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.
Can you show all working, I'm not sure where to begin? Thanks.

Question

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.
Can you show all working, I'm not sure where to begin? Thanks.

turingtest · Answer

the lattice energy is given by$$E_{lattice}=\frac{N_AMQ_1Q_2e^2}{4\pi\epsilon_0r_0}\left(1-\frac1n\right)$$where $N_A$ is Avagadro's number, $M$ is the Madelung constant, $Q1,Q_2$ are the magnitude of the charge of the ions, and $r_0$ is the total separation of the ion centers (which is the sum of the radii of each ion)
Use the given info to solve for the equilibrium separation $r_0$ and use the fact that the ionic radius of Cl is given to find the ionic radius of Cs.

turingtest · Answer

that energy is in kJ/mol btw, so you will have to put it back into Joules to get the radius

anonymous · Answer

oh okay thank you

turingtest · Answer

welcome!

anonymous · Answer

what do I use in the denominator??
it's 4 * 3.142 * ? * ?

turingtest · Answer

$\epsilon_0\approx8.85\times10^{-12}$ is the permittivity of free space, and since you are solving for $r_0$ you should have $E_{lattice}=633\times10^3$ as the other number

anonymous · Answer

I got the answer to be 1.67!!! Thank-you for the help!!! :) I appreciate it!

turingtest · Answer

welcome!

anonymous · Answer

What is $$Q _{1} and Q _{2}$$?