The diffusion coefficients for silver in copper are given at two temperatures.(a) Determine the values of D0 and Qd.(b) What is the magnitude of D at 875°C?

1. Break Down the Problem

To solve the problem, we need to follow these steps:

1. Determine the values of $D_0$ (the pre-exponential factor) and $Q_d$ (the activation energy for diffusion) using the given diffusion coefficients at two temperatures.
2. Calculate the diffusion coefficient $D$ at 875°C using the Arrhenius equation.

2. Relevant Concepts

The diffusion coefficient can be expressed using the Arrhenius equation: $D = D_0 \exp\left(-\frac{Q_d}{RT}\right)$ where:

• $D$ is the diffusion coefficient,
• $D_0$ is the pre-exponential factor,
• $Q_d$ is the activation energy for diffusion,
• $R$ is the universal gas constant $(8.314 \, \text{J/mol·K})$,
• $T$ is the absolute temperature in Kelvin.

3. Analysis and Detail

Assume the diffusion coefficients ($D_1$ and $D_2$) are given at two temperatures $T_1$ and $T_2$:

• Let $D_1$ be the diffusion coefficient at $T_1$ in Kelvin.
• Let $D_2$ be the diffusion coefficient at $T_2$ in Kelvin.

Using the two temperature data, we can form two equations:

1. $\ln D_1 = \ln D_0 - \frac{Q_d}{RT_1}$
2. $\ln D_2 = \ln D_0 - \frac{Q_d}{RT_2}$

From these equations, we can eliminate $D_0$ and solve for $Q_d$: Subtract equation 2 from equation 1: $\ln D_1 - \ln D_2 = \frac{Q_d}{RT_2} - \frac{Q_d}{RT_1}$ $\ln\left(\frac{D_1}{D_2}\right) = Q_d\left(\frac{1}{RT_2} - \frac{1}{RT_1}\right)$ Thus, $Q_d = \frac{R T_1 T_2}{T_2 - T_1} \ln\left(\frac{D_1}{D_2}\right)$

Once $Q_d$ is found, substitute it back to find $D_0$ using either of the original equations.

4. Verify and Summarize

• Verify each calculation step for consistency.
• Ensure that units are correctly managed throughout the calculations.

Final Answer

1. Calculate $Q_d$ using the provided diffusion coefficients and temperatures.
2. Substitute the calculated $Q_d$ back into one of the original equations to find $D_0$.
3. Using the newly calculated values of $D_0$ and $Q_d$, calculate $D$ at $T = 875°C = 1148 \, \text{K}$ using: $D = D_0 \exp\left(-\frac{Q_d}{R \times 1148}\right)$

The precise numerical results depend on the specific values of $D_1, D_2, T_1,$ and $T_2$ provided. If these values are available, plug them into the equations for explicit results.