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Classical Mechanics Level 4

If you were floating around in empty space, how long would it take before most of your body freezes into ice, assuming that no biochemical processes take place in the body?

Details and Assumptions :

The human body can be modeled as $70 \text{ kg}$ of water, around $310 \text{ K}$ , that is $37\text{ K}$ above the freezing point of water, radiating like a blackbody.
The surface area of the body is about $2\ \text{m}^2$ .
Useful constants include $c_w = 4180\ \text{J kg/K}$ and $\sigma = 5.67\times 10^{-8}\ \text{W/m}^2\ \text{K}^4$ .

Several seconds Several minutes Several hours Several days

1 solution

Arjen Vreugdenhil
May 16, 2016

At a temperature around 300 K, a black body radiates with an intensity of $I = \sigma T^4 \sim 5.67\times 10^{-8} \cdot 300^4 \sim 500\ \text{W/m}^2.$ The rate at which the body would lose heat is $P = I\ A \sim 500\cdot 2 \sim 1000\ \text{W}.$ The corresponding temperature drop is $dT/dt = \frac{P}{mc} \sim \frac{1000}{70\cdot 4180} \sim 0.003\ \text{K/s}.$ At this rate, losing $37\ \text{K}$ would take $t \sim \frac{37}{0.003} \sim 12\:000\ \text{s} \approx 3\ \text{h}.$ This brings the body to the freezing point. Freezing the whole body takes as much heat loss as dropping the temperature by $80\ \text{K}$ , which would take about 6 additional hours.

The answer, then, is $\boxed{\text{several hours}}$ .

It's a jungle out there

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