Question

Consider again the system of two large, identical Einstein solids treated in Problem $2.22$.

(a) For the case $N={10}^{23}$, 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?

Short answer

(a) The entropy of this method , assuming that each one of the microstates are allowed $S_{\text{total}}=3.826\mathrm{J}\cdot {\mathrm{K}}^{-1}$

(b) The entropy again, assuming that the system in its presumably macro state $S_{mp}=3.826\mathrm{J}\cdot {\mathrm{K}}^{-1}$

(c) The entropy is given by , but over while scales it's given by which is ever slightly larger than .$S_{mp}$

(d) Placea superb excellent resistance between two surfaces, keeping them from transmitting energy. In effect, this shows that the model is stuck in its current condition.

Step-by-step solution

Einstein Spheres (a)

(a) Suppose two big, similar Einstein spheres, every with $N$clocks and $q=2N$amount of energy. We had (from issue $2.22$) for the state during which all business can make are able:

$\begin{array}{c}{\mathrm{\Omega }}_{\text{total}}\approx \frac{2^{4N}}{\sqrt{8\pi N}}\end{array}$

This complete agency's volatility is:

$\begin{array}{c}{S}_{\text{total}}=k\ln \left({\mathrm{\Omega }}_{\text{total}}\right)\\ \end{array}$

If we modify ${\mathrm{\Omega }}_{\text{total}}$with , we get:

$S_{\text{total}}=k\ln \left(\frac{2^{4N}}{\sqrt{8\pi N}}\right)$

$\ln \left(ab\right)=\ln \left(a\right)+\ln \left(b\right),\ln \left(\frac{a}{b}\right)=\ln \left(a\right)-\ln \left(b\right)\text{and}\ln \left(a^b\right)=b\ln \left(a\right)$

$S_{\text{total}}=k\left[4N\ln \left(2\right)-\frac{1}{2}\ln \left(8\pi N\right)\right]$

$k=1.38\times {10}^{-23}\mathrm{J}\cdot {\mathrm{K}}^{-1}\text{and}N={10}^{23}\text{:}$

$\begin{array}{c}{S}_{\text{total}}=\left(1.38\times {10}^{-23}\right)\left[4\times {10}^{23}\times \ln \left(2\right)-\frac{1}{2}\times \ln \left(8\pi \times {10}^{23}\right)\right]\\ \end{array}$

$S_{\text{total}}=3.826\mathrm{J}\cdot {\mathrm{K}}^{-1}$

Two solids (b)

(b) Its presumably situation, that the energy is spread evenly between the $2$ solids, had a redundancy (from problem $2.22$):

$\begin{array}{c}{\mathrm{\Omega }}_{mp}\approx \frac{2^{4N}}{4\pi N}\end{array}$

The volatility of its most frequent state was even as follows:

$\begin{array}{c}{S}_{mp}=k\ln \left({\mathrm{\Omega }}_{mp}\right)\\ \end{array}$

Assuming we swap ${\mathrm{\Omega }}_{mp}$, we get:

$S_{mp}=k\ln \left(\frac{2^{4N}}{4\pi N}\right)$

$S_{\text{total}}=k[4N\ln (2)-\ln (4\pi N\left)\right]$

$k=1.38\times {10}^{-23}\mathrm{J}\cdot {\mathrm{K}}^{-1}\text{and}N={10}^{23}$

$\begin{array}{c}{S}_{mp}=\left(1.38\times {10}^{-23}\right)\left[4\times {10}^{23}\times \ln \left(2\right)-\times \ln \left(4\pi \times {10}^{23}\right)\right]\\ \end{array}$

$S_{mp}=3.826\mathrm{J}\cdot {\mathrm{K}}^{-1}$

Hence, whether object is viewed as either an unified location with all feasible microstates or as $2$ independent systems for his or her most plausible shape, the energy is virtually defined solely more by $2^{4N}$variable.

Entropy (c) and (d)

(c) The entity is liberal to pursue each one of the microstates included in $S_{\text{total}}$, over longer timeframes, yet it's extremely likely to be in or near its most frequent state at any given time. Thus, we may claim that the entropy is supplied by $S_{mp}$ over small scales, while it's given by $S_{\text{total}}$ very long time scales, which has become so marginally bigger than $S_{mp}$.

(d) Think about a situation within which we elect some extent t so when system is in its possibly efficient than one (i.e., the entropy is $S_{mp}$) and so place a wonderful resistance between two surfaces, keeping them from transmitting energy. In effect, this shows that the model is stuck in its current condition.