Question

if air temperature is 20°c and the body cools for 20 min from 140°c to 80°c.find when temperature will be 35°c?​

Answer 1

A body kept in the air at 25 degrees Celsius cools from 140 degrees Celsius to 80 degrees Celsius in 20 minutes. Find when the body cools down to 35 degrees Celsius.

The time is at 35°c is 37.5min.

Given:

If the air temperature is 25°c and the body cools for 20 min from 140°c to 80°c.

To Find:

The time at 35°c.

Solution:

To find the temperature at 35°c we will follow the following steps:

According to Newton's law of cooling the rate at which an object cools is directly proportional to the difference in the temperature of the object and surroundings in a condition where the temperature difference is small.

$mc \frac{dT}{dt} = k(t - t0)$

Here, t is the average temperature of the object and t0 is the temperature of the surrounding. k is the constant that depends on the nature of the object.

So,

The rate of cooling is directly proportional to temperature difference.

Now,

$mc \frac{t1 - t2}{t} = k[ \frac{(t1 + t2)}{2} - t0)]$

$\frac{mc}{k} = \frac{[ \frac{(t1 + t2)}{2} - t0)] }{ \frac{t1 - t2}{t}}$

Now, putting values in the above formula we get,

$\frac{mc}{k} = \frac{ \frac{140 + 80}{2} - 35}{ \frac{140 - 80}{25} }$

= $\frac{75}{2.4}$

Now, rearranging and putting values in above equation we get,

$mc \frac{t1 - t2}{k} = t[ \frac{(t1 + t2)}{2} - t0)]$

$\frac{75(140-80)}{2.4}= t[ \frac{(140 + 80)}{2} - 25)]$

= $\frac{30×75}{25×2.4}$

7.5min

Henceforth, the time is at 35°c is 37.5min.

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