Compute the entropy of a mole of helium at room temperature and atmospheric pressure, pretending that all the atoms are distinguishable. Compare to the actual entropy, for indistinguishable atoms, computed in the text.
The actual entropy indistinguishable atoms is
The Sackur-Tetrode formula by ideal gas is
Where,represents volume, represents energy, represents the number of molecules, represents the mass of a single molecule, and represents Planck's constant.These are some of the assumptions used to generate this formula is that the molecules are indistinguishable, therefore altering any of the molecules makes no change in any arrangement of the molecules in position and momentum space. This assumption inserts the component into the multiplicity function's denominator.
The logarithm factor loses its , we get
The mole mass of helium is ,the mass of helium molecule is
From ideal gas law, the pressure of and temperature of one mole occupies a volume of
The monatomic gas of internal energy is
Helium is monatomic gas so .
By degree of freedom,
Substituting the values of , we get
Because there are many more molecular orbitals accessible to the system if the molecules are distinct, the entropy is substantially larger.
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.
Rather than insisting that all the molecules be in the left half of a container, suppose we only require that they be in the leftmost (leaving the remaining completely empty). What is the probability of finding such an arrangement if there are molecules in the container? What if there are molecules? What if there are ?
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