A latex balloon, wall thickness 3.091 x 10-4 m, contains helium at a concentration of 0.35 kg m-3. Under these conditions the total surface area of the balloon is 0.57 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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what is the formula?
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. Fick's Law states that the rate of diffusion (effusion in this case) is proportional to the concentration gradient and the surface area, and inversely proportional to the thickness of the material. The equation for Fick's Law is as follows:
r = -D * (A / x) * (dc / dx)
Where:
r = rate of diffusion (effusion)
D = diffusion coefficient
A = surface area of the balloon
x = thickness of the balloon wall
dc/dx = concentration gradient
Let's calculate step by step:
1. Convert the thickness of the balloon wall from scientific notation to a decimal:
3.091 x 10^(-4) m = 0.0003091 m
2. Plug in the values into the Fick's Law equation:
r = - (4.9 x 10^(-9) m^2/s) * (0.57 m^2) / (0.0003091 m) * (0.35 kg/m^3)
3. Simplify and convert units:
r = -8.418 g/s
4. Convert the rate to grams per hour:
r = (-8.418 g/s) * (3600 s/hr) = -30262.8 g/hr
The negative sign in the result indicates that helium is effusing out of the balloon.
Therefore, the rate of helium effusion from the balloon is approximately 30262.8 grams per hour.