Question

4. A mysterious document had been discovered in F¨uhrerbunker in Berlin and Robert Langdon was urgently called to investigate. As part of his top secret mission, at midnight he goes into a concealed archival chamber, which is sealed off from the rest of the facility by a hermetical door that will be reopened at exactly 6 am. The chamber has the volume of 2000 m3 and a primitive air circulation system with air pumped in and out at the rate of 1000 L/min. Prof. Langdon only needs two hours to examine the document. Unbeknownst to him, an agent of the powerful Rote Hakenkreuz secret society had infiltrated the security, and has managed to connect a source of carbon monoxide (CO) to the air intake. The poisonous gas is dispersed uniformly and the mixture is continuously removed via the air outflow. (a) Assuming that the concenration of CO in the air inflow is 0.5%, or 5000 ppm (parts per million), write down a differential equation for the amount A(t) of CO in the chamber as a function of time. (b) At midnight, the concentration of CO in the chamber is already 200 ppm. Set up and solve the initial value problem for A(t). (c) As a former diver, Langdon can tolerate up to 480 ppm of CO while maintaining his mental acuity and up to 1000 ppm before he passes out. Will the professor survive this ordeal and complete his mission?

Answer 1

Rate of inflow:

Amount of CO inflow:

Amount of CO (in fraction) at any time in the room:

Amount of outflow: where V is the volume of the room

Solving the equation:

Integrating both sides,

K is the integration constant

At t=0, A(0)=0.0002 (200 ppm)

Now, we will start plugging in the values.

Therefore,

For completion of mission,

taking ln on both sides

Prof. Langdon requires 2 hours to complete his mission. So he will be just about done.

He passes out at 1000 ppm,

taking ln on both sides

He has 12 AM to 6 AM (6 hours) which means he'll be fine. He has had closer calls anyway :p