 We've started talking about the partition function, which remember is the sum of all the Boltzmann factors, the sum of eta minus energy over kt for all the energies that the system can have. When we go to implement this in practice, it's pretty common to find that multiple states will have the same energy as each other. For example, we've seen most recently for butane, if we draw an energy ladder, the anti-state has a lower energy than the Gauss states, but there's two Gauss states, Gauss plus and Gauss minus that have the same energy as each other, so those two states have the same energy. Another example would be, for example, the hydrogen atom, where I can take electrons in a hydrogen atom and I can put them in the 1s orbital, the 2s orbital, the 2p orbital and so on. Under ordinary circumstances, the energy of an electron, the 2px orbital, the 2py orbital, the 2pz orbital, those energies are all the same, so energy of the 2px and 2py and 2pz orbitals are the same as each other, so a couple of examples of how sometimes the energies turn out to be exactly the same as each other, usually for symmetry reasons like the three identical p orbitals with different orientations or the two equivalent conformations of butane that are rotated in opposite directions. So what this means is when we go to write out the partition function, let's take the butane example. We have two ways to do it. If I take this expression literally, the sum of e to the minus energy over kt for all the different states, the states are anti and Gauss plus and Gauss minus, that partition function looks like e to the minus e anti over kt plus a Boltzmann factor for the Gauss plus state, e to the minus e Gauss over kt, a different Boltzmann factor for the Gauss minus state, and because the energies are the same, they both look like e to the minus e Gauss over kt, the energy for the two Gauss states are the same as each other. So that's taking this expression literally, writing out one term for each state of the system, but of course algebraically, this term and this term are the same, so we could also say the partition function is e to the minus e anti over kt plus twice e to the minus e Gauss over kt. Of course those two expressions are the same, doesn't matter which one we use, we'll get the same result for the partition function. We run into that situation fairly often, so these two different ways of writing the partition function, the first one, I'll just rewrite this expression. If we take the sum of a Boltzmann factor for each state of the system, let me emphasize here that we're summing over all the different states that the system can have, Gauss plus and Gauss minus being two different states, or if we think about a sum that would correspond to that second way of writing things, if I want to sum e to the minus energy over kt, sometimes having a pre-factor in front, what this factor means is there's a two here because there's two states that have that energy, there's a one here because there's only one state with that energy e anti. So if I write this term g sub e, that's a quantity we'll call the degeneracy. If a state is degenerate, meaning if there's more than one state that has the same energy, if that energy level is degenerate with two different states having the same energy level, then I can write just e to the energy, don't worry about an index for the different states and just multiply that energy, the Gauss energy gets multiplied by two, and now instead of summing over all the states of the system, I'm just going to sum over all the possible energies that the system can have. So this is a little bit different in concept than this sum, we can sum over all the states and include each of them once, even if the energies are the same as each other, or I can sum over energy levels, the entire energy level and the Gauss energy level only do the sum over the two energy levels and include a coefficient to include how many states that are at each energy level. So that's this idea of degeneracy. The degeneracy is the number of states that have the same energy, and I could say that a system is degenerate or has degeneracy if there are states that have multiple states that have the same energy as each other. So that's important to introduce this idea of degeneracy so that we can use either one of these expressions for describing the partition function as we move ahead, and sometimes this one will be more convenient, sometimes this one will be more convenient, but now we have the ability to think about either one of them when we want to.