An isolated system consists of an incompressible substance o

An isolated system consists of an incompressible substance of total mass m with constant specific heat C(t)=kt confined to a box. Initial 1/3 of the mass is the left hand side of the box at temperature T₁ and 2/3 of the mass in the right hand side at temperature T₂. Eventually the system attains an equilibrium, when it is well mixed with a final temperature T_f.

a) Derive the expression for T_f (in terms of T₁ and T₂).

Solution

(a) let T₁is greater then T₂

then (m/3) kt (T₁-Tf ) =(2m/3) kt (Tf-T₂ )

m/3(T₁ +2T₂ ) =m/3(3m) Tf

Tf= (T₁+2T₂ /3m)

(b) we know entropy S= Q/T