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A cylinder with a movable piston contain...

A cylinder with a movable piston contains 3 moles of hydrogen at standard temperature and pressure. The walls of the cylinder are made of a heat insulator, and the piston is insulated by having a pile of sand on it. By what factor does the pressure of the gas increase if the gas is compressed to half its original volume ?

Text Solution

Verified by Experts

Initial volume of gas `V_1=V`
Final volume `V_2=V/2`
Since hydrogen is a diatomic,
`gamma=7/5`=1.4
Heat is not exchanged in adiabatic process, (Walls and cylinder are insulators )
`therefore P_1V_1^gamma =P_2V_2^gamma`
`therefore (P_1V_1^gamma)/V_2^gamma=P_2`
`therefore P_2=P_1 ((V)/(V/2))^gamma`
`therefore P_2=P_1(2)^(1.4)`
`therefore P_2/P_1=(2)^(1.4)`
`therefore "log"P_2/P_1=1.4 log2`
=1.4(0.3010)
-0.4214
Taking antilog
`P_2/P_1`=2.638
`therefore P_2/P_1 approx 2.64`
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