2. A Bacterium with size R in a lake, Oxygen concentration c(r) is steady, c(inf
ID: 3308044 • Letter: 2
Question
2. A Bacterium with size R in a lake, Oxygen concentration c(r) is steady, c(inf-co. This bacterium is consuming all oxygen around it, i.e. c(R)-0. Under this condition, there is a maximum rate of Oxygen supply rate: amount of oxygen can diffuse to the bacterium per second: I. We showed in the class that I = 4DRCo. The unit of l depends on the unit of Co, for example, if oxygen concentration Co is in the unit of mole/m, then the unit of I will be mole/sec. Assume the diffusion constant for O: is D-10 mls. a)Assume the metabolic consumption of O: for certain bacteria is 1.4x103 mole/kg/sec. i.e. amount of oxygen needed per unit of its mass per sec. Notice that its mass is proportional to its volume. Assume its mass density is the same as water. Find out the maximum radius of such spherical bacterium can grow, when the lake oxygen level Co 0.3 mole/m Can you think of some way for a bacterium to overcome such limit? b)Explanation / Answer
(a) Since the mass of the becterium is proportional to the volume of the sphere, therefore, the mass density is equal to the volume density, therefore, to calculate the spherical radius, we use the formula,
I = 4pi DR , where I is the maximum oxygen supply rate, D is the diffusion constant and C0 is mass density, therefore, since I = 1.4 x 10-3 mol/kg/sec, we get,
1.4 x 10-3 = 4 x 3.14 x 10-9 x R
R = 8.9714 x 10-6 m
(b) Since the relation of the maximum of the consumption of the oxygen is given by I = 4pi D R , where we can see that I is directly proportional to R , so the overcome this limit of oxygen consumption, the size or the spherical radius of the becterium must be large, so that it can overcome this limit and can consume more oxyegn according to it's size.
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