Question

A black body at ${327}^{0}C$, when suspended in black enclosure at ${27}^{0}C$ cools at a certain rate. Find the temperature of body at which its rate of cooling will be half of above rate. Assume the other conditions remain unchanged.

Answer 1

Rate of cooling $\frac{dT}{dt}=\frac{k}{ms}\left(T^4\right)$

Or $\frac{dT}{dt}\propto \left(T^4\right)$

Or $\frac{(dT/dt{)}_1}{(dT/dt{)}_2}=\frac{\left(T_1^4\right)}{\left(T_2^4\right)}$

Or $\frac{1}{2}=(\frac{T_1}{600}{)}^4$

$⟹T_1=600\times \frac{1}{2^{1/4}}=316K$