# Newton's Law of Cooling

Written by Jerry Ratzlaff on . Posted in Thermodynamics

## Newton's Law of Cooling

This formula calculates the temperature of an object as it loses heat at any given time.  The law states that the rate of heat loss of a body is directly porportional to the difference in the temperature between the body and its surrounding provided the temperature difference is small and the nature of radiating surface remains the same.

### Newton's Law of Cooling Formula

Newton's law of cooling is given by

$$\large{ \frac { dT } { dt } = -\; k \left( T_t \; - \; T_a \right ) }$$

Where:

$$\large{ dt }$$ = change in time

$$\large{ dT }$$ = change in temperature of object

$$\large{ k }$$ = is a constant (decay constant)

$$\large{ T_t }$$ = temperature at time t

$$\large{ T_a }$$ = ambient temperature (temperature of environment)

Newton's Law of Cooling Formula is given by

$$\large{ T \left( t \right ) = T_a \; +\; \left( T_0 \; - \; T_a \right ) e ^{ -kt} }$$

Where:

$$\large{ e }$$ = exponential $$e$$ implies a continious rate of cooling

$$\large{ k }$$ = cooling constant temperature of object

$$\large{ t }$$ = time taken to cool

$$\large{ T \left(t \right ) }$$ = temperature of object at a given time $$\large{ t }$$

$$\large{ T_a }$$ = ambient temperature (temperature of environment)

$$\large{ T_0 }$$ = initial temperature of object