Newtons Law of Cooling Objectives Use Newtons Law

Newton’s Law of Cooling

Objectives � Use Newton’s Law of Cooling to determine the time of death in a controlled environment.

Temperature Change Newton’s Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the temperature of its surroundings (i. e. the ambient temperature ).

Temperature Change � An object’s temperature over time will approach the temperature of its surroundings � The greater the difference between the object’s temperature and the temperature of its surroundings, the greater the rate of change of the object’s temperature � This change is a form of exponential decay T 0 Ts

Newton’s Law of Cooling The rate at which an object cools is proportional to the difference in temperature between the object and the temperature of its surrounding: T – TS = (TO – TS) e-kt Where T is the final temperature of the object TO is initial temperature of the object k is a cooling constant TS is the temperature of the surroundings t is time A coroner uses this to help determine the time of death and is seen in every “Crime” TV series from Dragnet to CSI.

Newton’s Cooling Equation � T – TS = (TO – TS) e-kt

Example: A potato is taken out of a 350 o F oven and left to cool in a room at 72 o F. The potato has cooled to a temperature of 300 o F in 45 minutes. Find the value of k. Use the equation solved for k Plug in your values for each variable and perform each operation on your calculator. The value for k (cooling constant) for this potato in these surroundings is:

Forensics Use of Newton’s Equation Now the variables have to be changes to match the field of forensics. Now: Tlater is the temperature of the body at the second reading Tdisc is initial temperature when the body was discovered k is a cooling constant Tamb is the ambient temperature of the room (surroundings) Δt is the time difference between the first and second temperature readings What do you notice about the color coding from this set of variables and the variables used earlier in the presentation.

Forensics Use of Newton’s Equation The formula used in forensics now looks like this: Where: Tlater is the temperature of the body at the second reading Tdisc is initial temperature when the body was discovered k is a cooling constant Tamb is the ambient temperature of the room (surroundings) Δt is the time difference between the first and second temperature readings

Forensics Use of Newton’s Equation

Forensics Use of Newton’s Equation Now the equation is set up to find change in time since death substitute in the variable (Δtdeath) for (Δt) and substitute the body temperature at the time of death 98. 6 o F (Tdeath) for (Tlater). So…… Becomes: Once the value of k is found simply plug in the data collected and solve with your calculator.

Forensics Use of Newton’s Equation Assume that the value of k is. 132647814 (do not round this number) and the body temperature at the time of discovery was 90. 4 o F with the temperature of 680 Find the time since death. Plug-in: Results in: This number will let you know how many minutes or hours ago the person died.

Newton’s Law of Cooling Example: The great detective Sherlock Holmes and his assistant, Dr. Watson, are discussing the murder of actor Cornelius Mc. Ham 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 Mc. Ham at 10: 06 p. m. , at which time the coroner recorded a body temperature of 77. 9°F and noted that the room thermostat was set to 72°F. At 11: 06 p. m. the coroner took another reading and recorded a body temperature of 75. 6°F. Since Mc. Ham’s normal temperature was 98. 6°F, 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?