Coagulating value of the electrolytes AlCl$$_3$$ and NaCl for As$$_2$$S$$_3$$ are 0.09 and 50.04 respectively. The coagulating power of AlCl$$_3$$ is x times the coagulating power of NaCl. The value of x is _____.

We are given the coagulating values of two electrolytes for $$As_2S_3$$ sol:

Coagulating value of $$AlCl_3$$ = 0.09

Coagulating value of $$NaCl$$ = 50.04

Coagulating power is the reciprocal of coagulating value — a lower coagulating value means a higher coagulating power (less electrolyte is needed to cause coagulation). So we have:

$$\text{Coagulating power} = \frac{1}{\text{Coagulating value}}$$

Now, coagulating power of $$AlCl_3$$:

$$P_{\text{AlCl}_3} = \frac{1}{0.09}$$

And coagulating power of $$NaCl$$:

$$P_{\text{NaCl}} = \frac{1}{50.04}$$

Since the coagulating power of $$AlCl_3$$ is $$x$$ times the coagulating power of $$NaCl$$:

$$x = \frac{P_{\text{AlCl}_3}}{P_{\text{NaCl}}} = \frac{1/0.09}{1/50.04} = \frac{50.04}{0.09} = 556$$

This is consistent with the Hardy-Schulze rule: $$Al^{3+}$$ (trivalent cation) has much greater coagulating power than $$Na^+$$ (monovalent cation) for the negatively charged $$As_2S_3$$ sol.

Hence, the value of $$x$$ is $$556$$.