## Wednesday, February 9, 2011

### But Was It...MURDER???? [1]

"Meanwhile, Dr. Johnson’s cursory examination revealed the body was not quite cold; he concluded that death had occurred three to four hours earlier."
—E. J. Wagner's "A Murder in Salem" from the Nov. 2010 issue of Smithsonian

While stuck in Philadelphia International Airport after missing a connecting flight, I came across E. J. Wagner's "A Murder in Salem" in the Smithsonian magazine.  I was struck by the simple physical analysis implicit in Dr. Johnson's examination of the murdered man's body.  It seemed Sherlock Holmes-ian.  How much would a dead body cool in four hours?

In principle, a dead body loses most of its heat through the skin.2  There is a well known physics equation describing the amount of heat flowing through a material from a hot temperature to a cold temperature.  In symbols, this equation is written as

dQ / dt = (k · A / d) · ( Thot -Tcold )

Here, dQ is the heat energy that passes through the surface in some small amount of time dt.  The variables k, A, d, Thot, and Tcold represent the thermal conductivity, the surface area of the material, the thickness of the material, and the temperatures of the hot side and cold side, respectively.  Immediately before death, a healthy human body has a temperature of 37 °C (~98.6 °F).  The average April temperature in Salem, MA is 8.7 °C (~47.6 °F), but room temperature is 25 °C (~72 °F).3   I'll assume the temperature difference between a healthy body and air in the room is about 12 °C.  A large percentage of the energy our bodies use gets converted to heat.  We consume about 2000 Calories per day or about 100 W, so we can estimate the heat energy as

dQ / dt = 100 W.

Dividing dQ/dt by the temperature difference, we can estimate the ratio of k·A/d

k · A / d = ( dQ / dt ) / ( Thot -Tcold )
= (100 W) / (12 °C)
= 8.3 W/°C

The thermal conductivity equation is helpful, but we still need to know how much heat energy is stored in the dead man's body.  An object's capacity for storing heat is called the "heat capacity".  This heat capacity can be described by the equation

dQ = m · Cp · dT

where dQ is the heat energy lost when temperature decreases, m is the mass of the body that stores the heat, Cp is the heat capacity, and dT is the change in temperature.  The heat capacity of a human body is about 3470 J/kg·°C and I'll assume the deceased man weighed 60 kg.  Now you might think we could just solve these equations to obtain the time it takes for all the heat to flow out of the body, but there's one problem.  Since the body's cooling, the temperature difference between the body and the air is always changing.  To deal with rates of change, we generally use calculus.  To do this, we combine the two equations and integrate to get

TfinalTair + (Thealthy body - Tair) exp(- C t),

where,

C =k · A / [ m · Cp · d ]
= ( 8.3 W/°C)  / [ (60 kg) · (3470 J/kg·°C) ]
= 0.00004 Hz.

Plugging in, we can plot the body temperature as a function of time:

We get a final body temperature of 32°C.  This is still above room temperature but not yet cool, consistent with Dr. Johnson's observations.