Calculate the cohesive energy of potassium iodide (KI). The ionic radii of K+ and I−, are 152 pm and 206 pm, respectively. Assume the Born exponent n is 10. Assume a Madelung constant of 1.7. Please report your answer in kJ/mol.
To calculate the cohesive energy of potassium iodide (KI), we can use the Born-Lande equation, which is commonly used to calculate the lattice energy of ionic compounds:
E = (k * Q1 * Q2) / (r * n) * (1 - (1/2)^(1/n))
where:
E is the cohesive energy (lattice energy)
k is the Coulomb's constant (8.9875517923 x 10^9 N m^2/C^2)
Q1 and Q2 are the charges of the ions (in this case, K+ and I-)
r is the sum of the ionic radii (r = r1 + r2)
n is the Born exponent
First, we need to calculate the ionic radii sum (r) for K+ and I-. Using the values given:
r = r1 + r2
= 152 pm + 206 pm
= 358 pm
Next, we can calculate the cohesive energy using the Born-Lande equation:
E = (k * Q1 * Q2) / (r * n) * (1 - (1/2)^(1/n))
From the given information, we can determine the charges of K+ and I-:
Q1 = +1 (charge of K+)
Q2 = -1 (charge of I-)
Substituting the values into the equation:
E = (8.9875517923 x 10^9 N m^2/C^2 * +1 * -1) / (358 pm * 10) * (1 - (1/2)^(1/10))
Now we need to convert the value of r from picometers (pm) to meters (m):
1 pm = 1 x 10^-12 m
So, r = 358 pm * (1 x 10^-12 m / 1 pm) = 358 x 10^-12 m
Substituting this value into the equation:
E = (8.9875517923 x 10^9 N m^2/C^2 * +1 * -1) / (358 x 10^-12 m * 10) * (1 - (1/2)^(1/10))
Simplifying further:
E = (8.9875517923 x 10^9 N m^2/C^2) / (358 x 10^-12 m * 10) * (1 - (1/2)^(1/10))
E = (8.9875517923 x 10^9 N m^2/C^2) / (358 x 10^-11 m) * (1 - (1/2)^(1/10))
Now, we can calculate the cohesive energy (E) using a Madelung constant of 1.7 (as mentioned in the question):
E = (8.9875517923 x 10^9 N m^2/C^2) / (358 x 10^-11 m) * (1 - (1/2)^(1/10)) * 1.7
E ≈ -826 kJ/mol
Therefore, the cohesive energy of potassium iodide (KI) is approximately -826 kJ/mol.