A latex balloon, wall thickness 3.091 x 10-4 m, contains helium at a concentration of 0.5 kg m-3. Under these conditions the total surface area of the balloon is 0.72 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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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 can use Fick's Law of diffusion, which states that the rate of diffusion is proportional to the surface area, concentration difference, and the diffusion coefficient.
The formula for Fick's Law is:
Rate of diffusion = Surface area × Concentration difference × Diffusion coefficient
First, let's determine the concentration difference. The concentration inside the balloon is given as 0.5 kg/m^3. To find the concentration outside the balloon, we need to consider that helium diffuses through the latex balloon, and the latex has a wall thickness of 3.091 x 10^-4 m.
To calculate the concentration outside the balloon, we can assume that the concentration is negligible after it has diffused through the entire wall thickness. So the concentration outside the balloon is effectively 0 kg/m^3.
Now we can calculate the rate of diffusion.
Rate of diffusion = Surface area × (Concentration inside - Concentration outside) × Diffusion coefficient
Given:
Surface area = 0.72 m^2
Concentration inside = 0.5 kg/m^3
Concentration outside = 0 kg/m^3
Diffusion coefficient = 4.9 x 10^-9 m^2/s
Rate of diffusion = 0.72 m^2 × (0.5 kg/m^3 - 0 kg/m^3) × 4.9 x 10^-9 m^2/s
Simplifying this equation, we have:
Rate of diffusion = 0.72 × 0.5 × 4.9 x 10^-9
= 0.1764 x 10^-9 g/s
To convert this value to g/hr, we need to multiply it by the appropriate conversion factors. There are 3600 seconds in an hour and 1 g = 1000 mg.
Rate of helium effusion = (0.1764 x 10^-9 g/s) × (3600 s/hr) × (1000 mg/g)
= 635.04 mg/hr
Therefore, the rate of helium effusion from the balloon is approximately 635.04 mg/hr.