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If I had to describe the purpose of these series as succinctly as possible it would be these three words: to handle complexity. What do I mean by this? Lets suppose we need to describe some function $\mathbb{R}^M\to\mathbb{R}^N$. The cardinality of the set of all such functions is $2^{\beth_1} = 2^{\left(2^{\aleph_0}\right)}$: the cardinality of the ...

1

I do not think Maclaurin series has any special physical significance that is different from the Taylor series. But then for some functions it is desirable to have a series expansion at the origin even though it is singular at the origin. Maclaurin series requires that the function is analytic everywhere within the circle of convergence so it can not be ...

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If the ring is stable it is in equilibrium even if it is not in the center of your force table. In that case, while the forces will be given by the $mg$ values, the angles that you are reading are not the actual vector angles. The ring will move so that the forces balance out. The angles you read on the edge of the table are correct only if the ring is ...

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You forgot to multiply $T \sin{\theta}$ by the distance from the wall to the end of the bar in the torque balance. When you do that, you get an extra factor of 4 in the first term for the expression for x, $x = \frac{8\sin{\theta}}{\mu_s \cos{\theta} + \sin{\theta}} - 2$, which is positive. (PS: I didn't check your math, I just added the factor of 4, so I ...

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The best vacuum we can make is about 10^-12 Pa. Atmospheric pressure is about 10^5 Pa, so it's 17 orders of magnitude lower pressure. The best vacuum recorded is the intergalactic void, at about 10^-17 Pa. Even if you managed to remove all matter, there would still be energy from any light or electric fields.

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Assuming you have a liquid water reservoir available and the liquid is heated along with the gas, water will evaporate to keep the partial pressure of water equal to the vapor pressure of water at the current temperature. As the total gas mix is maintained at a given pressure, the rest of the gases must expand more than Charles' law would indicate, reducing ...

1

What do you mean by temperature? Let's say you have your gas hooked up to a very tiny thermometer. It has a pointer that jiggles and bumps around. What you'd find is that the pointer fluctuations are mostly related to the mechanical evolution of the thermometer itself, not the system under consideration. But that's beside the point. Temperature is not just ...

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Depends what sort of precision you're looking for and the scales of the volumes you are considering. Assuming you mean volumes on the scale of your hand or larger, yes. You can do some really neat things with lasers. However, at some point (getting smaller and smaller) the heisenberg uncertainty principle will start to get in the way.

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The conclusion is correct: for any system containing a single reaction, there exists an equilibrium (with detailed balance) for any values of the rate constants. I'm not sure what's meant exactly by "equilibrium tools," but it's true to say that there's no real difference in the kinetics of an equilibrium or non-equilibrium system if it only contains one ...

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No, existence of equilibrium state is a separate basic assumption of thermodynamics, necessary to formulate both first and second law. The second law stated in terms of entropy talks only about equilibrium states, so they need to be introduced before the 2nd law is derived. More in the papers by Uffink & Brown: ...

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