Two block with heat capacities `C_(1)` and `C_(2)` are connected by a rod of length l, cross-sectional area A and heat conductivity K. Initial temperature difference between the two blocks I `T_(0)` . Assuming the entire system to be isolated from surroundings , heat capacity of the rod of be negligible. The temperature difference between the blocks as a function of time is.

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`