A bar of pure nickel is coupled with a bar of pure iron (interfaced). The diffusion couple is then heated to a temperature of 1000°C. (a) How long will it take for the concentration of nickel to reach 0.1 wt%, 1.0 μm below the interface? (b) How long will it take for the concentration of nickel to reach 0.1 wt%, 1.0 mm below the interface? (c) What does the comparison of the two answers show?
To determine the time required for the concentration of nickel to reach a specific level at different distances below the interface, you can use Fick's second law of diffusion. Fick's second law describes how the concentration of a diffusing species changes with time and position.
The equation for Fick's second law is:
dC/dt = D * (d^2C/dx^2)
Where:
dC/dt is the rate of change of concentration with time,
D is the diffusion coefficient, and
(d^2C/dx^2) is the second derivative of concentration with respect to position.
In this case, we want to find the time required for the nickel concentration to reach 0.1 wt% at different distances below the interface.
(a) To find the time required for the concentration of nickel to reach 0.1 wt%, 1.0 μm below the interface, we need to determine the diffusion coefficient of nickel in the iron-nickel system at the given temperature. Let's assume the diffusion coefficient is D = 1 x 10^-9 cm^2/s. We also need to know the initial concentration of nickel at the interface, which we can assume is the bulk concentration of nickel in the pure nickel bar.
Using the equation for Fick's second law, we can write:
(dC/dt) = D * (d^2C/dx^2)
Integrating this equation with appropriate boundary and initial conditions, we can solve for the time required. Since the initial concentration of nickel at the interface is known, we can substitute the concentration of 0.1 wt% and the distance of 1.0 μm into the equation and solve for time.
(b) To find the time required for the concentration of nickel to reach 0.1 wt%, 1.0 mm below the interface, we repeat the same process as in (a) using the new distance value.
(c) The comparison of the two answers (time for reaching the same concentration at different distances) shows that the diffusion process is slower at a greater distance from the interface. This is because the concentration gradient decreases as the distance from the interface increases, resulting in slower diffusion. Therefore, the time required for the concentration to reach a certain level increases as the distance from the interface increases.