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But what if the pressure in the balloon increases? Doesn't it make sense that the balloon would want to expand? That is, that as pressure increases, volume increases. This seems to contradict Boyle's Law. In simple words: If you increase the pressure in the balloon and let it expand, then the pressure in the balloon is not really increasing, as you are ...

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The two scenarios you mentioned both are correct, “the pressure $p$ has different sign from other generalized force, if we increase the pressure, the volume increases, whereas if we increase the force, $Y$, for all other cases, the extensive variable, $x$, decreases”.[1] There is no conflict between the two scenarios. [1] L.E.Reichl, A Modern Course in ...

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It is reasonably easy: the balloon will want to maintain pressure equilibrium with its surroundings, i.e. $P_{in} = P_{out}$. This occurs because any pressure imbalance can be redressed on the sound-crossing time scale, i.e. the time it takes a sound wave to cross the balloon's diameter. This can easily be checked to be less than a millisecond, thus on ...

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Ahh, I spent quite some time reading this problem, the problem with applying Dalton's Law of Partial Pressures is that we shouldn't be multiplying moles of $CO2$ with the total Pressure, rather we should multiply the mole fraction of $CO2$ with the total Pressure, in this case however, since the initial quantity/moles of oxygen is not known, it is not ...

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Actually, it does make sense that $\mu \rightarrow - \infty$ Given the ideal chemical potential for in ideal gas: $$\mu = -k_B T\ln \left( \frac{V}{N} \left(\frac{mk_B T}{2 \pi \hbar}\right)^{3/2} \right )$$ so $$\mu \beta \sim - \ln(T) \\ \:\\ \therefore \lim_{T \rightarrow \infty} -\mu \beta >> 1$$

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There are indeed other assumptions in the derivation you quote. Namely, Chandler considers the classical limit of an ideal quantum mechanical gas with average particle number $<N> = \sum_j <n_j> = \sum [ e^{\beta (e_j - \mu)} ± 1 ]^{-1}$ (plus for Fermi-Dirac, minus for Bose-Einstein statistics). In the classical limit (low density) there ...

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