We are going to estimate the spacing between states for a system of 1028 air molecules in a room that is 5 meters across. We will use the fact that a particle confined to a certain region of space is represented by a standing wave, and only certain wavelengths fit. The momentum of an air molecule is given by h/λ and its energy by (1/2m)p2, where m = 4.8 × 10−26 kg.
(a) What is the smallest possible energy of an air molecule in this room? (Hint: What is the longest possible wavelength?)
(b) What is the next smallest possible energy?
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(c) Suppose we were to add a tiny amount of energy to the air and that this added energy just happened to be taken by one of the slowest moving molecules in the room. What is the smallest amount of energy we could add?