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Here's a start: Write the hydrostatic equation as dP = - p g dz write the ideal gas law as P = p K T / u M Divide one by the other to get dP / P = (- u M g / K T) dz (the p's cancel out) Now integrate both sides. Note that this assumes that g does not vary with z, nor does T. Neither is true in real life. Also note that, in the equation you give, the ...

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If you rearrange the ideal gas law to be expressed in terms of pressure $$P=\frac{NRT}{V}\qquad\Rightarrow\qquad \frac{T_1}{V_1}=\frac{T_2}{V_2}$$ where the right hand equation assumes it is an isobaric process with no mass exchange. So, in an isobaric process temperature and volume vary inversely. If the volume decreases then the temperature must go up. ...

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Yes, Temperature is halved. You are keeping the pressure constant. You are just imagining pressing a piston in which pressure is not constant. Volume is reducing. But we have to keep the pressure same. So we must reduce kinetic energy of our particles to keep the pressure same. The rest is obvious.

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Yes - you can have a state where increasing the pressure would create a supercritical fluid See Phase Diagram

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Let's look at the terms of $PV=nRT$ before and after: Before After P 2.6x10^5 P2 V V V (the question says "Assume that the volume has not changed") n n n (the number of molecules of gas is unchanged - nothing has got out or in) R R R (universal constant) T(C) ...

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For isochoric processes $$\frac{p}{T}=const$$ which means the answer for your question: $$p_2=\frac{p_1*T_2}{T_1}$$ where all temperatures are in Kelvin. For more information see http://de.wikipedia.org/wiki/Isochore_Zustands%C3%A4nderung (unfortunatelly english version is not so usefull this time)

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You are using the equation $V=RT/P$ but you seem to have left out $N$. This is likely (one reason) why your answer doesn't make sense. Since you are working with a tire that is presumable sealed, you know $N$ is constant. You are also told $V$ is constant. So consider $PV=NRT$. You can rearrange to get $N/V=\frac{P}{RT}.$ Since the LHS of the equation ...

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