Question

A block of steel is heated at ${100}^{o}C$ is left in room to cool. Which of the curves shown in figure represents the correct cooling behavior?

• A. A
• B. B
• C. C
• D. D

Answer 1

The correct option is A A

Newton's law of cooling gives us the relation:

$\frac{d\theta }{dt}=-bA(\theta -{\theta }_0)$

i.e. $\frac{d\theta }{(\theta -{\theta }_0)}=-Kdt$

Integrating between initial temperature and final temperature ${\theta }_{i}to{\theta }_f$ in time 0 to t.

$ln\left(\frac{{\theta }_f-{\theta }_0}{{\theta }_i-{\theta }_0}\right)=-kt$

${\theta }_f-{\theta }_0=({\theta }_i-{\theta }_0)e^{-kt}$

${\theta }_f={\theta }_0+({\theta }_i-{\theta }_0)e^{-kt}$

Here initial temperature $={100}^{\circ }C$ and final temperature is '$T$' substituting this

$T={\theta }_0+(100-{\theta }_0)e^{-kt}$

Graph of option A matches the curve we obtain.