The vacancy fraction of a particular metal is found to double as the temperature of the metal is increased from 700 degrees C to 850 degrees C. Calculate the enthalpy of vacancy formation for this metal. Express your answer in units of eV.
To calculate the enthalpy of vacancy formation, we need to use the following equation:
Hv = -k * T * ln(V2/V1),
where Hv is the enthalpy of vacancy formation, k is the Boltzmann constant (8.6173 x 10^-5 eV/K), T is the absolute temperature, V2 is the vacancy fraction at the higher temperature, and V1 is the vacancy fraction at the lower temperature.
First, we need to find the values of V2 and V1.
Given:
- The vacancy fraction doubled as the temperature increased from 700 degrees C to 850 degrees C.
- Let's denote the vacancy fraction at 700 degrees C as V1, and at 850 degrees C as V2.
Since the vacancy fraction doubled, we can write:
2V1 = V2.
Next, we convert the temperatures to Kelvin:
T1 = 700 degrees C = 700 + 273 = 973 K,
T2 = 850 degrees C = 850 + 273 = 1123 K.
Now, we can calculate the enthalpy of vacancy formation using the equation:
Hv = -k * T * ln(V2/V1).
Plugging in the values:
Hv = -8.6173 x 10^-5 eV/K * 973 K * ln(V2/V1).
Since we established that 2V1 = V2, we can substitute:
Hv = -8.6173 x 10^-5 eV/K * 973 K * ln(2V1/V1).
Simplifying:
Hv = -8.6173 x 10^-5 eV/K * 973 K * ln(2).
Evaluating ln(2):
Hv = -8.6173 x 10^-5 eV/K * 973 K * 0.693147.
Calculating the result:
Hv ≈ -5.01 x 10^-2 eV.
Therefore, the enthalpy of vacancy formation for this metal is approximately -5.01 x 10^-2 eV.