An isolated system consists of an incompressible substance of total mass m with

Question

An isolated system consists of an incompressible substance of total mass m with constant specific heat C(T) = kT confined to a box. Initially 1/3 of the mass is in the left hand side of the box at temperature T_1 and 2/3 of the mass in the right hand side at temperature T_2. Eventually the system attains an equilibrium, when it is well mixed with a final temperature T_f. Derive the expression for T_f (in terms of T_1 and T_2). Derive the expression for sigma the entropy produced during this process (in terms of m, k, T_1, T_2).

Explanation / Answer

Let Ti is the heat exchange to attain left hand side box and Tf is the heat exchange to attain right hand side of the box.

m1c1(Tf - T1) = m2 C2(T2 - Tf)

c1 = C2

1/3 m (Tf - Ti) = 2/3 m (T2 - Tf)

Tf - Ti = 2(T2-Tf)

2Tf =( 2T2 + T1) / 3

dS generation = ( ds)1/3m + (ds) 2/3m

= m1c ln (Tf / T1) + m2 c ln (Tf / T2)

= 1/3 m c ln ( Tf / T1) + 2 / 3 mc ln (Tf / T2)

= 1/ 3 mc ( ln Tf / T1 + ln ( Tf / T2)

= 1/3 mc ln ( Tf3 / T1 T22 ) Answer

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