When was the Victim Murdered? Newton's Law of Cooling

Unfortunately, each of his ex-wives and all of his children are known for living extravagent lifestyles well above their means, and were eagerly counting on their share of his estate to pay off their numerous debts before the repossession of their mansions, oil paintings, stamp collections, and tiaras. They all have a motive to get their hands on his money, and weren't happy that he was so philanthropic. He had told all of them that they were in his will, but refused to disclose the amounts.

It's summer, and the air conditioner in the mansion library has been on all day, and the temperature was a steady $72^{o}F$ throughout the mansion. This information will be useful later in our story.

The coroner indicates that the victim was stabbed multiple times. The assailant apparently wore gloves, left no blood traces, fingerprints, shoe prints, or any other useful forensic clues. But according to the coroner, the victim's body temperature, at $10:50$ p.m was $85.4^{o} F$. This will also be useful.

The police are almost certain that one of his children or ex-wives is the murderer, so it would be helpful if they could narrow down the list of potential suspects. Some may have alibis, so knowing the victim's time of death is critical. But when was the victim murdered?

Before forensic scientists had more accurate, contemporary methods, scientist Isaac Newton came to their rescue, with his Law of Cooling.

$T(t) = T_e + (T_o - T_e)e^{kt}$

But what does that mean?

At first this law looks daunting, but determining the time of death is fairly simple. because we know the victim's initial body temperature, $T_o$ and the room temperature, $T_e$.

$k$ is simply a constant that depends on the properties of the object. In this case, ours is a dead body. Previous research has determined that a good estimate for the value of $k$ for this situation is about $-0.134/hour$.

We will make one assumption, that the victim's initial body temperature was $98.6^{o} F$.

We are trying to determine $t$, the amount of time the victim's body has been cooling, which should tell us approximately how long he has been dead.

The important numbers we need are:

$T(t) = 85.4^{o} F$ (victim body temperature taken by the coroner at 10:50 p.m., temperature at time $t$)

$T_e = 72^{o} F$ (the temperature of the room, the environment temperature)

$T_o = 98.6^{o} F$ (the victim's temperature before he was murdered, his original body temperature)

$k = -0.134/hour$ (sources provide different values but we'll use this one from [3])

Using Newton's Law of Cooling,

$T(t) = T_{e} + (T_{o} - T_{e})e^{kt}$

we can now substitute those numbers, and to make things simple, we'll leave off the units:

$85.4 = 72 + (98.6 - 72)e^{-0.134t}$

And doing a little simplification:

$85.4 = 72 + 26.6e^{-0.134t}$

Subtract $72$ from both sides of the equation:

$13.4 = 26.6e^{-0.134t}$

Divide both sides of the equation by $26.6$:

$0.504 = e^{-0.134t}$

We can do a little algebra and use a natural log rule, and rewrite this as:

$\ln (0.504) = -0.134t$

Then with a little assistance from your calculator and the LN key:

$t = \frac { \ln (0.504)}{-0.134}$

$t = 5.11 \; hours$

$0.11 \; hours = 0.11 \; hours \; x \; \left ( \frac{60 \; minutes}{1 \; hour} \right ) = 6.6 \; minutes$

We'll round that to $7$ minutes. To calculate the victim's time of death we need to subtract $5$ hours and $7$ minutes from the time the coroner recorded the victim's body temperature:

$10:50 \; p.m. - \; 5 \; hours - \; 7 \; minutes \; = \; 5:43 \; p.m.$

So the victim was murdered at about $5:43 \; p.m.$

If we don't want to use all of those steps, we can use this equation to determine $t$:

$t = \frac{1}{k} \ln \left ( \frac{T(t) - T_e}{T_o - T_e} \right )$

$t = \frac{1}{-0.134} \ln \left ( \frac {85.4 - 72}{98.6 - 72} \right )$

$t = \frac{1}{-0.134} \ln \left ( \frac {13.4}{26.6} \right )$

$t = \frac{1}{-0.134} \ln (0.504)$

$t = 5.11 \; hours$

That verifies that the time of death was about 5:43 p.m.

You and the police have a lot of work to do. At 5:43 p.m, the children may not have started their Saturday night partying, and likely don't have alibis. The ex-wives might have dinner and social engagements, and could be home selecting the evening's jewelry. Now you and the police can use investigative skills and the victim's time of death to determine the most likely suspects.

Sources:

Denise Meeks, dmeeks@email.arizona.edu