Newton's Law of Cooling

We are considering a situation when we take out ice-cream from the freezer.

Suppose just after taking out the ice-cream, our mother asks us to do some work.

By using the Newton's law of cooling, we can find approximately till what time, the ice-cream will not melt.

And thus, we can say to our Mom, wait for let say, minutes after calculating the time otherwise the ice-cream will melt.

Similar to this, we can find the rate of cooling of any body using the Newton's law of cooling.

Let's understand what is Newton's law of cooling and subsequently, derive its expression.

Suppose a body at temperature is kept in an environment which is at a temperature of .

There will be radiation as well as absorption of heat. But, the body will radiate more energy than that it will absorb.

As the temperature of the body is greater than that of .

As per Stefan's Boltzman law, the emissive power of the body, let say, can be given by,

Similarly, the absorptive power of the body, let say , can be given by,

At thermal equilibrium, the absorptive and emissive power becomes equal and thus,we have got equal to .

We can write the net power emission of the body, let say, as,

Power is the rate of change of energy. so, we can write as,

The heat can also be written as,

We write the rate of change of the heat as shown here. The negative sign indicates continuous decrease in temperature of the body

Thus, we have got the rate of cooling or the rate at which the temperature of body is decreasing w.r.t. time.

If is close to , then as per Newton's law of cooling, rate of cooling is directly proportional to the temperature difference .

So, we can also define it as,

Now, let's find out the exact expression of the Newton's law of cooling.

For this, we shall again write the expression obtained from the Stefan Boltzman Law.

We can further write as,

Let's expand the polynomial obtained binomially and ignore the higher orders of as this will be very small.

Hence, we have arrived at the expression of Newton's law of cooling which is shown here.

The limitation of this law is that it is valid only when the temperature difference is very small.

So, till now, we have understood the Newton's law of cooling and its limitation. Let's once recall whatever we studied.

Revision

If the temperature difference is very small, then the rate of cooling is directly proportional to the temperature difference. This is the Newton's law of cooling.

We can also define it mathematically as,

The End