Differential Equation on a Mixture Problem

The air in a room whose volume is 10000 cu ft tests 0.15% CO2. Starting at t=0, outside air testing 0.05% CO2 is admitted at the rate of 5000 cu ft per minute. (a) What is the percentage of CO2 in the air in the room after 3 minutes? (b) When does the air in the room test 0.1% CO2?

I can usually setup similar differential equations with respect to mixture problems, but this one stomps me. Can anyone help me setup the differential equation?

Let y = amount of CO2 in room measured in cubic feet

dy/dt = amount entering - amount exiting

dy/dt = 5000 * 0.0005 - 5000 * y / 10000

dy/dt = 2.5 - y / 2

y (t) = c e^(-0.5 t) + 5

. . . given y (0) = 10000 * 0.0015 = 15
. . . 15 = c + 5
. . . c = 10

y (t) = 10 e^(-0.5 t) + 5 <=== amount of CO2 after t minutes (not concentration in %)

Let P = percent concentration of CO2

P(t) = y (t) / 10000

P(t) = (10 e^(-0.5t) + 5) / 10000

P(t) = 0.001 e^(-0.5 t) + 0.0005

P(3) ~ 0.000723 = 0.0723% <==== a

0.001 = 0.001 e^(-0.5 t) + 0.0005

e^(-0.5 t) = 0.5

t = - ln(0.5) / 0.5 ... about 1.386 minutes

Still the air does not exits, so there is no "OUT part of the equation" .. but ...

Report Abuse

Since the gas has not an intrinsic volume,
the problem has to be solved considering masses ;
This infers knowing the relative densities of air and CO2 ,
or the specific gravity for CO2 , namely 1.52 ;

``````````````
I cannot develop any further right now,
but I'll be back to it tomorrow if the problem is still on.