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Ei kT
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where i is the degeneracy of the ith energy level The two ways of dealing with degeneracy are equivalent The method we choose will depend on which is easier to use or more beneficial for the particular situation Summing over energy levels is mathematically simpler, so we will use that method most commonly unless there is a specific benefit to summing over states In the case of the example of a molecule with seven conformational states, the difference between the two methods is the difference between the following two equations: Summing over states, Z = e kT + e kT + e kT + e kT + e kT + e kT + e kT Summing over energy levels, Z = e kT + e kT + 4 e kT + e kT
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E1 E2 E3 E4 E1 E2 E3 E3 E3 E3 E4
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One final note: Sometimes energy levels are referred to as energy states They are the same thing The key is the use of the word energy to distinguish between all possible states of the system, including those states with the same energy, versus only the energy states (ie, levels) of the system and using degeneracy to account for other states of the system In an effort to keep the distinction clear, we will usually use the word levels when talking about energy and the word states when including all configurations of the system (not just energy states, but conformations, vibrational modes, rotational modes, etc) Sometimes, however, it will make sense for us to use the term energy states Just keep in mind that energy states and energy levels mean the same thing
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The Reference State and Relative Energy
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It is often convenient to define a reference state of the system and define all energy levels relative to the energy of the reference state For example, when using statistical mechanics to examine the unwinding of the DNA double helix,
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chapter 5 s tat i s t i c a l M e c h a n i c s
we might define the fully twisted, closed double helix as the reference state All other relevant states of the system then involve partially or fully unwinding the DNA double helix Typically the reference state is chosen as the lowest energy state of the system and is therefore sometimes also called the ground state The energy terms in our statistical mechanical calculations are then always the difference between the energy of a particular state and the ground state Since these involve energy differences, we write them as DEi, where DEi = Ei Eref and Eref is the energy of the reference state One of the advantages of defining a reference state is that we don t have to know Eref We only need to know or measure the energy change in going from one state to the next, something that experimentally is often much simpler to do than measuring the absolute energy of a particular state With a reference state defined this way, we write the partition function as follows: Z = 1+
LMAX i =1
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ie
Ei kT
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Notice the introduction of 1 as the first term in the equation This comes from the fact that the energy difference between the reference state and itself is zero: DEref = Eref Eref = 0, and any number raised to the zeroth power is equal to 1 Notice that we are still counting from i = 1 We do this simply as a matter of convenience by defining the reference energy level as level zero Thus, we can write the energy of the reference state as E0 and DEi = Ei E0 We have also defined the reference state as having no degeneracy (again a choice of convenience), but if it did have degeneracy, we could write Z = 0 +
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