An activation energy of 2.0 eV is required to form a vacancy in a metal. At 800¡èC there is one vacancy for every 10^-4 atoms. At what temperature will there be one vacancy for every 1000 atoms?
To find the temperature at which there will be one vacancy for every 1000 atoms, we first need to determine the relationship between the number of vacancies and the temperature. We can use the concept of the Arrhenius equation to do that.
The Arrhenius equation relates the rate constant (k) of a reaction to the activation energy (Ea) and the temperature (T) using the formula:
k = A * e^(-Ea/RT)
Where:
k = rate constant
A = pre-exponential factor (constant)
R = gas constant (8.314 J/mol·K)
T = temperature in Kelvin
In our case, since we are interested in the number of vacancies, we will consider the rate constant as the rate of formation of vacancies.
Now, let's proceed step by step to find the temperature at which there will be one vacancy for every 1000 atoms:
Step 1: Getting the rate constant at 800°C
Convert 800°C to Kelvin by adding 273 to it:
T1 = 800 + 273 = 1073 K
We are given that at 800°C, there is one vacancy for every 10^-4 atoms. This can be considered as the fraction of vacancies (f) at that temperature, which is given by:
f1 = 1 vacancy / 10^-4 atoms
= 10^4 vacancies / 1 atom
= 10^4 vacancies
We can also express the fraction of vacancies using the equation for the rate constant:
k1 = A * e^(-Ea/RT1)
= f1 / 1
= 10^4
So, at 800°C, the rate constant (k1) is equal to 10^4.
Step 2: Using the ratio of vacancies to solve for the temperature
Now, let's find the temperature (T2) at which there will be one vacancy for every 1000 atoms.
We know that the fraction of vacancies (f2) at T2 is given by:
f2 = 1 vacancy / 1000 atoms
= 10^3 vacancies / 1 atom
= 10^3 vacancies
Now we can set up a ratio equation using the rate constants:
k1 / k2 = f1 / f2
Substituting the values we have:
10^4 / k2 = 10^4 / 10^3
Simplifying the equation, we find:
k2 = 10^3
Step 3: Finding the temperature at which there will be one vacancy for every 1000 atoms
Now, let's find the temperature (T2) using the known value of k2.
k2 = A * e^(-Ea/RT2)
Substituting the value of k2 = 10^3:
10^3 = A * e^(-Ea/RT2)
Since the pre-exponential factor (A) and the activation energy (Ea) are constant, we can solve for the temperature (T2) by isolating it in the equation:
e^(-Ea/RT2) = 10^3 / A
Take the natural logarithm (ln) on both sides to remove the exponential function:
-Ea/RT2 = ln(10^3 / A)
Multiply both sides by (-R):
Ea / RT2 = -R * ln(10^3 / A)
Finally, solve for T2 by isolating it:
T2 = -Ea / (R * ln(10^3 / A))
Plug in the values for the activation energy (Ea = 2.0 eV) and the gas constant (R = 8.314 J/mol·K), and convert the pre-exponential factor (A) from eV to J using the conversion 1 eV = 1.602 x 10^-19 J.
T2 = (-2.0 eV) / (8.314 J/mol·K * ln(10^3 / (A in J)))
= (-2.0 * 1.602 x 10^-19 J) / (8.314 J/mol·K * ln(10^3 / (A in J)))
Once you know the value of the pre-exponential factor (A) for the specific material you're studying, you can substitute that value into the equation to obtain the temperature (T2) in Kelvin.
Note that the value of the pre-exponential factor (A) can vary depending on the specific material and the type of vacancy being studied.