An insulated cylinder is divided into three parts A, B and C. Pistons 1 and 2 are connected by a rigid rod and can slide without friction inside the cylinder. Piston 1 is perfectly conducting while piston 2 is perfectly insulating. Equal quantity of an ideal gas is filled in three compartments and the state of gas in every part is same `(P_(0) V_(0) T_(0))`. Adiabatic exponent of the gas is g = 1.5. The compartment B is slowly given heat through a heater H such that the final volume of gas in part C becomes `(4V_(0))/(9)` (a) Calculate the heat supplied by the heater. (b) Calculate the amount of heat flow through piston 1. (c) If heater were in compartment A, instead of B how would your answers to (a) and (b) change?

An insulated cylinder is divided into three parts A, B and C. Pistons

1.	An insulated cylinder is divided into three parts A, B and C. Pistons 1 and 2 are connected by a rigid rod and can slide without friction inside the cylinder. Piston 1 is perfectly conducting while piston 2 is perfectly insulating. Equal quantity of an ideal gas is filled in three compartments and the state of gas in every part is same `(P_(0) V_(0) T_(0))`. Adiabatic exponent of the gas is g = 1.5. The compartment B is slowly given heat through a heater H such that the final volume of gas in part C becomes `(4V_(0))/(9)` (a) Calculate the heat supplied by the heater. (b) Calculate the amount of heat flow through piston 1. (c) If heater were in compartment A, instead of B how would your answers to (a) and (b) change?
Answer» Correct Answer - (i) `P_(A) = P_(C) = (27)/(8)P_(0) , P_(B) = (21)/(4)P_(0)` (ii) `T_(A) = T_(B) = (21)/(4)T_(0), T_(C) = (3)/(2)T_(0)` (iii) `18P_(0)V_(0) (iv) W_(A) = P_(0)V_(0) , W_(B)= 0` (v)`(17)/(2)P_(0)V_(0)` For compartment `C` `P_(0)V_(0)^(gamma) = P((4V_(0))/(9))^(gamma)` `" "` `implies P = (27)/(8)P_(0)` `implies P_(0)T_(0)^(gamma//1-gamma) = PT^(gamma//1-gamma)` `P_(0)T_(0)^(-3) = ((27)/(8)P_(0)) xx T^(-3) implies T = (3)/(2)T_(0)` For compartment `A` `P_(A) = (27)/(8)P_(0)` `(P_(0)V_(0))/(RT_(0)) = (P_(1)V_(1))/(RT_(1))` `implies T_(1) = ((27)/(8)P_(0)(2V_(0)-(4V_(0))/(9)))/(R) xx (RT_(0))/(P_(0)V_(0)) = T_(1) = (21)/(4)T_(0)` For compartment `B` `(P_(0)V_(0))/(T_(0)) = (P_(1)V_(1))/(T_(1)) implies (P_(0)V_(0))/(T_(0))= (P_(1)V_(0))/(((21)/(4)T_(0)))implies P_(1) = (21)/(4)P_(0)` Final pressure in `A (27)/(8)P_(0)` Final pressure in `B (21)/(4)P_(0)` Final pressure in `C (3)/(2)P_(0)` (ii) Final temperature in `A = (21)/(4) T_(0)` Final temperature in `B = (21)/(4) T_(0)` Final temperature in `C = (3)/(2) T_(0)` (iii) Heat supplied by heater = (`DeltaU + W)` ` Q = (DeltaU_(1) + W_(1)) + (DeltaU_(2) + W_(2)) + 0` =(`DeltaU_(1) - W_(3)) + (DeltaU_(2) + 0)` =`DeltaU_(1) + DeltaU_(2) - W_(3)` = ` (n_(1)RDeltaT)/(gamma_(1) - 1) + (n_(2)RDeltaT)/(gamma_(2)-1) - ((n_(3) RDeltaT)/(gamma_(3)-1))` =`(17)/(2)P_(0)V_(0) + (17)/(2)P_(0)V_(0) + P_(0)V_(0) = 18 P_(0)V_(0)` (iv) Work done by gas in chamber `B =0` Work done by gas in chamber `C = -DeltaU = (-nRDeltaT)/(gamma-1) = -P_(0)V_(0)` Work done by gas in chamber `A = (-) W_(chamber) = -(P_(0)V_(0)) = P_(0)V_(0)` (v) Heat flowing across piston-I = `DeltaU_(2)= (17)/(2)P_(0)V_(0)`

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