low-temperature reservoir, where it can no longer be used (unless an even lower-temperature reservoir is available). By dumping heat into the low-temperature reservoir, the engine wastes much of the energy that was put into it. This means that the efficiency of a heat engine is limited, as I will explain presently.

The efficiency of an engine is the amount of work it does, divided by the energy it uses:

efficiency =
work you get from the engine
energy you put into the engine

The maximum possible efficiency of a heat engine is determined by the temperatures of the hot and cold reservoirs:

maximum efficiency = 1 -
cold temperature in kelvins
hot temperature in kelvins
What is the maximum possible efficiency of an engine that uses hot and cold reservoirs at 100 °C and 0 °C, respectively? Before using the above equation, we must convert these temperatures to kelvins: 373.15 K and 273.15 K. The maximum possible efficiency for this heat engine is:
1 -
273.15 K
373.15 K
= 0.268 = 26.8%
Not very efficient! And that’s the maximum possible efficiency allowed by the laws of physics. In real life, heat engines tend to be much less efficient than the theoretical maximum.

The only way to get 100% efficiency would be to find a cold reservoir whose temperature is absolute zero (-273.15 °C, or 0 K). Unfortunately, no such thing exists.

All of the above limitations on the transfer of thermal energy are consequences of the second law of thermodynamics. The modern formulation of the second law will be introduced on the next page.