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Chapter 12: 29P (page 509)
Suppose that the entropy of a certain substance (not anEinstein solid) is given by, where ais a constant. Whatis the energy Eas a function of the temperature T?
The energy as a function of the temperature can be given as, .
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List explicitly all the ways to arrange 2 quanta among 4 one-dimensional oscillators.
The reasoning developed for counting microstates applies to many other situations involving probability. For example, if you flip a coin 5 times, how many different sequences of 3 heads and 2 tails are possible? Answer: 10 different sequences, such as HTHHT or TTHHH. In contrast, how many different sequences of 5 heads and 0 tails are possible? Obviously only one, HHHHH, and our equation gives , using the standard definition that 0! is defined to equal 1.
If the coin is equally likely on a single throw to come up heads or tails, any specific sequence like HTHHT or HHHHH is equally likely. However, there is only one way to get HHHHH, while there are 10 ways to get 3 heads and 2 tails, so this is 10times more probable than getting all heads. Use the expressionto calculate the number of ways to get 0 heads, 1 head, 2 heads, 3 heads, 4 heads, or 5 heads in a sequence of 5 coin tosses. Make a graph of the number of ways vs. the number of heads.
Approximately what fraction of the sea-level air density is found at the top of Mount Everest, a height of 8848 m above sea level?
Figure 12.57 shows a one-dimensional row of 5 microscopic objects each of mass , connected by forces that can be modeled by springs of stiffness 15 N/m. These objects can move only along the x axis.
(a) Using the Einstein model, calculate the approximate entropy of this system for total energy of 0, 1, 2, 3, 4, and 5 quanta. Think carefully about what the Einstein model is, and apply those concepts to this one-dimensional situation. (b) Calculate the approximate temperature of the system when the total energy is 4 quanta. (c) Calculate the approximate specific heat on a per-object basis when the total energy is 4 quanta. (d) If the temperature is raised very high, what is the approximate specific heat on a per-object basis? Give a numerical value and compare with your result in part (c).
In order to calculate the number of ways of arranging a given amount of energy in a tiny block of copper, the block is modeled as containing independent oscillators. How many atoms are in the copper block?
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