using Newton’s Law of Cooling to calculate the time of death

1. Problem
The body of a monkey was discovered in a lab at
9am, and its body temperature measured at 9am and
11am. The temperature of the room changed from
30°C to 20°C at an unknown time between 6pm and
6am. Additionally, the body temperature of the
monkey could have varied 2°C . Using this
information, determine the time of death.
Facts
• Temperature of a living
Capuchin monkey is 40 ±
2°C.
• Body temperature at 9am
was 30°C.
• Body temperature at 11am
was 28°C.
• The room temperature
dropped 10°C during the
night.
• k = ~ .011157 (Equation
II)
• Body temp at 6am =
~33.97542 (Equation IV)
Assumptions
• Time for the lab to cool to the
new temperature setting is
negligent
• Monkey models as a particle,
thus we can use Newton’s Law
of Cooling to determine the
time of death
• Monkey’s body temperature
does not affect the room
temperature
Newton’s Law of Cooling:
I. 𝑇 = 𝑇𝐴 + (𝑇 𝐻 − 𝑇𝐴)𝑒−𝑘𝑡
II. 𝑘 =
ln
𝑇−𝑇 𝐴
𝑇 𝐻−𝑇 𝐴
−𝑡
III. 𝑡 =
ln
𝑇−𝑇 𝐴
𝑇 𝐻−𝑇 𝐴
−𝑘
IV. 𝑇 𝐻 = 𝑇𝐴 +
𝑇−𝑇 𝐴
𝑒−𝑘𝑡
T – Temperature of the body (°C)
TH – Initial temperature of the monkey
(°C)
TA – Ambient temperature (°C)
t- Time (hours)
k – A constant
Earliest and Latest
Times of Death
• Equation III to
determine time
of death in hours
before a given
time (t)
• Assume the ambient temp dropped 10°C at 6am or
6pm, so room was either 30°C or 20°C through the
night (TA)
• Temperature of the body at 6am (earliest) and 11am
(latest). (T)
• Monkey’s initial temperature is 42°C (earliest) or
38°C (latest). (TH)
9.90192 = 9 hours and 54 minutes before 6am (earliest:
20:06)
7.26824 = 7 hours and 16 minutes before 11am (latest:
3:44)
Body Temp Earliest Latest
38 23:44 3:44
40 21:44 2:47
42 20:06 1:56
Linear time of Death
Equation III to determine the time of death in hours before 6am.
(t)
Assume ambient temperature changes linearly, so the room’s
temperature decreases by –(
10
12
)°C per hour. (TA)
Temperature of the monkey at 6am. (T)
Monkey’s initial temperature can be
40 ± 2°C. (TH)
Body Temp Time of Death
38 3:38
40 2:30
42 1:30