An activationenergy of 2.0 eV is required to form a vacancy in a metal. At 800°C there is one vacancy for every 104 atoms. At what temperature will there be one vacancy for every 1000 atoms?
1200 RIGHT
To solve this problem, we need to use the concept of the Arrhenius equation, which relates the rate constant (k) of a reaction to the temperature (T) and the activation energy (Ea). The Arrhenius equation is given by:
k = A * exp(-Ea/RT)
Where:
- k is the rate constant,
- A is the pre-exponential factor (or frequency factor)
- Ea is the activation energy,
- R is the ideal gas constant, and
- T is the temperature in Kelvin.
In this case, we are not interested in the rate constant itself, but rather the ratio of vacancies to atoms, which can also be related to the rate constant.
The ratio of vacancies to atoms (N_vacancy/N_atoms) can be expressed as:
N_vacancy/N_atoms = k * t
Where t represents time. By rearranging the equation, we have:
k = (N_vacancy/N_atoms) / t
Now, we can substitute the expression for k from the Arrhenius equation:
(N_vacancy/N_atoms) / t = A * exp(-Ea/RT)
We are given that at 800°C (which is 1073 K), there is one vacancy for every 104 atoms. So, we can write:
(1/104) / t = A * exp(-2.0 eV / (R * 1073 K))
Next, we need to find the value of the pre-exponential factor (A). Unfortunately, this value is not provided in the question. The pre-exponential factor depends on various factors such as the specific metal and the specific process leading to the formation of vacancies. Therefore, we need to make some assumptions or obtain the value of A through experimental data or other means.
Let's assume a reasonable value for A. Typically, the pre-exponential factor of diffusion processes is around 10^13 to 10^15 s^-1. For simplicity, we will assume A = 10^14 s^-1.
With this assumption, we can proceed to find the time (t) required for one vacancy per 1000 atoms, which is (1/1000):
(1/1000) / t = 10^14 s^-1 * exp(-2.0 eV / (R * T))
Now, we can rearrange the equation to isolate time (t):
t = (1/1000) / (10^14 s^-1 * exp(-2.0 eV / (R * T)))
To find the temperature (T) at which there will be one vacancy for every 1000 atoms, we can use numerical methods or computer software that can iteratively solve equations.