Новое решение
en/5.8.4.md
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| + | ### Statement | ||
| + | |||
| + | $5.8.4.$ [Insert the problem statement] | ||
| + | |||
| + | ### Solution | ||
| + | |||
| + |  | ||
| + | |||
| + | What should happen is the same number of molecules that are already in $V=1 mm^3$ should appear in $V.$ | ||
| + | Total of molecules in the room (my bathroom after i take a shower $:)$ ): | ||
| + | $$N=N_{A} \cdot \frac{PV_0}{RT}\approx N_{A} \cdot \frac{10^5 \cdot 10}{8.3 \cdot 300}\approx10^{26}$$ | ||
| + | We need:$$ n=N\cdot \frac{V}{V_0} \approx 10^{16}$$Probability of existing of $n$ molecules is $1$. | ||
| + | Probability of existing in $V$ for $1$ molecule in $p= \frac{V}{V_0}.$ | ||
| + | |||
| + | So final answer is | ||
| + | $$ P= (\frac{V}{V_0})^n=(\frac{V}{V_0})^{N \cdot \frac{V}{V_0}}=x^{Nx}, x=\frac{V}{V_0}$$In our case $\frac{V}{V_0}=10^{-10}$ and $P=10^{-10^{16}}$, which is close to Savchenko's estimation. The upper power depends strongly on temperature and volume. | ||
| + | |||
| + | We can plot $P(x)$ and see that $P=10^{-4}% $ if $x=\frac{V}{V_0}\approx10^{-27}$, so | ||
| + | $V=10^{-17} mm^3$ | ||
| + | |||
| + | Intereating to notice, that with decreasing of $x$, $P$ goes to $1$. | ||
| + | |||
| + | Explanation in terms of our model is that if we consider a volume less than length of free fly of molecule, there should be all pairs of colliding molesules, so double density. | ||
| + | |||
| + | But much more truthful is the next fact: | ||
| + | |||
| + | The key postulate of thermodynamix say that we consider a system that has many enough partickles, so fluctuations of $n$ are negligible. But with decreasing of volume, we can no longer have this fact true. | ||
| + | |||
| + | #### Answer | ||
| + | |||
| + | [Insert a concise answer or boxed result] | ||
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| ### Statement | |||
| $5.8.4.$ [Insert the problem statement] | |||
| ### Solution | |||
|  | |||
| What should happen is the same number of molecules that are already in $V=1 mm^3$ should appear in $V.$ | |||
| Total of molecules in the room (my bathroom after i take a shower $:)$ ): | |||
| $$N=N_{A} \cdot \frac{PV_0}{RT}\approx N_{A} \cdot \frac{10^5 \cdot 10}{8.3 \cdot 300}\approx10^{26}$$ | |||
| We need:$$ n=N\cdot \frac{V}{V_0} \approx 10^{16}$$Probability of existing of $n$ molecules is $1$. | |||
| Probability of existing in $V$ for $1$ molecule in $p= \frac{V}{V_0}.$ | |||
| So final answer is | |||
| $$ P= (\frac{V}{V_0})^n=(\frac{V}{V_0})^{N \cdot \frac{V}{V_0}}=x^{Nx}, x=\frac{V}{V_0}$$In our case $\frac{V}{V_0}=10^{-10}$ and $P=10^{-10^{16}}$, which is close to Savchenko's estimation. The upper power depends strongly on temperature and volume. | |||
| We can plot $P(x)$ and see that $P=10^{-4}% $ if $x=\frac{V}{V_0}\approx10^{-27}$, so | |||
| $V=10^{-17} mm^3$ | |||
| Intereating to notice, that with decreasing of $x$, $P$ goes to $1$. | |||
| Explanation in terms of our model is that if we consider a volume less than length of free fly of molecule, there should be all pairs of colliding molesules, so double density. | |||
| But much more truthful is the next fact: | |||
| The key postulate of thermodynamix say that we consider a system that has many enough partickles, so fluctuations of $n$ are negligible. But with decreasing of volume, we can no longer have this fact true. | |||
| #### Answer | |||
| [Insert a concise answer or boxed result] | |||