1. Newton’s Law of Cooling

2. Use Newton’s Law of Cooling to solve problems
Objectives

3. Initial condition – allows the user to find the particular solution from a family of solutions
Equilibrium – a steady state condition with neither growth nor decay
Vocabulary

4. Temperature Change
An object’s temperature over time will approach the temperature of its surroundings (the medium)
The greater the difference between the object’s temperature and the medium’s temperature, the greater the rate of change of the object’s temperature
This change is a form of exponential decay T0 Tm

5. CSI: Newton’s Law of Cooling
The rate at which an object cools is proportional to the difference in temperature between the object and the surrounding medium: Where T is temperature of the object, k is a proportionality constant, T0 is the temperature of the surrounding medium and t is time A coroner uses this to help determine the time of death and is seen in every “Crime” TV series like CSI. dT
---- = -k( aT – T0) dt

6. Newton’s Cooling Equation
Given by separating variables we get the following equation dT
---- = -k( aT – T0) dt

7. Example
Example: A potato is taken out of a 300o F oven and left to cool in a room at 75o F. Write a differential equation expressing the change in rate of the temperature of the potato, T, with respect to time, t. dT
---- = - k(T – T0) dt dT
---- = - k(300 – 75) dt dT
---- = - 225k dt 7

8. Example cont
Example: A potato is taken out of a 300o F oven and left to cool in a room at 75o F. Write a differential equation expressing the change in rate of the temperature of the potato, T, with respect to time, t. dT
= - k(T – T0) dt
T = T0 + Ke –kt
T = 75 + (300 – 75)e –kt
T = e –kt
Use intermediate condition to find k

9. Newton’s Law of Cooling
Example: The great detective Sherlock Holmes and his assistant, Dr. Watson, are discussing the murder of actor Cornelius McHam. McHam was shot in the head, and his understudy, Barry Moore, was found standing over the body with the murder weapon in hand. Let’s listen in:
Watson: Open-and-shut case, Holmes. Moore is the murderer.
Holmes: Not so fast, Watson – you are forgetting Newton’s Law of Cooling!
Watson: Huh?
Holmes: Elementary, my dear Watson. Moore was found standing over McHam at 10:06 p.m., at which time the coroner recorded a body temperature of 25.5°C and noted that the room thermostat was set to 22.2°C. At 11:06 p.m. the coroner took another reading and recorded a body temperature of 24.2°C. Since McHam’s normal temperature was 37°C, and since Moore was on stage between 6:00 p.m. and 8:00 p.m., Moore is obviously innocent. Ask any calculus student to figure it out for you.
How did Holmes know that Moore was innocent?

10. CSI Solution T = T0 + Ke –kt
T is temperature of the body at t hours since death
T0 = 22.2 temperature of the room T = 37 temperature of the body
t would represent the hours since death
But we don’t know the time of death. We can use the coroner's temperature readings to determine k.
T(10:06) = 25.5
T(11:06) = so T(1) = 24.2 = (25.5 – 22.2)e–k
2 = 3.3e–k
k = -ln(2/3.3) =
T(t=0) = 37
T(t, 10:06) = 25.5 = e t
3 = 14.8e t
t = ln(3/14.8) / = hours since death
so death occurred at about 7 pm.

11. Black coffee served at McDs is said to be 800 C which is to hot to drink/ A customer notes that after 2 minutes 30 seconds it has cooled to 76.00C. How long will it take to cool to C if the surrounding temperature is 18?