Answer:
The distance from the higher concentration side is [tex]= 4.06*10^{-3}m[/tex]
Explanation:
From the question we are told that
The thickness of the steel is [tex]D = 5.0mm = \frac{5}{1000} = 5*10^{-3} m[/tex]
The temperature is [tex]T = 1000^oC[/tex]
The diffusion coefficient of nitrogen in steel is [tex]D = 1.95 *10^{-10}m^2/s[/tex]
The diffusion flux is [tex]J = 1.2 *10^{-7}m^2 s[/tex]
The concentration of nitrogen in steel is [tex]M = 3kg/m^3[/tex]
The concentration at distance d is [tex]M_d = 0.5kg/m^3[/tex]
Generally Fick's first law show the relationship between diffusion flux and concentration under an assumption of steady state and this can be represented mathematically as
[tex]J = -D \frac{dC}{dx}[/tex]
Where D is the diffusion coefficient and [tex]\frac{dC}{dx}[/tex] is the concentration gradient
and J is the diffusion flux
Now if we are considering two concentration the equation for concentration gradient becomes
[tex]\frac{dC}{dx} = \frac{C_B - C_A }{x_B - x_A}[/tex]
Where [tex]C_A[/tex] is the concentraion at high pressure while [tex]C_B[/tex] is concentration at low pressure
[tex]x_A[/tex] is the position at the high concentration side
[tex]x_B[/tex] is the position at the low concentration side
Now sustituting values into the formula for concentration gradient
[tex]\frac{dC}{dx} = \frac{0.5 - 3}{x_B -x_A}[/tex]
[tex]\frac{dC}{dx} = \frac{-2.5}{x_B -x_A}[/tex]
Now substituting values into equation for Fick's law
[tex]1.2*10^{-7} =- 1.95 *10^{-10} \frac{-2.5}{x_B -x_A}[/tex]
[tex]1.2*10^{-7} =\frac{4.875*10^{-10}}{x_B -x_A}[/tex]
[tex]x_B - x_A = 4.06 *10^{-3}m[/tex]
[tex]x_B = x_A + 4.06 *10^{-3}m[/tex]
Since the position the higer concentration side from origin is [tex]x_A[/tex] the from the equation we see that the distance of the sheet from the higher concentration side is [tex]= 4.06*10^{-3}m[/tex]
