Figureshows a largetankof waterat constant temperatureT_(C)and a smallvessel containingclodwaterof mass m at initial temperatureT_(1) .A metal rodof lengthL., area of crossectionA andthermalconductivityk connects the two vessesl to becomes T_(2) (T_(C) lt T_(2) ltT_(1)). Thespecificheat capacityof water is c and all otherheat capacitiesare negligible .

Figureshows a largetankof waterat constant temperatureT_(C)and a small

1.	Figureshows a largetankof waterat constant temperatureT_(C)and a smallvessel containingclodwaterof mass m at initial temperatureT_(1) .A metal rodof lengthL., area of crossectionA andthermalconductivityk connects the two vessesl to becomes T_(2) (T_(C) lt T_(2) ltT_(1)). Thespecificheat capacityof water is c and all otherheat capacitiesare negligible .
Answer» Solution :Letat any time ,tthe temperatureof SMALLER vessel be Tand after a small timeinternal dt , it becomes T+ dT. Theheat TRANSFERRED from tank to small vesselin small timeinterval dt is. `DQ = KA((T_(c) - T))/(l) dt ` But dQ = me dT. `therefore` m`dT = KA(T_(C) - T)/(l) dt "" therefore (dT)/(T_(c) - T) = (KA)/(mcl) dt ` LET `tau ` be the timerequire d , thenIntegrating. `int_(T_(c))^(T_(2)) (dT)/(T_(C) - T)= (KA)/(mcl) [t]_(0)^(tau) "" [log_(e) (T_(C) - T)]_(T_(1))^(T_(2)) = (KA)/(mcl) tau rArr log_(e) ((T_(c)-T_(2))/(T_(c) - T_(1))) = (KA)/(mcl) tau ` `thereforetau = (mcl)/(KA) log_(e)((T_(c) - T_(2))/(T_(c) - T_(1)))`