1. **State the problem:** We have a copper-zinc alloy with 90 wt% copper and 10 wt% zinc coupled with pure copper. The diffusion couple is heated to 1000 °C. We want to find the time it takes for the zinc concentration to reach 0.2 wt% at a depth of 2.5 mm below the interface. 2. **Formula used:** Diffusion in solids can be described by Fick's second law. For a semi-infinite solid with a constant surface concentration, the concentration profile is given by: $$ C(x,t) = C_s + (C_0 - C_s) \operatorname{erf}\left(\frac{x}{2\sqrt{Dt}}\right) $$ where: - $C(x,t)$ is the concentration at depth $x$ and time $t$, - $C_s$ is the surface concentration (at the interface), - $C_0$ is the initial concentration in the bulk, - $D$ is the diffusion coefficient, - $\operatorname{erf}$ is the error function. 3. **Known values:** - $C_s = 10$ wt% zinc (surface concentration from alloy side), - $C_0 = 0$ wt% zinc (pure copper side), - $C(x,t) = 0.2$ wt% zinc (desired concentration), - $x = 2.5$ mm = 0.0025 m, - Temperature = 1000 °C. 4. **Find diffusion coefficient $D$ for zinc in copper at 1000 °C:** Typical diffusion coefficient follows Arrhenius equation: $$ D = D_0 e^{-\frac{Q}{RT}} $$ where $D_0$ and $Q$ are material constants, $R$ is gas constant, $T$ is absolute temperature. For zinc in copper, approximate values are: - $D_0 \approx 2.1 \times 10^{-4}$ m$^2$/s, - $Q \approx 1.3 \times 10^{5}$ J/mol, - $R = 8.314$ J/mol·K, - $T = 1000 + 273 = 1273$ K. Calculate $D$: $$ D = 2.1 \times 10^{-4} e^{-\frac{1.3 \times 10^{5}}{8.314 \times 1273}} $$ Calculate exponent: $$ \frac{1.3 \times 10^{5}}{8.314 \times 1273} = \frac{130000}{10587.722} \approx 12.27 $$ So: $$ D = 2.1 \times 10^{-4} e^{-12.27} $$ Calculate $e^{-12.27} \approx 4.7 \times 10^{-6}$ Therefore: $$ D \approx 2.1 \times 10^{-4} \times 4.7 \times 10^{-6} = 9.87 \times 10^{-10} \text{ m}^2/\text{s} $$ 5. **Rearrange the concentration formula to solve for $t$:** $$ \frac{C(x,t) - C_s}{C_0 - C_s} = \operatorname{erf}\left(\frac{x}{2\sqrt{Dt}}\right) $$ Plug in values: $$ \frac{0.2 - 10}{0 - 10} = \operatorname{erf}\left(\frac{0.0025}{2\sqrt{9.87 \times 10^{-10} t}}\right) $$ Calculate left side: $$ \frac{-9.8}{-10} = 0.98 $$ So: $$ 0.98 = \operatorname{erf}\left(\frac{0.0025}{2\sqrt{9.87 \times 10^{-10} t}}\right) $$ 6. **Find the inverse error function:** $$ \operatorname{erf}^{-1}(0.98) \approx 1.821 $$ 7. **Solve for $t$:** $$ 1.821 = \frac{0.0025}{2\sqrt{9.87 \times 10^{-10} t}} $$ Rearranged: $$ 2\sqrt{9.87 \times 10^{-10} t} = \frac{0.0025}{1.821} $$ Calculate right side: $$ \frac{0.0025}{1.821} \approx 0.001373 $$ Square both sides: $$ (2\sqrt{9.87 \times 10^{-10} t})^2 = (0.001373)^2 $$ $$ 4 \times 9.87 \times 10^{-10} t = 1.885 \times 10^{-6} $$ Simplify: $$ 3.948 \times 10^{-9} t = 1.885 \times 10^{-6} $$ Divide both sides: $$ t = \frac{1.885 \times 10^{-6}}{3.948 \times 10^{-9}} $$ $$ t \approx 477.7 \text{ seconds} $$ **Final answer:** It will take approximately 478 seconds for the zinc concentration to reach 0.2 wt% at 2.5 mm below the interface at 1000 °C.