Question

Two cylinders A and B fitted with pistons contain equal amounts of an ideal diatomic gas at 300 K. The piston of A is free to move, while that of B is held fixed. The same amount of heat is given to the gas in each cylinder. If the rise in temperature of the gas in A is 30 K, then the rise in temperature of the gas in B is

• A. 30 K
• B. 18 K
• C. 50 K
• D. 42 K

Answer 1

Answer: (D)

42 K

Answer 2

A is free to move, therefore, heat will be supplied at constant
pressure
$\therefore \, \, \, \, \, \, \, dQ_A=nC_p dt_A \, \, \, \, \, \, \, \, \, \, \, \,$ ...(i)
B is held fixed, therefore, heat will be supplied at constant
volume.
$\therefore \, \, \, \, \, \, \, \, \, \, \, \, \, dQ_B=nC_vdT_B \, \, \, \, \, \, \, \, \, \, \, \,$ ...(ii)
But $\, \, \, \, \, \, \, \, \, \, \, \, dQ_A =dQ_B \, \, \, \, \, \, \, \, \, \, \, \, \, \,$ (given)
$\therefore \, \, \, \, \, \, \, \, \, \, \, \, \, _nC_pdt_A=nCvdT_B$
$\therefore \, \, \, \, \, \, \, \, \, \, dT_B=\big(\frac{C_p}{C_v}\bigg)dT_A$
$\, \, \, \, \, \, \, \, \, \, \, \, \, \, =\lambda(dT_A) \, \, \, \, \, [\gamma =1.4 (diatomic)]$
$\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, (dT_A=30 K)$
$\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, =(1.4)(30 K)$
$\therefore \, \, \, \, \, \, \, \, \, \, dT_B=42 K$