An adiabatic cylinder ie perfectly insulated from surroundin

An adiabatic cylinder( i.e. perfectly insulated from surroundings) is divided into two chambers A and B by a moving wall that moves without friction does not conduct heat. Each of the two chambers initially contains 20 liters of an ideal gas at 25C (298K) and 1 atmosphere. Gas in chamber A is slowly heated by means of a heat resistance until pressure in chamber B is equal to 2 atmosphere. Determine: Final volume and temperature of gas in chamber B. Work transferred from Chamber A to Chamber B. Final temperature in chamber A. Heat given by the heat resistance to gas in Chamber A. Use C_v = 20.80 J.K^-1 mol^-1 and gamma = 1.4 reminder R/C_v = C_p/C_v - 1 = gamma - 1

Solution

initial pressure P1= 1 atm initial volume V1 = 20 Ltr. initial temp T1 = 298K

final pressure P2 = 2 atm

as the chamber is insulated from external environment, no heat transfer takes place to external environment and it is an adiabatic compression

P1V1 = const.

1/2 = (V2/20)1.4

V2 = (0.5)^1/1.4 *20 = 12.19 Ltr.

PV = nRT

T2 = T1*(P2v2/P1V1) = 298*(2*12.19/20)

       = 363.26 K

Work is done on the gas, work done in an adiabatic process is given by

W = (P1V1-P2V2)/(-1) = (101e+3*20e-3 -2*101e+3*12.19e-3)/0.4

      = -1106 J

-ve sign indicates work is done on the gas.

Chamber A:

Heat supplied is utilized to in expanding the gas and to increase the internal energy by heating the gas

as the wall is free to move, the pressure on both sides of the wall would be equal at any time of equilibrium position.

initial pressure P1 = 1atm, volume V1 = 20 Ltr. temp T1 = 298K

Final volume V2 = 40-12.19 = 27.81 Ltr.

heat is supplied to the gas in chamber A.

Final pressure in A is P2 = 2 atm

T2 = T1(P2V2/P1V1) = 298* (2*27.81/1*20) = 828.74 K

Change in internal energy

U = nC_vdT

n = PV/RT = 101e+3*20e-3/8.54*(298) = 0.794 moles

U = 0.794*20.8*(828.74-298) = 8765.3 J

work done in expanding the gas in chamber A is equal to the work done in compressing the gas in chamber B. In a work is done by the gas abd in B work is done on the gas.

Total heat energy supplied = 8765.3 + 1106 = 9871 J