1. Use Faraday's first law of electrolysis
   The mass deposited is $m = ZQ$ where $Z$ is the electrochemical equivalent and $Q$ is the charge passed.

2. Given charge
   Here, $Q = 1$ coulomb.

3. Substitute into the formula
   $m = Z \times 1 = Z$

4. Interpretation
   The mass of silver deposited by passing 1 coulomb through $\mathrm{AgNO_3}$ solution is exactly one electrochemical equivalent of silver.

5. Check options

   • A: $0.1$ g-atom of silver $\rightarrow$ incorrect. 1 g-atom means 1 mole, which is far too large.
   • B: 1 chemical equivalent of silver $\rightarrow$ incorrect. One chemical equivalent of Ag is deposited by 1 faraday of charge, not by 1 coulomb.
   • C: 1 g of silver $\rightarrow$ incorrect. Actual deposited mass for 1 coulomb is much smaller.
   • D: 1 electrochemical equivalent of silver $\rightarrow$ correct.

6. Extra verification
   For silver, $\mathrm{Ag^+ + e^- \to Ag}$ so 1 faraday deposits 1 gram-equivalent of Ag $\approx 108$ g. Therefore 1 coulomb deposits $\frac{108}{96500} \text{ g} \approx 1.12 \times 10^{-3} \text{ g}$ which is indeed the electrochemical equivalent of silver, not 1 g or 1 gram-equivalent.

Therefore, the correct answer is D.