Answer:
844°C
Explanation:
The problem can be easily solve by using Fick's law and the Diffusivity or diffusion coefficient.
We know that Fick's law is given by,
![J = - D \frac{\Delta c}{\Delta x}](https://tex.z-dn.net/?f=J%20%3D%20-%20D%20%5Cfrac%7B%5CDelta%20c%7D%7B%5CDelta%20x%7D)
Where
is the concentration of gradient
D is the diffusivity coefficient
and J is the flux of atoms.
In the other hand we have, that
![D= D_0 e^{\frac{E_d}{RT}}](https://tex.z-dn.net/?f=D%3D%20D_0%20e%5E%7B%5Cfrac%7BE_d%7D%7BRT%7D%7D)
Where
is the proportionality constant,
R is the gas constant, T the temperature and
is the activation energy.
Replacing the value of diffusivity coefficient in Fick's law we have,
![J = -D_0 ^{\frac{E_d}{RT}}\frac{\Delta c}{\Delta x}](https://tex.z-dn.net/?f=J%20%3D%20-D_0%20%5E%7B%5Cfrac%7BE_d%7D%7BRT%7D%7D%5Cfrac%7B%5CDelta%20c%7D%7B%5CDelta%20x%7D)
Rearrange the equation to get the value of temperature,
![T=\frac{Ed}{Rln(\frac{J\Delta x}{D_0 \Delta c})}](https://tex.z-dn.net/?f=T%3D%5Cfrac%7BEd%7D%7BRln%28%5Cfrac%7BJ%5CDelta%20x%7D%7BD_0%20%5CDelta%20c%7D%29%7D)
We have all the values in our equation.
![\Delta c = 0.664-0.339 = 0.325 C. cm^{-1}](https://tex.z-dn.net/?f=%5CDelta%20c%20%3D%200.664-0.339%20%3D%200.325%20C.%20cm%5E%7B-1%7D)
![\Delta x = 9.7*10^{-3}m](https://tex.z-dn.net/?f=%5CDelta%20x%20%3D%209.7%2A10%5E%7B-3%7Dm)
![E_d = 82000J](https://tex.z-dn.net/?f=E_d%20%3D%2082000J)
![D_0 = 6.5*10^{-7}m^2/s](https://tex.z-dn.net/?f=D_0%20%3D%206.5%2A10%5E%7B-7%7Dm%5E2%2Fs)
![J = 3.2*10^{-9}m^2/s](https://tex.z-dn.net/?f=J%20%3D%203.2%2A10%5E%7B-9%7Dm%5E2%2Fs)
![R= 8.31Jmol^{-1}K](https://tex.z-dn.net/?f=R%3D%208.31Jmol%5E%7B-1%7DK)
Substituting,
![T=\frac{Ed}{Rln(\frac{J\Delta x}{D_0 \Delta c})}](https://tex.z-dn.net/?f=T%3D%5Cfrac%7BEd%7D%7BRln%28%5Cfrac%7BJ%5CDelta%20x%7D%7BD_0%20%5CDelta%20c%7D%29%7D)
![T=-\frac{-82000}{(8.31)ln(\frac{3.2*10^{-9}(9.7*10^{-3})}{6.5*10^{-7} (0.325)})}](https://tex.z-dn.net/?f=T%3D-%5Cfrac%7B-82000%7D%7B%288.31%29ln%28%5Cfrac%7B3.2%2A10%5E%7B-9%7D%289.7%2A10%5E%7B-3%7D%29%7D%7B6.5%2A10%5E%7B-7%7D%20%280.325%29%7D%29%7D)
![T=1118.07K=844\°C](https://tex.z-dn.net/?f=T%3D1118.07K%3D844%5C%C2%B0C)