Oxygen diffuses from the surface of insects to the interior through tiny tubes called tracheae. An average trachea is about 2mm long and has cross-sectional area of 2×10−9m^2. Assuming the concentration of oxygen inside is half what it is outside in the atmosphere. Find the concentration of oxygen in the air (assume 21% is oxygen) at 20∘C.Calculate the diffusion rate JJ. Assume the diffusion constant is 1×10−5m^2.Estimate the average time for a molecule to diffuse in.
To find the concentration of oxygen in the air, we can start by calculating the partial pressure of oxygen in the atmosphere at 20∘C.
1. Convert the percentage of oxygen in air to a decimal: 21% = 0.21.
2. Find the atmospheric pressure at 20∘C, which is approximately 101.3 kPa.
3. Multiply the atmospheric pressure by the decimal representing the percentage of oxygen to get the partial pressure of oxygen: 0.21 x 101.3 kPa = 21.273 kPa.
Now that we have the partial pressure of oxygen in the air, we can calculate the concentration of oxygen inside the tracheae using the given information:
1. Divide the partial pressure of oxygen inside the tracheae by the atmospheric pressure to get the concentration: 21.273 kPa / 101.3 kPa = 0.210.
The concentration of oxygen inside the tracheae is 0.210 or 21% of the atmospheric concentration.
Next, let's calculate the diffusion rate (J) using Fick's Law, which states that the diffusion rate is directly proportional to the concentration difference and the surface area, while inversely proportional to the distance:
J = -D * A * (dC/dx),
where:
- J is the diffusion rate,
- D is the diffusion constant (given as 1×10−5 m^2/s),
- A is the cross-sectional area (2×10−9 m^2),
- dC/dx is the concentration gradient.
Since we know that the concentration inside is half of the concentration outside, the concentration gradient can be calculated as follows:
dC/dx = (C_inside - C_outside) / x
= (0.210 - 0.21) / (2×10^-3 m)
= 0 / (2×10^-3 m)
= 0.
Therefore, the diffusion rate (J) is 0.
Now, let's estimate the average time for a molecule to diffuse in. The average time can be calculated using the equation:
t = x^2 / (2D),
where:
- t is the average time,
- x is the distance (2 mm = 2×10^-3 m),
- D is the diffusion constant (1×10^-5 m^2/s).
Plugging in the values:
t = (2×10^-3 m)^2 / (2 * 1×10^-5 m^2/s)
= 4×10^-6 / 2×10^-5
= 0.2.
Therefore, the average time for a molecule to diffuse in is 0.2 seconds.