Mission Improbable

I watched Mission: Impossible yesterday and was reminded of two things: (1) there was a time when people thought Tom Cruise wasn't crazy and (2) there was a time when people thought the internet was.  Long before anyone dreamed of Facebook, Twitter, or rage comics, the internet was a pretty novel concept, and many people, myself included, weren't ready to dive in head first.  I haven't thought about this for awhile, but I'm pretty sure Mission: Impossible was the reason I decided to get a computer.  After all, if super spies used the internet, then it was probably pretty darn cool.  Who knows?  If I hadn't seen it, maybe I wouldn't be blogging this now.¹  Still, there's one thing that bugged me in my second viewing.  In the iconic scene where Cruise is dangling from a rope while trying to steal sensitive government data, an alarm will go off and ruin the mission if he touches the floor, makes a sound, or raises the temperature of the room by 1° F.  There were a few close calls on these fronts, but I was particularly skeptical of the temperature restriction.  How long would it take your body heat to raise the temperature of the room by 1° F?

Human bodies release heat at a rate of about P = 100 W.²  This heat will spread throughout the room, which has a volume of roughly 15 ft by 15 ft by 30 ft = 200 m³.   With a density of 1 kg/m³, the total mass of air in the room will be about m = 200 kg.  The specific heat of water is about c = 1 J/g·K.  From this we can fin the time it will take to raise the room by 1° F (~0.6 K),  

 t = m c ΔT / P

= (200 kg) · (1 J/g K) · (1° F) / (100 W)

= 20 minutes.

Apparently, they would have time as long as they were fairly quick about it.³

[1] Admittedly, if I had never seen the movie, I couldn't be writing this post right now, but you get the idea.
[2] See the post titled "Igloos Rock!"
[3] I've neglected the fact that the room is temperature controlled.  As the temperature rises, the temperature control will turn on the AC to return it to normal.  This may have given them even more time.