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 between 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.

Options

A. `T_(0) e^((-KA(C_(1) + C_(2))t)/(C_(1)C_(2)))`
B. `T_(0) e^((-KA(C_(1) + C_(2)))/(C_(1)C_(2)))`
C. `T_(0) e^((KA(C_(1) + C_(2))t)/(C_(1)C_(2)))`
D. `T_(0) e^((KA(C_(1) + C_(2))t^(2))/(C_(1)C_(2)))`

Answer

A

Explanation

Let T be the tempeature difference between two block at time t.

Heat transferred per second,
`(dQ)/(dt) = (KAT)/(l) "….."(i)`
Also, `dT = dT_(1) + dT_(2)"….."(ii)`
Heat lost by one block is equal to the heat gained by the other, `C_(1)dT_(1) = C_(2)"......."(iii)`
From equations, (ii) and (iii), we get
`dT = ((C_(1) + C_(2))/(C_(2))) dT_(1)"......"(iv)`
If a block loses heat, `dQ = - C_(1)dT_(1)`,
From equation, (i), `(dQ)/(dt) = - C_(1) (dT)/(dt) = - C_(1)(dT_(1))/(dt) = (KAT)/(l) "....."(v)`
From equations (iv) and (v), we get
`(-C_(1)C_(2))/(C_(1) + C_(2)) (dT)/(dt) = (KA)/(l) T`
`(dT)/(t) = (-KA(C_(1) + C_(2)))/(C_(1)C_(2))dt`
or `underset(T_(0))overset(T)(int) (dT)/(dt) = (-KA(C_(1)+C_(2)))/(C_(1)C_(2)) underset(0)overset(t)(int)dt`
`"ln"(T)/(T_(0)) = ((-KA(C_(1) + C_(2)))/(C_(1)C_(2)))t`
or `T = T_(0) "exp" ((-KA(C_(1)+C_(2))t)/(C_(1)C_(2)))`