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## University of Liverpool

Skip to 0 minutes and 3 seconds In the depths of an Antarctic winter, you can find male adult emperor penguins huddle together on the open ice, in colonies of several hundred birds. Each penguin has a precious single egg, balanced on its feet, and sheltered by a flap of fat. In the Antarctic winter, the temperature can drop as low as minus 60 degrees centigrade, with wind speeds of 200 kilometres per hour. The penguins huddle very close together, with up to 10 of these 40 kilogramme birds in a square metre. If we take penguins as an analogy for atoms or molecules, then this is a very ordered, and hence, low entropy state. We have a good chance of guessing where to find a particular penguin.

Skip to 1 minute and 0 seconds In spring, when the temperature rises, the penguins begin to shuffle away from the huddle, and the probability of being able to locate a particular penguin decreases.

Skip to 1 minute and 12 seconds This corresponds to a less ordered, high entropy state. In summer, when the temperature rises further, to about 10 degrees on the coast on the sea ice melts, the penguins disperse as they hunt for fish, and we have a very small chance of finding a particular penguin. This summer dispersion corresponds to a disordered state with high entropy. So we can begin to see that there might be a connection between entropy and the Boltzmann distribution, which describes the distribution of molecules over energy states, or perhaps even the location of Emperor Penguins. At higher temperatures, molecules have distributed over more energy states, just as in summer, penguins are distributed far and wide in the Antarctic Ocean.

Skip to 2 minutes and 7 seconds Ludwig Boltzmann, who was born in 1844 in Vienna, Austria, realised this connection and expressed it in his famous equation. S, for entropy, is equal to k, for Boltzmann’s constant, multiplied by the logarithm of W, where W is a measure of the number of ways in which the molecules can be arranged to achieve the same total energy content. Boltzmann made this equation famous by having it carved on his gravestone when he died in September 1906, nearly 110 years ago. If we take the temperature even lower than in an Antarctic winter, to absolute zero, then all molecules will organise themselves into a single arrangement, known as the ground state. So in this state, the number of possible arrangements, W, equals one.

Skip to 3 minutes and 7 seconds And the logarithm of one is zero. So the entropy of matter at absolute zero is zero, according to Boltzmann, because S equals k times the logarithm of one, which is k times zero. This means we can be absolutely certain of guessing the energy state of the molecule at absolute zero temperature. This seems all very neat and tidy. Unfortunately, a few substances have more than one ground state due to molecular rotations that can occur. These are known as degenerates and have a residual non-zero entropy at absolute zero temperature. Examples include water and carbon dioxide. Boltzmann’s constant, k, is the same one that we met earlier in the course in week one, in the video called, “A Little Statistical Thermodynamics.”

Skip to 4 minutes and 7 seconds Boltzmann’s constant is pretty small. It’s only 1.38 times 10 to the minus 23 joules per Kelvin. It turns out that it’s the ratio of the molecular gas constant and Avogadro’s number.

# Penguins and entropy

Ludwig Boltzmann realised the link between entropy and probability which he encapsulated in an equation that is engraved on his tomb. Eann uses the behaviour of Emperor penguins in Antarctica as an analogy to explain the ideas.