What is the energy of a bond formed between a potassium (K+) cation and an iodide (I−) anion? The ionic radii of K+ and I−, are 152 pm and 206 pm, respectively. Assume the Born exponent n is 10. Please report your answer in joules.
x=
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.
y=
To determine the energy of a bond formed between a potassium (K+) cation and an iodide (I-) anion, we can use the Born-Lande equation:
E = (Madelung constant * Charge of Ion1 * Charge of Ion2) / (4 * π * ε0 * Bond Length) * (1 - (1/n))
To find the bond energy between K+ and I-, we need to calculate the bond length first. The bond length can be estimated using the sum of the ionic radii of the ions:
Bond Length = Ionic Radii of K+ + Ionic Radii of I-
In this case, the ionic radii of K+ and I- are given as 152 pm and 206 pm, respectively.
Bond Length = 152 pm + 206 pm
Bond Length = 358 pm
Now we can plug the values into the formula:
E = (1.7 * (Charge of K+) * (Charge of I-)) / (4 * (π * ε0) * Bond Length) * (1 - (1/n))
The charge of K+ is +1, and the charge of I- is -1.
E = (1.7 * (1) * (-1)) / (4 * (π * ε0) * 358 pm) * (1 - (1/10))
Simplifying the equation:
x = (1.7 * (-1)) / (4 * (π * ε0) * 358 pm) * (1 - 0.1)
x = -0.0661764706 / (4 * (π * ε0) * 358 pm) * 0.9
Now, to convert x into joules, we need to consider the units involved. The unit pm represents picometers, and ε0 is the permittivity of free space with units of C^2/(J·m).
To convert pm into meters, we need to divide by 1×10^12 (1 picometer = 1×10^-12 meters). Similarly, to convert C^2/(J·m) to J, we need to multiply by 9×10^9 (1 C^2/(J·m) = 9×10^9 J).
x = -0.0661764706 / (4 * 3.14159 * 9×10^9 * (358 pm * 1×10^-12) m) * 0.9
Now we can calculate x:
x ≈ -1.96438101×10^-20 J
Therefore, the energy of the bond formed between a K+ cation and an I- anion is approximately -1.96438101×10^-20 joules (J).
Moving on to calculating the cohesive energy of potassium iodide (KI), we can use the following formula:
Cohesive Energy = Madelung constant * (Charge of K+)^2 / (4 * π * ε0 * Bond Length)
Given that the Madelung constant is 1.7 and the bond length is 358 pm (which we calculated earlier), we can calculate the cohesive energy:
y = 1.7 * (Charge of K+)^2 / (4 * π * ε0 * 358 pm * 1×10^-12 m)
The charge of K+ is +1, so:
y = 1.7 * 1^2 / (4 * π * π * 9×10^9 * (358 pm * 1×10^-12) m)
Simplifying the equation:
y = 1.7 / (4 * π * π * 9×10^9 * (358 pm * 1×10^-12) m)
Now we can calculate y:
y ≈ 4.20662951×10^10 J/mol
Therefore, the cohesive energy of potassium iodide (KI) is approximately 4.20662951×10^10 joules per mole (J/mol).