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I have three vessels shown in the figure. The ground area of each vessel is A. The vessels are in diathermic contact with each other by the disk on which they are mounted. Each vessel is closed by a piston that is mounted to the long arm with a distance from the center of the long arm shown in the figure. Last there is no external pressure acting on the vessels. I should calculate the ratio of the pressures and the temperatures inside the vessels.

What I would do now is to assume some angle $\alpha$ representing the angle of the envelope of the long arm to which the pistons are mounted. The next thing I would do is to assume that the distances from the pistons to the long arm are constant and out of this I would compute the accessible volumes of the vessels what would tell me the pressure ratios and temperature ratios. But my feeling tells me that this is wrong. Can anybody give me a hint a to consider this problem or how to start??

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  • $\begingroup$ What is the fluid in each vessel and how much of this fluid do each of them contain? $\endgroup$ – Chet Miller Mar 18 '18 at 12:30
  • $\begingroup$ @ChesterMiller The vessels are filled with some gas. Which is not stated only that it can vary from vessel to vessel. How much each vessel contains is also not given $\endgroup$ – zodiac Mar 18 '18 at 12:43
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So I figured out an answer now. Since the question can only be answered if the System is in thermal equilibrium because otherwise the long arm would fluctuate until it reaches equilibrium. Therefore the temperatures $T_{1}$, $T_{2}$ and $T_{3}$ have to be the same. This is because the vessels are able to exchange heat and they will do this until the temperatures are the same in all of them. The ratio of pressures can be obtained by the lever rule. That states balance of torques: $\sum_{i=1}^{N} r_{i}F_{i}=0$ with F being the forces and r the distance from the mounting. For the considered system the following is then true: $$r_{2}F_{2}+r_{1}F_{1}=r_{3}F_{3}.$$ By inserting the values of the figure and dividing by the area A that is constant for all the vessels: $$2P_{2}+P_{1}=3P_{3}.$$ And this equation determines the relation between their pressures.

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