A solid body A of mass m and specific heat capacity ‘s’ has temperature `T_(1) = 400 K`. It is placed, at time `t = 0`, in atmosphere having temperature `T_(0) = 300 K`. It cools, following Newton’s law of cooling and its temperature was found to be `T_(2) = 350 K `at time `t_(0)`. At time `t_(0)`, the body A is connected to a large water bath maintained at atmospheric temperature `T_(0)`, using a conducting rod of length L, cross section A and thermal conductivity k. The cross sectional area A of the connecting rod is small compared to the overall surface area of body A. Find the temperature of A at time `t = 2t_(0)`.

Explanation

The solid sphere loses heat due to radiation. Then, according to Newton.s law of cooling,
` (dT)/(dt) = - k(T - T_R)`
Integrating above equation from t = 0 to `t = t_1`
`int_(T_0)^(T_1) (dT)/(T - T_R) = - k int_(0)^(t_1) dt`
where `T_1 `= 350 K is the temperature of sphere after time `t_1` and `T_0` = 400 K is the initial temperature of sphere at t=0.
` rArr kt_1 = -ln ( (T_1 -T_R)/(T_0 - T_R) )`
` = -ln ( (350 - 300)/(400 - 300) ) = ln 2`
` rArr k = (ln2)/(t_1)`
Also, rate of heat loss is
`(dQ_1)/(dt) = -ms (dT)/(dt) = - c (dT)/(dt)`
` = c k (T - T_R) `
` (c (T - T_R )ln2)/(t_1) ` ...(i)
After time `t_1` , heat is lost to the atmosphere by radiation and to the large body. The rate of heat loss by sphere due to conduction is
`(dQ_2)/(dt) = KA (T - T_R) //L ` ...(ii)
Equations (i) and (ii) give the total rate of heat loss by sphere
`(dQ)/(dt) = (dQ_1)/(dt) + (dQ_2)/(dt)`
` = (c(T - T_R ln2 )/(t_1) + (KA(T - T_R ) )/(L) ` ....(iii)
Also, the rate of temperature change of sphere is related to its total rate of heat by
` (dQ)/(dt) = - c (dT)/(dt) ` ....(iv) Equating (iii) and (iv), we get
`(dT)/(dt) = -[(ln 2)/(t_1) + (KA)/(Lc) ] (T - T_R)`
Integrating above equation from `t=t_1` to `t = 2t_1` , we get
` int_(T_1)^(T_2) (dT)/(T - T_R) = ( (ln 2)/(t_1) + (KA)/(Lc) ) int_(t_1)^(2t_1) dt`
`rArr ln ( (T_2 - T_R)/(T_1 - T_R) ) = 2ln 2 - (KAt_1)/(Lc)`
` ln (T_1 - 300) - ln 50 = - ln 2 - (KAt_1)/(Lc)`
` = ln 50/2 - (KAt_1)/(Lc)`
` = ln 25 - (KAt_1)/(Lc)`
`ln [ (T_2 - 300 )/(25) ] = - (KAt_1)/(Lc)`
` T_2 - 300 = 25e^((KAt_1)/(Lc))`
` rArr T_2 = 300+ 25e^((KAt_1)/(Lc) )`