A slab of plate glass containing dissolved helium is placed in a vacuum furnace at a temperature of 400 C to remove the helium from the glass. The concentration of helium is constant throughout the glass before vacuum heat treatment. After 10 minutes in vacuum at 400 C, at what depth from the surface of the glass has the concentration of helium decreased to 1/3 of its initial value? Express your answer in meters. The diffusion coefficient of helium in plate glass at 400 C is 3.091 x 10-6 cm2/s
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To find the depth from the surface at which the concentration of helium has decreased to 1/3 of its initial value, we can use Fick's second law of diffusion:
∂C/∂t = D * (∂²C/∂x²),
where C is the concentration of helium, t is time, D is the diffusion coefficient, and x is the depth from the surface.
First, let's convert the diffusion coefficient from cm²/s to m²/s:
D = 3.091 x 10⁻⁶ cm²/s = 3.091 x 10⁻¹⁰ m²/s.
Next, we substitute the given values into Fick's second law:
∂C/∂t = (3.091 x 10⁻¹⁰) * (∂²C/∂x²).
Since the concentration of helium is constant throughout the glass before heat treatment, the left side of the equation can be simplified to:
∂C/∂t = 0.
Now, we are left with:
0 = (3.091 x 10⁻¹⁰) * (∂²C/∂x²).
Since the concentration of helium has decreased to 1/3 of its initial value, we can express it as:
C = (1/3) * C₀,
where C₀ is the initial concentration of helium.
Now, let's integrate the equation above to find the depth x at which the concentration reaches (1/3) * C₀.
Integrating twice with respect to x, we get:
C = (1/3) * C₀ = (2/3) * C₀ * (1 - 2 * (x/L)),
where L is the depth of the glass slab.
Solving for x:
(1 - 2 * (x/L)) = (1/2),
-2 * (x/L) = -1/2,
x/L = 1/4,
x = (1/4) * L.
Finally, we can multiply the total time (10 minutes) by the appropriate conversion factors to find the depth in meters:
10 minutes * (60 seconds/1 minute) * (3.091 x 10⁻¹⁰ m²/s) = 1.8546 x 10⁻⁷ meters.
Therefore, after 10 minutes in vacuum at 400°C, the concentration of helium has decreased to 1/3 of its initial value at a depth of 1.8546 x 10⁻⁷ meters from the surface of the glass.