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er of equally probable val sibilities on your whiteboa rrower. For the very large e value of a measuremer lition th e size of pesak litic the size the

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Please solve 3 and 4 only in 30 minutes
For the types of systems that we will examine in thermodynamics, there are large numbers of particles in the system (~Avogadro's number of particles) each with very large numbers of possible states (1020 possible values). We can look at the list of possible arrangements on your whiteboard as a plot of a number of possible microstates (horizontal) versus the macrostates (vertical). As the number of equally probable values grows for each of many particles, the peak that you see in the possibilities on your whiteboard gets larger and larger, and at the same time it gets narrower and narrower. For the very large numbers in real thermodynamic systems, the peak that represents the value of a measurement is narrower than any normal experimental error can detect. In addition, the size of the peak approaches a probability of 100%. Now that we see the role of many different microstates of a system resulting in the same value for a macrostate measurement, we will look at how this might cause a particular outcome for a real problem. We will now look at an example from D. Schroeder's book, Thermal Physics. Imagine a system that has three atomic-size harmonic oscillators. In PHYS121, we learned that these oscillators have energies of AE = hf = hw/2TT = ħw where ħ = 1.05 x 10-34 J.s You can draw the energy as a ladder with equally spaced rungs the lowest being at E = 0, the next at E = ħw, the next at E = 2hw, and so on. 1) On a scratchpad draw three ladders (side-by-side) with E0 = 0, E1 = ħw, E2 = 2ħw, E3 = 3ħw as the steps for each ladder. If we had the box of oscillators with zero total energy, there is only one way for this to occur. We can use a set of ordered triplets (osc1, osc2, osc3) to list the various ways that the energy one add up to zero in these oscillators. The total energy of this system would be (0, 0, 0). See how that works with your scratchpad ladders. 2) Now, if the box had a total energy of one unit, how many different ways could this occur in the set of three oscillators in our box? Use the ordered triplets and list below the various ways that this could occur. Notice how that might look with your scratchpad ladders. 3) Now, if the box had a total energy of two units, how many different ways could this occur in the set of three oscillators in our box? Use the ordered triplets and list below the various ways that this could occur. Notice how that might look with your scratchpad ladders. 4) Finally, if the box had a total energy of three units, how many different ways could this occur in the set of three oscillators in our box? Use the ordered triplets to list the various ways that this could occur.
Transcribed Image Text:For the types of systems that we will examine in thermodynamics, there are large numbers of particles in the system (~Avogadro's number of particles) each with very large numbers of possible states (1020 possible values). We can look at the list of possible arrangements on your whiteboard as a plot of a number of possible microstates (horizontal) versus the macrostates (vertical). As the number of equally probable values grows for each of many particles, the peak that you see in the possibilities on your whiteboard gets larger and larger, and at the same time it gets narrower and narrower. For the very large numbers in real thermodynamic systems, the peak that represents the value of a measurement is narrower than any normal experimental error can detect. In addition, the size of the peak approaches a probability of 100%. Now that we see the role of many different microstates of a system resulting in the same value for a macrostate measurement, we will look at how this might cause a particular outcome for a real problem. We will now look at an example from D. Schroeder's book, Thermal Physics. Imagine a system that has three atomic-size harmonic oscillators. In PHYS121, we learned that these oscillators have energies of AE = hf = hw/2TT = ħw where ħ = 1.05 x 10-34 J.s You can draw the energy as a ladder with equally spaced rungs the lowest being at E = 0, the next at E = ħw, the next at E = 2hw, and so on. 1) On a scratchpad draw three ladders (side-by-side) with E0 = 0, E1 = ħw, E2 = 2ħw, E3 = 3ħw as the steps for each ladder. If we had the box of oscillators with zero total energy, there is only one way for this to occur. We can use a set of ordered triplets (osc1, osc2, osc3) to list the various ways that the energy one add up to zero in these oscillators. The total energy of this system would be (0, 0, 0). See how that works with your scratchpad ladders. 2) Now, if the box had a total energy of one unit, how many different ways could this occur in the set of three oscillators in our box? Use the ordered triplets and list below the various ways that this could occur. Notice how that might look with your scratchpad ladders. 3) Now, if the box had a total energy of two units, how many different ways could this occur in the set of three oscillators in our box? Use the ordered triplets and list below the various ways that this could occur. Notice how that might look with your scratchpad ladders. 4) Finally, if the box had a total energy of three units, how many different ways could this occur in the set of three oscillators in our box? Use the ordered triplets to list the various ways that this could occur.
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