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

can you give steps please

To calculate the enthalpy of vacancy formation for the metal, we need to apply the Arrhenius equation. The Arrhenius equation gives us the relationship between the vacancy concentration and temperature for a given metal.

The Arrhenius equation is given by:

k = A * exp(-Q/RT)

Where:
k is the rate constant,
A is the pre-exponential factor,
Q is the activation energy,
R is the gas constant (8.314 J/(mol·K)),
T is the temperature in Kelvin.

In our case, since we want to calculate the enthalpy of vacancy formation, we can assume that the activation energy (Q) is related to the enthalpy (ΔH) through the equation:

Q = ΔH + RT

Now, let's define our problem. Given that the vacancy fraction doubles when the temperature is increased from 700 degrees C (T1) to 850 degrees C (T2), we can write the equation as follows:

k(T2) = 2 * k(T1)

Using the Arrhenius equation, we have:

A * exp(-Q/T2) = 2 * A * exp(-Q/T1)

Canceling out the pre-exponential factor (A) and taking the natural logarithm of both sides, we get:

ln(2) = -Q/T1 + Q/T2

Now, we can substitute Q = ΔH + RT into the equation:

ln(2) = -ΔH/RT1 + ΔH/RT2

Next, we'll rearrange the equation to isolate ΔH:

ln(2) = ΔH/R * (1/T2 - 1/T1)

Finally, we can solve for ΔH:

ΔH = R * ln(2) * (T2 - T1) / (T1 * T2)

Now we can plug in the values:

T1 = 700 °C = 973.15 K
T2 = 850 °C = 1123.15 K
R = 8.314 J/(mol·K)

Calculating ΔH:

ΔH = 8.314 J/(mol·K) * ln(2) * (1123.15 K - 973.15 K) / (973.15 K * 1123.15 K)

Using a scientific calculator, we find:

ΔH ≈ 0.673 eV

Therefore, the enthalpy of vacancy formation for this metal is approximately 0.673 eV.