Estimate the radius of the cesium ion, Cs+. The lattice energy of cesium chloride, CsCl, is 633 kJ/mol. For CsCl, the Madelung constant M is 1.763, and the Born exponent n is 10.7. The radius of Cl− is known to be 1.81 A˚. Express your answer in Angstroms.
To estimate the radius of the cesium ion, Cs+, we can use the Born-Haber cycle and the relationship between lattice energy and ionic radii.
The Born-Haber cycle relates the lattice energy (LE) to the ionic radii of the ions involved. The formula is given by:
LE = (k * z+ * z- * M) / (r+ + r-) * exp[-n * (r+ + r-) / R]
Where:
LE = Lattice energy
k = proportionality constant
z+ = charge of the cation (Cs+ = +1)
z- = charge of the anion (Cl- = -1)
M = Madelung constant
r+ = radius of the cation (Cs+)
r- = radius of the anion (Cl-)
n = Born exponent
R = gas constant
Given values:
LE = 633 kJ/mol
M = 1.763
n = 10.7
r- (Cl-) = 1.81 Å (Angstroms)
We need to solve for r+ (Cs+), so let's rearrange the equation to isolate r+:
r+ = (LE * (r- + 1.81)) / (k * z+ * z- * M * exp[-n * (r- + 1.81) / R])
Now, substitute the given values into the equation. Note that the gas constant, R, is 8.314 J/(mol*K).
r+ = (633 * (1.81 + 1.81)) / (k * 1 * (-1) * 1.763 * exp[-10.7 * (1.81 + 1.81) / 8.314])
Simplifying further:
r+ = 3.62 / (k * 1.763 * exp[-10.7 * 3.62 / 8.314])
To solve for the constant k, we need additional information. Unfortunately, the information provided in the question is not sufficient to determine the exact value of the radius of the cesium ion, Cs+.