If the diffusivity is observed to double when the temperature is increased by 10 degrees C, what is the activation energy for diffusion? Express your answer in units of eV.
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To calculate the activation energy for diffusion, we can use the Arrhenius equation:
D = D0 * exp[-Ea / (RT)]
Where:
D is the diffusivity at a given temperature,
D0 is the diffusivity at a reference temperature,
Ea is the activation energy,
R is the gas constant (8.314 J/(mol·K)),
T is the absolute temperature in Kelvin.
We are given that the diffusivity doubles when the temperature increases by 10 degrees Celsius. This implies that the ratio of diffusivities is 2:
D2 / D1 = 2
Taking the natural logarithm of both sides:
ln(D2 / D1) = ln(2)
Using the Arrhenius equation, we can write the diffusivities in terms of the activation energies and temperatures:
D2 = D0 * exp[-Ea / (R * (T1 + 10))]
D1 = D0 * exp[-Ea / (R * T1)]
Now substituting these values into the ratio:
ln(D0 * exp[-Ea / (R * (T1 + 10))]) / (D0 * exp[-Ea / (R * T1)]) = ln(2)
Simplifying:
-ln(D0 / D0) * Ea / (R * (T1 + 10)) + ln(2) = 0
-ln(1) * Ea / (R * (T1 + 10)) + ln(2) = 0
Ea / (R * (T1 + 10)) = ln(2)
Ea = ln(2) * (R * (T1 + 10))
Now we can substitute the value of R (8.314 J/(mol·K)) and convert the temperature to Kelvin:
Ea = ln(2) * (8.314 J/(mol·K)) * (T1 + 10 + 273.15)
Finally, we convert the answer from Joules to electron volts (eV):
1 eV = 1.6 x 10^-19 J
Ea (eV) = Ea (J) / (1.6 x 10^-19)
So, you can use the above equation to calculate the activation energy for diffusion in eV. Just plug in the value of the initial temperature (T1) in Celsius and perform the calculations.