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Chapter 2: Q. 2.14 (page 62)
Write in the form , for some
The value in the form as
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Problem . Suppose you were to shrink Figureuntil the entire horizontal scale fits on the page. How wide would the peak be?
For either a monatomic ideal gas or a high-temperature Einstein solid, the entropy is given by times some logarithm. The logarithm is never large, so if all you want is an order-of-magnitude estimate, you can neglect it and just say . That is, the entropy in fundamental units is of the order of the number of particles in the system. This conclusion turns out to be true for most systems (with some important exceptions at low temperatures where the particles are behaving in an orderly way). So just for fun, make a very rough estimate of the entropy of each of the following: this book (a kilogram of carbon compounds); a moose of water ; the sun of ionized hydrogen .
According to the Sackur-Tetrode equation, the entropy of a monatomic ideal gas can become negative when its temperature (and hence its energy) is sufficiently low. Of course this is absurd, so the Sackur-Tetrode equation must be invalid at very low temperatures. Suppose you start with a sample of helium at room temperature and atmospheric pressure, then lower the temperature holding the density fixed. Pretend that the helium remains a gas and does not liquefy. Below what temperature would the Sackur-Tetrode equation predict that is negative? (The behavior of gases at very low temperatures is the main subject of Chapter .)
Use a computer to plot formula directly, as follows. Define =, so that =. Then, aside from an overall constant that we'll ignore, the multiplicity function is , where ranges from toand the factor of ensures that the height of the peak is equal to for any . Plot this function for=, and . Observe how the width of the peak decreases asincreases.
For an Einstein solid with four oscillators and two units of energy, represent each possible microstate as a series of dots and vertical lines, as used in the text to prove equation .
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