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Chapter 2: Q. 2.15 (page 63)
Use a pocket calculator to check the accuracy of Stirling's approximation for . Also check the accuracy of equation for .
By using pocket calculator as,
and
By using Stirling's approximation as,
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Consider again the system of two large, identical Einstein solids treated in Problem .
(a) For the case , compute the entropy of this system (in terms of Boltzmann's constant), assuming that all of the microstates are allowed. (This is the system's entropy over long time scales.)
(b) Compute the entropy again, assuming that the system is in its most likely macro state. (This is the system's entropy over short time scales, except when there is a large and unlikely fluctuation away from the most likely macro state.)
(c) Is the issue of time scales really relevant to the entropy of this system?
(d) Suppose that, at a moment when the system is near its most likely macro state, you suddenly insert a partition between the solids so that they can no longer exchange energy. Now, even over long time scales, the entropy is given by your answer to part (b). Since this number is less than your answer to part (a), you have, in a sense, caused a violation of the second law of thermodynamics. Is this violation significant? Should we lose any sleep over it?
A black hole is a region of space where gravity is so strong that nothing, not even light, can escape. Throwing something into a black hole is therefore an irreversible process, at least in the everyday sense of the word. In fact, it is irreversible in the thermodynamic sense as well: Adding mass to a black hole increases the black hole's entropy. It turns out that there's no way to tell (at least from outside) what kind of matter has gone into making a black hole. Therefore, the entropy of a black hole must be greater than the entropy of any conceivable type of matter that could have been used to create it. Knowing this, it's not hard to estimate the entropy of a black hole.
Use dimensional analysis to show that a black hole of mass should have a radius of order , where is Newton's gravitational constant and is the speed of light. Calculate the approximate radius of a one-solar-mass black hole .
In the spirit of Problem , explain why the entropy of a black hole, in fundamental units, should be of the order of the maximum number of particles that could have been used to make it.
To make a black hole out of the maximum possible number of particles, you should use particles with the lowest possible energy: long-wavelength photons (or other massless particles). But the wavelength can't be any longer than the size of the black hole. By setting the total energy of the photons equal to , estimate the maximum number of photons that could be used to make a black hole of mass . Aside from a factor of , your result should agree with the exact formula for the entropy of a black hole, obtained* through a much more difficult calculation:
Calculate the entropy of a one-solar-mass black hole, and comment on the result.
Consider a two-state paramagnet with elementary dipoles, with the total energy fixed at zero so that exactly half the dipoles point up and half point down.
(a) How many microstates are "accessible" to this system?
(b) Suppose that the microstate of this system changes a billion times per second. How many microstates will it explore in ten billion years (the age of the universe)?
(c) Is it correct to say that, if you wait long enough, a system will eventually be found in every "accessible" microstate? Explain your answer, and discuss the meaning of the word "accessible."
The mathematics of the previous problem can also be applied to a one-dimensional random walk: a journey consisting of steps, all the same sic, cache chosen randomly to be cither forward or backward. (The usual mental image is that of a drunk stumbling along an alley.)
(a) Where are you most likely to find yourself, after the end of a long random walk?
(b) Suppose you take a random walk of steps (say each a yard long). About how far from your starting point would you expect to be at the end?
(c) A good example of a random walk in nature is the diffusion of a molecule through a gas; the average step length is then the mean free path, as computed in Section Using this model, and neglecting any small numerical factors that might arise from the varying step size and the multidimensional nature of the path, estimate the expected net displacement of an air molecule (or perhaps a carbon monoxide molecule traveling through air) in one second, at room temperature and atmospheric pressure. Discuss how your estimate would differ if the clasped time or the temperature were different. Check that your estimate is consistent with the treatment of diffusion in Section
Consider an ideal monatomic gas that lives in a two-dimensional universe ("flatland"), occupying an area instead of a volume . By following the same logic as above, find a formula for the multiplicity of this gas, analogous to equation .
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