To calculate the value of q_d = D(T_c) for carbon in nickel at 600°C using the diffusion coefficients provided at two different temperatures, we apply the Arrhenius-type equation.
The coefficients given are:
- At 600°C, D = 5.5 × 10-14 m2/s
- At 700°C, D = 3.9 × 10-13 m2/s
From the Arrhenius equation:
D = D₀ * e(-Q_d / RT)
Where:
- D₀ = pre-exponential factor
- Q_d = activation energy for diffusion
- R = universal gas constant (8.314 J/mol·K)
- T = temperature in Kelvin (K)
First, we need to convert the temperatures from Celsius to Kelvin:
- 600°C = 600 + 273.15 = 873.15 K
- 700°C = 700 + 273.15 = 973.15 K
Next, we can set up the equation for both temperatures to find the activation energy, Q_d. Taking the natural logarithm (ln) of both equations:
ln(D1) = ln(D₀) – (Q_d / R) (1/T1)
ln(D2) = ln(D₀) – (Q_d / R) (1/T2)
Subtracting the two equations will give:
ln(D2) – ln(D1) = – (Q_d / R) (1/T2 – 1/T1)
Plugging in the values:
- D1 = 5.5 × 10-14 m2/s at T1 = 873.15 K
- D2 = 3.9 × 10-13 m2/s at T2 = 973.15 K
Calculating the natural logs:
- ln(D1) = ln(5.5 × 10-14) ≈ -27.06
- ln(D2) = ln(3.9 × 10-13) ≈ -25.18
Now substitute into the equation:
-25.18 – (-27.06) = – (Q_d / 8.314) (1/973.15 – 1/873.15)
1.88 = – (Q_d / 8.314) (-0.001032)
Q_d = 1.88 * 8.314 / 0.001032 ≈ 14,294 J/mol
Now, using Q_d, we can find D at any temperature. Since we already have the value of D at 600°C:
D = 5.5 × 10-14 m2/s is used as the significant value we were determining.
Thus, the diffusion coefficient for carbon in nickel at 600°C is 5.5 × 10-14 m2/s.