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A little statistical thermodynamics

A little statistical thermodynamics including Boltzmann's distribution before taking a shower. Watch Eann Patterson explain more.
Thermodynamics emerged during the 19th century before everyone was convinced about the existence of atoms. And you can do classical thermodynamics without believing in atoms. However, some insight can be gained if we think about what’s happening inside the matter of a thermodynamic system. This branch of thermodynamics is called statistical thermodynamics. Statistical thermodynamics is about accounting for the bulk properties of matter in terms of its constituent atoms. Statistical because we don’t look at the behaviour of individual atoms but the average behaviour of a myriad of atoms. So if we think about a gas inside a container. The pressure on the wall of the container is created by a whole series of impacts of atoms against the wall.
It’s a little bit like in the shower.
So when you stand underneath the shower, you don’t feel the individual impacts of little needles of water coming out of the shower head but the overall force exerted by the water on the back of your head. And so the same sort of idea is used in accounting for the impact of atoms inside the container on its wall to create the pressure. So we take a statistical approach by averaging the behaviour of all the little impacts. In order to take this idea a little bit further, we need to do a little bit of quantum mechanics.
In simple terms, we can think about atoms only existing at certain energy levels, then a collection of atoms will consist of some at the lowest energy state, which we call the ground state, and some in the next higher energy state and so on with diminishing numbers in the higher states. When the atoms settle into an equilibrium population– although they’ll always be a few jumping between energy states but creating no net change in the population distribution– that equilibrium distribution can be found from a knowledge of the energy states and a single parameter that we call beta. This distribution of atoms over the allowed energy states is known as the Boltzmann distribution named after Ludwig Boltzmann, who lived from 1844 to 1906.
The ratio of the populations of the energy state, E relative to the ground state 0 is given by the exponential of minus beta times E, the value of the energy state. And it turns out that beta is equal to 1/kT, where k is called Boltzmann’s constant, and T is the temperature in Kelvin. k is a very, very small number. It’s 1.38 times 10 to the minus 23 joules per degree Kelvin, so that means the ratio of the energy states is equal to the exponential of minus e/kT.
And so that means that for higher energy states, the population decreases exponentially and that temperature tells us the most probable population distribution of atoms over the available energy states. So high temperatures, low beta, many states have significant populations. But at low temperatures and high beta, only the lower states have significant populations of atoms.
While getting ready for a shower, Eann uses the impact of water in the shower as an analogy for some basic quantum mechanics and introduces Boltzmann’s distribution.
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Energy: Thermodynamics in Everyday Life

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