An ideal monoatomic gas is confined in a horizontal cylinder by a spring loaded piston (as shown in the figure).Initially the gas is at temperature T _ 1 , pressure P _ 1 and volume V _ 1 and the spring is in its

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Correct answer: A. If V _ 2 = 2 V _ 1 and T _ 2 = 3 T _ 1 , then the energy stored in the spring is (1)/(4) P _ 1 V _ 1

(i) P = P _ 1 + (Kx)/(A) P _ 2 = (3)/(2) P _ 1 ⇒ x = V _ 1 A 3 P _ 1 2 = P _ 1 + (Kx)/(A) Kx = P _ 1 A 2 Energy of spring (1)/(2)K x ^ 2 = P _ 1 A 4 x = P _ 1 V _ 1 4 (ii) ∆ U = (f)/(2)( P _ 2 V _ 2 - P _ 2 V _ 1 ) = 3 P _ 1 V _ 1 (iii) P _ f = 4 P _ 1 3 KX = P _ 1 3 A X = 2 V _ 1 A W _ gas = - ( W _ P _ atm + W _ spring ) = ( P _ 1 Ax + (1)/(2)Kx . x) = + ( P _ 1 A . 2 V _ 1 A + (1)/(2) . P _ 1 A 3 . 2 V _ 1 A ) = 2 P _ 1 V _ 1 + P _ 1 V _ 1 3 = 7 P _ 1 V _ 1 3 (iv) ∆ Q = W + ∆ U = 7 P _ 1 V _ 1 3 + (3)/(2)( P _ 2 V _ 2 - P _ 1 V _ 1 ) = 7 P _ 1 V _ 1 3 + (3)/(2)((4)/(3) P _ 1 . 3 V _ 1 - P _ 1 V _ 1 ) = 7 P _ 1 V _ 1 3 + (9)/(2) P _ 1 V _ 1 = 41 P _ 1 V _ 1 6

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