Question

A body with an initial temperature ${\theta }_i$, is allowed to cool in surrounding which is at a constant temperature of ${\theta }_0$ $({\theta }_0<{\theta }_1)$. Assume that Newton's law of cooling is obeyed. Let $K$ = constant. The temperature of the body after time $t$ is best expressed by

• A. $({\theta }_i-{\theta }_0)e^{-Kt}$
• B. $({\theta }_i<{\theta }_0)ln\left(kt\right)$
• C. ${\theta }_0+({\theta }_i-{\theta }_0)e^{-Kt}$
• D. ${\theta }_i{e}^{-Kt}-{\theta }_0$

Answer 1

The correct option is C ${\theta }_0+({\theta }_i-{\theta }_0)e^{-Kt}$

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.

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

or

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

or

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