You wish to remove nitrogen from the near-surface region of a plate of cobalt which is 1.0 cm thick. The plate is placed in a furnace at 417∘ C where an atmosphere of H2 and NH3 reacts with the nitrogen in the cobalt and fixes the surface concentration to 111 ppm (parts per million by mass). If the initial concentration is 3091ppm, how long will it take to reduce the nitrogen concentration to 1662 ppm at a depth of 10μm? The diffusion of nitrogen in cobalt has an activation energy of 100 kJ/mol and a preexponential value (Do) of 0.01 cm2/sec.
Give your answer in units of seconds.
To determine the time required to reduce the nitrogen concentration in the cobalt plate, you can use Fick's second law of diffusion, which describes the diffusion of substances in a solid material. The equation is as follows:
∂C/∂t = D * (∂²C/∂x²)
Where:
∂C/∂t is the rate of change of concentration over time,
D is the diffusion coefficient of nitrogen in cobalt,
∂²C/∂x² is the second derivative of concentration with respect to distance.
To solve this equation, we need to use the boundary conditions. At the surface of the plate, where x = 0, the concentration is fixed at 111 ppm. At a depth of 10 μm, where x = 0.1 cm, we want the concentration to reach 1662 ppm.
Assuming that the concentration profile varies only in the x-direction and is independent of y and z, we can integrate Fick's second law under these conditions:
∫(∂C/∂t) dt = D * ∫(∂²C/∂x²) dx
Integrating both sides of the equation, we get:
∫(∂C/∂t) dt = D * ∫(∂²C/∂x²) dx
∫(∂C/∂t) dt = D * (dC/dx)
Integrating with respect to time and concentration, and plugging in the boundary conditions:
∫[C₀ - C(t)] / dt = D * ∫(dC / dx) dx
Where C₀ is the initial concentration (3091 ppm), and C(t) is the concentration at a depth of 10 μm (1662 ppm).
∫[C₀ - C(t)] / dt = D * ∫(dC / dx) dx
∫[3091 - C(t)] / dt = D * ∫(dC / dx) dx
Now, let's solve for the time required to reduce the concentration to 1662 ppm at a depth of 10 μm. To do this, we'll need to calculate the diffusion coefficient (D) using the given activation energy and preexponential value. The diffusion coefficient can be determined using the Arrhenius equation:
D = Do * exp(-Ea / (R * T))
Where:
Do is the preexponential value of the diffusion coefficient,
Ea is the activation energy,
R is the ideal gas constant (8.314 J/(mol·K)),
T is the temperature in Kelvin.
First, convert the given temperature from Celsius to Kelvin:
T = 417°C + 273.15 = 690.15 K
Then, substitute the values into the Arrhenius equation:
D = 0.01 cm²/sec * exp(-100,000 J/mol / (8.314 J/(mol·K) * 690.15 K))
Calculate the value of D using an appropriate scientific calculator or software.
Once you have the diffusion coefficient (D) and the concentration values, you can solve the integral equation numerically to determine the time required to reduce the concentration. This can be done using numerical methods such as finite difference or finite element methods. Alternatively, you can use software packages or programming languages with built-in numerical solvers.
Note: The exact approach for solving the integral equation will depend on the software or method you choose.