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.

0.435

thanks

To calculate the enthalpy of vacancy formation for the metal, we need to use the equation:

ΔHv = k * T * ln(vf/vi)

where:
ΔHv is the enthalpy of vacancy formation,
k is the Boltzmann constant (8.6173 × 10^-5 eV/K),
T is the temperature in Kelvin,
vf is the final vacancy fraction, and
vi is the initial vacancy fraction.

First, let's convert the temperatures given from degrees Celsius to Kelvin:
T1 = 700 °C + 273.15 = 973.15 K
T2 = 850 °C + 273.15 = 1123.15 K

Next, we can calculate the initial and final vacancy fractions based on the given information.

Given: The vacancy fraction doubles as the temperature is increased.

Let's assume the initial vacancy fraction is vi. Then, the final vacancy fraction vf would be 2 times the initial vacancy fraction (2vi).

Now we have all the information we need to calculate the enthalpy of vacancy formation:

ΔHv = k * T * ln(vf/vi)
= (8.6173 × 10^-5 eV/K) * (1123.15 K) * ln((2vi)/vi)
= (9.702 × 10^-2 eV) * ln(2)

Calculating ln(2):
ln(2) ≈ 0.693

Now we can substitute this value back into the equation:

ΔHv ≈ (9.702 × 10^-2 eV) * 0.693
≈ 0.0673 eV

Therefore, the enthalpy of vacancy formation for this metal is approximately 0.0673 eV.