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.
To calculate the cohesive energy of potassium iodide (KI), we can use the Born-Landé equation:
E_coh = (1 - 1 / n) * C * (z^+ * z^-) * e^2 / (4 * π * ε_0 * r_0),
where:
E_coh is the cohesive energy,
n is the Born exponent,
C is the Madelung constant,
z^+ and z^- are the charges of the cation and anion, respectively,
e is the elementary charge (approximately 1.602 × 10^−19 C),
ε_0 is the vacuum permittivity (approximately 8.854 × 10^-12 C^2/Nm^2),
r_0 is the sum of the ionic radii.
Given:
- The ionic radii of K+ and I-, which are 152 pm and 206 pm, respectively.
- The Born exponent n = 10.
- The Madelung constant C = 1.7.
First, let's convert the ionic radii from picometers (pm) to meters (m):
152 pm = 152 × 10^-12 m,
206 pm = 206 × 10^-12 m.
Next, let's calculate the sum of the ionic radii:
r_0 = (152 × 10^-12 m) + (206 × 10^-12 m).
Now, substitute the values into the formula:
E_coh = (1 - 1 / 10) * 1.7 * (1 * -1) * (1.602 × 10^-19 C)^2 / (4 * π * 8.854 × 10^-12 C^2/Nm^2 * r_0).
Simplify and calculate the value of r_0, then substitute the values and calculate E_coh:
E_coh = (9/10) * 1.7 * (1 * -1) * (1.602 × 10^-19 C)^2 / (4 * π * 8.854 × 10^-12 C^2/Nm^2 * r_0).
Finally, substitute the calculated value of E_coh to get the cohesive energy of potassium iodide (KI).