# Newton's Law of Cooling

Written by Jerry Ratzlaff on . Posted in Thermodynamics

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

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

Where:

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

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

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

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

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

Newton's Law of Cooling Formula Solving $$T_t$$

$$\large{ T_t = T_a + \left( T_0 - T_a \right ) e ^{ -k t} }$$

Where:

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

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

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

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

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

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