To determine the quantity of silver deposited when one coulomb of charge is passed through $$\mathrm{AgNO}_3$$ solution, we need to refer to Faraday's laws of electrolysis, especially to the concept of the electrochemical equivalent. The electrochemical equivalent (ECE) of a substance is the amount of the substance that is deposited or dissolved at an electrode during electrolysis by one coulomb of charge.

The electrochemical equivalent of a substance can be calculated using the formula:

$$E = \frac{M}{nF}$$

where:

• $E$ is the electrochemical equivalent of the substance in grams per coulomb ($\mathrm{g/C}$).
• $M$ is the molar mass of the substance in grams per mole ($\mathrm{g/mol}$).
• $n$ is the number of electrons involved in the oxidation or reduction reaction (the valency).
• $F$ is the Faraday constant, approximately $96485 \, \mathrm{C/mol}$, which is the charge of one mole of electrons.

In the case of silver ($\mathrm{Ag}$) being deposited from $ \mathrm{AgNO}_3 $, when silver ion ($\mathrm{Ag}^+$) is reduced to metallic silver ($\mathrm{Ag}$), the reaction involves one electron ($n = 1$). The molar mass of silver (Ag) is approximately $107.868 \, \mathrm{g/mol}$. Using this information:

$$E = \frac{107.868 \, \mathrm{g/mol}}{1 \cdot 96485 \, \mathrm{C/mol}} \approx 0.001118 \, \mathrm{g/C}$$

This means that for every coulomb of charge passed through the solution, $0.001118 \, \mathrm{g}$ of silver is deposited.

Given the options:

• Option A: $0.1$ g atom of silver - This is incorrect because the calculation shows a much smaller amount is deposited per coulomb.
• Option B: 1 chemical equivalent of silver - This does not directly relate to the amount deposited per coulomb.
• Option C: $1 \mathrm{~g}$ of silver - This is incorrect as it overestimates the amount deposited by a large margin.
• Option D: 1 electrochemical equivalent of silver - This is correct, as it directly relates to the amount of a substance deposited by one coulomb of charge, based on the formula and calculation shown.

Therefore, the correct answer is Option D: 1 electrochemical equivalent of silver.