Insects do not have lungs as we do, nor do they breathe through their mouths. In
ID: 1473537 • Letter: I
Question
Insects do not have lungs as we do, nor do they breathe through their mouths. Instead, they have a system of tiny tubes, called tracheae, through which oxygen diffuses into their bodies. The tracheae begin at the surface of the insect's body and penetrate into the interior. Suppose that a tracheae is 1.77 mm long with a cross-sectional area of 1.49 x 10-9m2. The concentration of oxygen in the air outside the insect is 0.602 kg/m3, and the diffusion constant is 1.97 x 10-5m2/s. If the mass per second of oxygen is diffusing through a trachea is 1.49 x 10-12 kg/s, then find the oxygen concentration at the interior end of the tube.
Explanation / Answer
You must use the equation m=(DA (Coutside-Cinside))t/L where m is mass, D is the diffusion constant, A is the area, C concentration which you are trying to find the interior concentration in this problem, and the exterior concentration is given, t is time and L is the length
L=1.77mm=.00177m A=1.49 x 10-9m2 D=1.97 x 10-5m2/s Coutside=.602 The only tricky thing is they give you the mass per second so you can just use t= 1s and m= 1.49 x 10-12 kg
m = [DA(C2-C1)]t/L
This looks like a rate equation if you move 't' to the left side:
m/t = DA(C2-C1)/L
Note that m/t, not m, equals 1.45E-12 kg/s.
Now if C1 is the high concentration I'd think it would be
m/t = DA(C1-C2)/L
Then
m/t*L/(DA) = C1-C2 ==>
C2 = C1-m/t*L/(DA) = 0.602-(1.49E-12*0.00177)/1.97E-5*1.49E-9) = 0.5121 kg/m^3
(In other words, the same difference from the outside density 0.602 kg/m^3 but downward not upward.)