Knowing that the energy associated with the formation of 1 mole of vacancies in copper (Cu) is 104 kJ, determine the vacancy density (nv/cm3) in Cu at 1197 K. Assume the pre-exponential factor is one.

for 1369 K answer is 9.10814144008e+18

To determine the vacancy density (nv/cm^3) in copper (Cu) at 1197 K, we can use the following equation:

nv = A * exp(-Qv / (RT))

Where:
nv is the vacancy density
A is the pre-exponential factor (given as 1)
Qv is the energy associated with the formation of 1 mole of vacancies (104 kJ)
R is the gas constant (8.314 J/(mol·K))
T is the temperature in Kelvin (1197 K)

First, we need to convert the energy from kJ to J:
Qv = 104 kJ * 1000 J/kJ = 104000 J

Now let's substitute the values into the equation:
nv = 1 * exp(-104000 J / (8.314 J/(mol·K) * 1197 K))

Simplifying the equation:
nv = exp(-104000 J / (9965098 J·K/mol))

Calculating the value:
nv = exp(-0.1044 mol^-1)

Using the exponential function, we find:
nv ≈ 0.9009

Therefore, the vacancy density in copper at 1197 K is approximately 0.9009 nm^-3 (or cm^-3).

To calculate the vacancy density in copper (Cu) at a given temperature, we need to use the Arrhenius equation and the energy associated with the formation of 1 mole of vacancies.

The Arrhenius equation is given by:
k = A * exp(-Q/RT)

Where:
k = rate constant
A = pre-exponential factor
Q = activation energy
R = gas constant
T = temperature in Kelvin

In this case, we assume the pre-exponential factor (A) is one. The activation energy (Q) is given as 104 kJ. The gas constant (R) is approximately 8.314 J/(mol*K).

To determine the vacancy density, we need to find the rate constant (k). The rate constant is related to the vacancy density (nv) by the equation:
k = nv * v

Where:
nv = vacancy density
v = vibrational frequency

The value of the vibrational frequency (v) is typically taken as 10^12 Hz.

Now, let's calculate the vacancy density (nv) using the given information:

1. Convert the activation energy (Q) from kJ to J:
Q = 104 kJ * 1000 J/kJ = 104,000 J

2. Convert the temperature from Celsius to Kelvin:
Given temperature = 1197 K

3. Plug the values into the Arrhenius equation:
k = 1 * exp(-104,000 J / (8.314 J/(mol*K) * 1197 K))

4. Calculate the rate constant (k).

5. Use the rate constant (k) and the vibrational frequency (v) to calculate the vacancy density (nv):
nv = k / v

Finally, calculate the value of nv in cm^-3 using appropriate conversion factors.