2.4

## University of Liverpool

Skip to 0 minutes and 4 secondsRemember last week, I talked about how thermodynamics was developed before everybody believed in the existence of atoms. But it can be informative to use statistical thermodynamics to think about what's happening inside matter. And I talked about Boltzmann's distribution, which is the distribution of atoms over the allowed energy states. Boltzmann distribution is that the ratio of the population of the atoms at energy state E relative to the population of the ground state is equal to the exponential of minus beta times E where E indicates the energy state we're interested in. Turns out that beta is equal to 1 upon kT where T is temperature in Kelvin, and k is Boltzmann's constant.

Skip to 0 minutes and 52 secondsSo that means that the ratio of the populations of the energy states is equal to the exponential of minus E over kT. k is a tiny number, 1.38 times 10 to the minus 23 joules per Kelvin. And so that means at high energy states, the population decreases exponentially. And temperature tells us the most probable population distribution of atoms over the available energy states. So at high temperatures, that's low beta, many states will have significant populations of atoms. And at low temperatures, that's high beta, only the lower energy states will have significant populations of atoms and molecules.

Skip to 1 minute and 38 secondsSo if we consider a swarm of molecules moving at different speeds in different directions, such as those coming out of the kettle here as it boils, then the speed is related to the kinetic energy or translational energy. And the Boltzmann distribution describes the distribution of those translational energies. We can relate the speed of the molecules to temperature, using the Maxwell-Boltzmann distribution. Thus temperature is a measure of the average molecular speed. So, for instance, on a warm day, say, 25 degrees centigrade, the average speed of the molecules in the air will be about 4% higher than on a cold day at, say, 0 degrees centigrade.

Skip to 2 minutes and 25 secondsIf we think about gas trapped in a container rather than free to billow around in the air, as the steam coming out the kettle was, then at a higher temperature, the molecules will collide more often and faster with the walls of the container. And that'll lead to higher pressures. In an ideal gas, that pressure will be equal to the density of the fluid times the gas constant times temperature. So pressure p is equal to rho RT. So let's recap for a moment. We can think of energy being stored as translational energy of atoms and molecules, according to the Boltzmann distribution.

Skip to 3 minutes and 5 secondsThe stored energy is known as the internal energy of the material of the matter, and we can use temperature as a measure of internal energy. The ability of matter to absorb and store energy is known as its heat capacity, and so if we plot that stored or internal energy against temperature, then heat capacity is the slope of the graph.

Skip to 3 minutes and 30 secondsIf we go back to our fluid trapped in a container, if the system is free to expand - perhaps we've got a frictionless piston at one end of the container so we can keep the pressure constant - then some of the energy we transferred into the container as heat will be used for the expansion to push the piston out, and we'll need more energy to achieve the same temperature rise. And so the heat capacity at constant pressure is always greater than the heat capacity at constant volume. Water has a high heat capacity. One joule of energy delivered as heat raises one gramme of water by just one fifth of a degree centigrade.

Skip to 4 minutes and 11 secondsAnd so that means it can absorb an awful lot of energy, and that makes it useful for heating systems. It's also why the sea is very slow to warm up in spring, and slow to cool down in autumn or the fall. So our oceans can act as a massive energy storage system, and that has a major impact on our weather systems. To read more about that, look at the next article on the storm in a computer.

# Natural energy storage

In his kitchen, Eann discusses some basic concepts of statistical thermodynamics including energy storage in matter, temperature as a measure of internal energy and heat capacity.