GREPhysics.NET: GR9277 #73

GR9277 #73

Problem

GREPhysics.NET Official Solution

\prob{73}
A system in thermal equilibrium at temperature T consists of a large number $N_0$ of subsystems, each of which can exist only in two states of energy $E_1$ and $E_2$ , where $E_2-E_1=\epsilon \gt 0$ . In the expressions that follow, k is the Boltzmann constant.

Which of the following is true of the entropy of the system?

It increases without limit with T from zero at T=0.
It decreases with increasing T.
It increases from zero at T=0 to $N_0k\ln 2$ at arbitrarily high temperatures.
It is given by $N_0k\left(\frac{5}{2}\ln T - \ln \rho + constant\right)$
It cannot be calculated from the information given.

Statistical Mechanics $\Rightarrow$ }Entropy

The third law of thermodynamics says that $S(T\rightarrow 0) \rightarrow 0$ .

Also, the statistical definition of entropy is just $S = Nk ln Z$ , where Z is the partition function. For this problem, one has $Z=e^{-\epsilon/kT}+e^{-2\epsilon/kT}$ . For high temperatures, one has $Z \rightarrow 1 + 1 = 2$ , since $e^{x}\approx 1+x$ for small x (and then $1+x \approx 1$ for very small x).

Thus the entropy behaves as in choice (C).

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Comments

Poop Loops 2008-10-25 17:38:11	You can also look at this conceptually. Remembering that $S \right 0$ as $T \right 0$ , you already eliminate all but 2 choices. It's tempting to think that Entropy can increase indefinitely, but think about what's going on here. Say this system is a perfect gas. As temperature increases, so does entropy because everything is getting more and more disordered. But, at really high temperatures, every particle is already zooming around everywhere really fast, so you can't really get more disorder. Reply to this comment
Imperate 2008-10-16 12:54:18	A simple way to see that C is the correct answer is that at low temps all the subsystems reduce into the lowest state E_1, and thus the number of accessible microstates of the whole system is just one at abs zero (so S=klnW=kln(1)=0). But at arb high temps, each system is equally likely to be in E_1 or E_2, so you have a two level system (just like the isolated spin-1/2 paramagnet at high temps), and to find the microstates of the whole system it's like finding how many ways to arrange O's and X's on a checkboard with N places. Thus there are W=2^N microstates at high temps, and entropy is S=kln2^N=kNln2. Reply to this comment
sharpstones 2007-03-26 18:39:34	Not to be too nitty gritty but $S = \frac{\partial}{\partial T}(Nk T ln Z)$ . In our case at high T where Z = 2 then it reduces to the right answer. Reply to this comment

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