c. Find the temperature in the lower chamber immediately before the piston... · OlympiadBench #1497

Physics · thermodynamics

Problem

c. Find the temperature

T_{2}

in the lower chamber immediately before the piston touches the diaphragm. Given constants: A hollow insulated cylinder of height

2 \mathrm{H}

and volume

2 \mathrm{~V}

is closed from below by an insulating piston. The cylinder is divided into two initially identical chambers by an insulating diaphragm of mass

m

. The diaphragm rests on a circular ledge and a gasket between them provides tight contact. Both chambers are filled with gaseous helium at pressure

p

and temperature

T

. A force is applied to the piston, so that it moves upwards slowly. <image_1> Context question: a. Find the volume of the lower chamber

V_{0}

when the gas starts to leak between the chambers. Context answer:

V(1+x)^{-3 / 5}

Context question: b. Find the temperature

T_{1}

in the upper chamber when the piston touches the diaphragm. Context answer:

\frac{T}{2}\left(\left(1+(1+x)^{3 / 5}\right)^{5 / 3}-x\right)