A latex balloon, wall thickness 3.091 x 10-4 m, contains helium at a concentration of 0.31 kg m-3. Under these conditions the total surface area of the balloon is 0.11 m2. The diffusion coefficient of He in latex at room temperature is 4.9 x 10-9 m2s-1. Calculate the rate of helium effusion (in g/hr) from the balloon.
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42.8
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Fick’s Law
M= -D•(Δρ/Δx)•A,
where
M = m/t = mass flux [kg/s],
D= 4.9•10 ⁻⁹ m ²/s is the diffusion coefficient (diffusivity) [m²/s],
Δρ/Δx is the gradient of density [kg/m⁴],
A is the area [m²].
M= - 4.9•10⁻⁹•(0 - 0.35)/3.091•10⁻⁴)•0.88 =
4.9•10⁻⁹•0.35•0.88/3.091•10⁻⁴=
=4.88•10⁻⁶ kg/s= 17.58 g/hr
To calculate the rate of helium effusion from the balloon, we need to use Fick's law of diffusion, which states that the rate of diffusion of a substance is proportional to the surface area and concentration gradient of that substance.
The formula for Fick's law is:
rate of diffusion = (diffusion coefficient * surface area * concentration gradient) / thickness
Let's break down the given information:
- Diffusion coefficient (D) = 4.9 x 10^-9 m^2/s
- Surface area (A) = 0.11 m^2
- Concentration gradient (∆C) = the difference in concentration between the inside and outside of the balloon
- Thickness of the balloon wall (d) = 3.091 x 10^-4 m
Since we are given the concentration of helium inside the balloon (0.31 kg/m^3), we need to find the concentration of helium outside the balloon to calculate the concentration gradient.
Assuming the concentration of helium outside the balloon is negligible (or zero), the concentration gradient (∆C) would be:
∆C = concentration inside the balloon - concentration outside the balloon
= 0.31 kg/m^3 - 0 kg/m^3
= 0.31 kg/m^3
Now we can calculate the rate of helium effusion using Fick's law:
rate of diffusion = (diffusion coefficient * surface area * concentration gradient) / thickness
= (4.9 x 10^-9 m^2/s * 0.11 m^2 * 0.31 kg/m^3) / (3.091 x 10^-4 m)
The unit of the rate of helium effusion should be in grams per hour. To convert from kg/s to g/hr, we can multiply the result by 3600 (seconds to hours) and 1000 (kg to g):
rate of helium effusion = [(4.9 x 10^-9 m^2/s * 0.11 m^2 * 0.31 kg/m^3) / (3.091 x 10^-4 m)] * 3600 * 1000
Now you can plug in the values into this equation to find the rate of helium effusion from the balloon in grams per hour.