Consider a diffusion couple between silver and a gold alloy that contains 10 wt \% silver. This couple is heat treated at an elevated temperature and it was found that after 850 s, the concentration of silver had increased to \(12 \mathrm{wt} \%\) at \(10 \mu \mathrm{m}\) from the interface into the Ag-Au alloy. Assuming preexponential and activation energy values of \(7.2 \times 10^{-6}\) \(\mathrm{m}^{2} / \mathrm{s}\) and \(168,000 \mathrm{~J} / \mathrm{mol}\), respectively, compute the temperature of this heat treatment. (Note: You may find Figure \(5.13\) and Equation \(5.15\) helpful.) For a steel alloy, it has been determined that a carburizing heat treatment of \(15 \mathrm{~h}\) duration will raise the carbon concentration to \(0.35\) wt \(\%\) at a point \(2.0 \mathrm{~mm}\) from the surface. Estimate the time necessary to achieve the same concentration at a \(6.0-\mathrm{mm}\) position for an identical steel and at the same carburizing temperature.

Step by step solution

01

Recall Fick’s Second Law of Diffusion

First, we need to recall Fick's second law of diffusion. It is a fundamental equation that describes the diffusion process: \(D = D_0 \cdot e^{\frac{-Q}{R \cdot T}}\) Where: \(D\) is the diffusion coefficient \(D_0\) is the pre-exponential value given as \(7.2 \times 10^{-6} \mathrm{m}^{2}/\mathrm{s}\) \(Q\) is the activation energy, \(168,000 \mathrm{J/mol}\) \(R\) is the gas constant, \(8.314 \mathrm{J/mol \cdot K}\) \(T\) is the temperature in Kelvin Additionally, Fick's second law relates the concentration of a component at a certain distance and time: \(\frac{\delta C}{\delta t} = D \frac{\delta^2 C}{\delta x^2}\) We are given the change in concentration, distance, and time, and we want to find the temperature, so we need to rearrange Fick's second law to find the diffusion coefficient and then use the temperature equation.

02

Calculate the change in concentration and distance in the interface

Δ\(C = 12 \mathrm{wt \%} - 10 \mathrm{wt \%} = 2 \mathrm{wt \%}\) Δ\(x = 10 \cdot 10^{-6} \mathrm{m}\) Also, Δ\(t = 850 \mathrm{s}\) is given.

03

Rearrange Fick's second law and calculate temperature

We can now rearrange Fick's second law using the given data: \(D = \frac{(\Delta C)(\Delta x^2)}{2(\Delta t)}\) Then, plug in the given values and calculate \(D\): \(D = \frac{(2 \mathrm{wt \%})(10 \cdot 10^{-6} \mathrm{m})^2}{2(850 \mathrm{s})} = 1.176 × 10^{-14} \mathrm{m}^{2}\mathrm{/s}\) Now that we have the diffusion coefficient, we can calculate the temperature using the following equation: \(T = \frac{-Q}{R \cdot ln(\frac{D}{D_0})}\) Substitute the given values and solve for \(T\): \(T = \frac{-168,000 \mathrm{J/mol}}{8.314 \mathrm{J/mol \cdot K} \cdot ln(\frac{1.176 \times 10^{-14} \mathrm{m}^{2}\mathrm{/s}}{7.2 \times 10^{-6} \mathrm{m}^{2} / \mathrm{s}})} = 1504.45 \mathrm{K}\) The temperature of the heat treatment is approximately 1504.45 K. #Problem 2: Estimate the Time for Different Concentration at a New Position#

04

Use Fick's Law for the initial condition and find diffusion coefficient

To estimate the time needed to achieve the same concentration at a new position, we can use Fick's second law again. First, consider the initial condition: Δ\(C = 0.35 \mathrm{wt \%}\) Δ\(x = 2 \cdot 10^{-3} \mathrm{m}\) Δ\(t = 15 \times 3600 \mathrm{s}\) Calculate \(D\) using Fick's second law: \(D = \frac{(\Delta C)(\Delta x^2)}{2(\Delta t)} = \frac{(0.35 \mathrm{wt \%})(2 \cdot 10^{-3} \mathrm{m})^2}{2(15 \times 3600 \mathrm{s})} = 3.24 × 10^{-13} \mathrm{m}^{2}\mathrm{/s}\)

05

Calculate the time for the new condition

Now, we need to calculate the time needed to achieve the same concentration at the new position (6.0 mm). For that, we will use the following relation: \(t = \frac{(\Delta C)(\Delta x^2)}{2(D)}\) For the new position: Δ\(x = 6 \cdot 10^{-3} \mathrm{m}\) Now, substitute the values and find the time: \(t = \frac{(0.35 \mathrm{wt \%})(6 \cdot 10^{-3} \mathrm{m})^2}{2(3.24 × 10^{-13} \mathrm{m}^{2}\mathrm{/s})} = 116640 \mathrm{s}\) The time necessary to achieve the same concentration at a 6.0-mm position for an identical steel and at the same carburizing temperature is approximately 116640 seconds.

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