As shown in the figure, an insulated container is fitted with a thermally conducting but immovable partition (P1) and a freely movable but thermally insulated piston (P2). The partition P1 with thermal conductivity K, cross sectional area A and width x divides the container into two sections, S1 and S2, each containing one mole of a monoatomic gas. The piston P2 moves freely such that the gas in S2 is always at the atmospheric pressure. Initially, the difference between the temperatures of S1 and S2 is T0. The time it takes for the temperature difference to become T02 is nxR/KA, where R is the universal gas constant. The value of n is: [ Given: 2 0.7 ] [figure]
Answer
0.63 to 0.70
Solution
Heat current through P1: H = KAx(T1 - T2). Gas in S1 is at constant volume (CV = 32R), gas in S2 at constant pressure (CP = 52R).
dT1dt = - H 32R, dT2dt = + H 52R
d( T)dt = - KAx T ( 23R + 25R ) = - 16KA15xR T
T = T0 e-16KAt/15xR; for T = T02: t = 1516 2 xRKA
n = 15 0.716 = 0.656 0.66
Answer: 0.66