Q.(a) Explain Pauling method to determine ionic radii.

(a) Pauling's method partitions the X-ray-measured inter-ionic distance between an isoelectronic cation and anion in inverse proportion to their Slater effective nuclear charges. (b) Ramsay and Rayleigh isolated the noble gases from air by chemically stripping out $O_2$ and $N_2$ (hot Cu, then hot Mg) after removing $CO_2$/moisture, leaving an inert residue that was further resolved by fractional distillation/selective adsorption.

(a) Pauling's method for ionic radii

Pauling considered ionic crystals in which the cation and anion are isoelectronic, i.e. both have an identical (usually noble-gas) electron configuration — for example $Na^+$ and $F^-$, both with the neon configuration (10 electrons). For such a pair he assumed that the radius of each ion is inversely proportional to its effective nuclear charge $Z^*$ (the nuclear charge actually 'felt' by the outermost electrons, obtained from Slater's screening rules, $Z^* = Z - \sigma$):

$r \propto \frac{1}{Z^*}$

Since both ions have the same number of electrons, the ion with the larger $Z^*$ (i.e. less screened, more strongly attracting its electron shell) is the smaller ion, and vice versa:

$\frac{r_{cation}}{r_{anion}} = \frac{Z^*_{anion}}{Z^*_{cation}}$

The sum of the two radii equals the observed inter-ionic (internuclear) distance $d$, measured experimentally by X-ray diffraction of the crystal:

$d = r_{cation} + r_{anion}$

Combining the two relations gives the individual radii:

$r_{cation} = \frac{Z^*_{anion}}{Z^*_{cation}+Z^*_{anion}}\,d \qquad r_{anion} = \frac{Z^*_{cation}}{Z^*_{cation}+Z^*_{anion}}\,d$

Worked illustration (NaF): $Na^+$ and $F^-$ are isoelectronic (Ne configuration). The observed $Na$–$F$ inter-ionic distance in the crystal is $d = 2.31\ \text{\AA}$. Using the Slater effective nuclear charges of $Na^+$ and $F^-$ in this relation, the individual radii come out to approximately $r(Na^+) \approx 0.95\ \text{\AA}$ and $r(F^-) \approx 1.36\ \text{\AA}$ (their sum reproduces the observed $2.31\ \text{\AA}$). By applying the same procedure to a series of isoelectronic ion pairs, Pauling built up a self-consistent table of 'univalent' ionic radii for the common ions.

(b) Ramsay–Rayleigh isolation of noble gases from air

Rayleigh's careful density measurements showed that nitrogen isolated from air was slightly denser than nitrogen prepared from chemical compounds — the difference being due to an unreacted inert gas mixed with the atmospheric nitrogen. Ramsay then isolated this residue by chemically removing every other constituent of air:

1. Remove $CO_2$ and moisture: air is passed through KOH solution (absorbs $CO_2$) and then dried over $P_2O_5$ or conc. $H_2SO_4$.
2. Remove oxygen: the purified air is passed repeatedly over red-hot copper turnings/gauze; $O_2$ combines with copper to form copper(II) oxide: $2Cu + O_2 \xrightarrow{\Delta} 2CuO$.
3. Remove nitrogen: the remaining gas is then passed repeatedly over red-hot magnesium turnings; $N_2$ combines with magnesium to form magnesium nitride: $3Mg + N_2 \xrightarrow{\Delta} Mg_3N_2$. …