Building the Periodic Table — Question 6

Question 6

A screening model predicts a contraction along an isoelectronic sequence

Three ions have ten electrons each and nuclear charges $Z=9,10,11$. In a supplied effective-charge model their common screening constant is $\sigma=6.0$, and a characteristic valence radius is $r=n^2a_0/(Z-\sigma)$ with $n=2$.

1. Find the ion charges and effective charges.

2. Rank their radii and find $r_{Z=11}/r_{Z=9}$.

Question 6 - Solution

A screening model predicts a contraction along an isoelectronic sequence

(a) Method and calculation

$$q/e=Z-10\Rightarrow\boxed{q/e=-1,0,+1}.$$$$Z_{\mathrm{eff}}=Z-6\Rightarrow\boxed{Z_{\mathrm{eff}}=3,4,5}.$$

(b) Interpretation and further calculation

$$\boxed{r_9=4a_0/3>r_{10}=a_0>r_{11}=4a_0/5}.$$$$\boxed{r_{11}/r_9=(4/5)/(4/3)=3/5}.$$

Physical check

The constant screening assumption isolates the effect of increased nuclear attraction at fixed electron count. These radii are model scales, not precision measured ionic radii.