Two chunks of metal with heat capacities `C_(1)` and `C_(2)`, are interconnected by a rod length `l` and cross-sectional area `S` and fairly low heat conductivity `K`. The whole system is thermally insulated from the environment. At a moment `t = 0` the temperature difference betwene the two chunks of metal equals `(DeltaT)_(0)`. Assuming the heat capacity of the rod to be negligible, find the temperature difference between the chucks as a function of time.

Answer

`DeltaT = (DeltaT)_(0)e^(-alphat)`, where `alpha = (1//C_(1)+1//C_(2)) SK//l`

Explanation

`(dQ)/(dt) = (Ks)/(l) (T_(1)-T_(2))`
where `T_(1)` and `T_(2)` are temperature of two chunks are function of time 't'. ltbRgt `-C_(1) (dT_(1))/(dt) =(Ks)/(l)(T_(1)-T_(2))`
`C_(1)(dT_(2))/(dt) =(Ks)/(l)(T_(1)-T_(2))`
or `-(dT_(1))/(dt) =(Ks)/(lC_(1)) (T_(1)-T_(2))`
or `(dT_(2))/(dt) = (Ks)/(lC_(2)) (T_(1)-T_(2))` ltbgt or `(-d(T_(1)-T_(2)))/(dt) = (Ks)/(l) ((1)/(C_(1))+(1)/(C_(2)))`
or `-overset(DeltaT)underset(DeltaT_(0))int (d(T_(1)-T_(2)))/((T_(1)-T_(2)))=(Ks)/(l)[(1)/(C_(1))+(1)/(C_(2))]overset(t)underset(0)intdt`
`DeltaT = DeltaT_(0)e^(-alphat)`
where `alpha = (1//C_(1) +- 1//C_(2)) SK//l`