The user wants me to solve the problem and provide a step-by-step solution in a specific JSON format.

Problem Analysis

We are given the atomic weight and density of silver, and a specific surface area. We need to find the number of silver atoms that would fit on this surface. The final answer should be expressed in scientific notation $y \times 10^x$, and we need to find the value of the exponent, $x$.

The core of the problem is to determine the area occupied by a single silver atom on a surface.

Step-by-Step Solution

1. Calculate the volume of one mole of silver (Molar Volume). The molar volume ($V_m$) can be calculated from the atomic weight (Molar Mass, $M$) and the density ($\rho$). Given:

   • Atomic weight, $M = 108$ g/mol
   • Density, $\rho = 10.5$ g/cm³

   $V_m = \frac{M}{\rho} = \frac{108 \text{ g/mol}}{10.5 \text{ g/cm}^3} \approx 10.286 \text{ cm}^3/\text{mol}$

2. Calculate the volume occupied by a single silver atom. We can find the volume of a single atom ($V_{atom}$) by dividing the molar volume by Avogadro's number ($N_A = 6.022 \times 10^{23}$ atoms/mol).

   $V_{atom} = \frac{V_m}{N_A} = \frac{10.286 \text{ cm}^3/\text{mol}}{6.022 \times 10^{23} \text{ atoms/mol}} \approx 1.708 \times 10^{-23} \text{ cm}^3/\text{atom}$

3. Estimate the surface area occupied by a single silver atom. To estimate the area an atom occupies on a surface, we can model the volume it occupies as a small cube. If the volume of this cube is $V_{atom}$, its side length 'a' would be $a = (V_{atom})^{1/3}$. The area this atom presents on a surface would be $a^2$. Therefore, the area of a single atom ($A_{atom}$) can be estimated as:

   $A_{atom} = (V_{atom})^{2/3} = (1.708 \times 10^{-23} \text{ cm}^3)^{2/3}$ $A_{atom} \approx 6.63 \times 10^{-16} \text{ cm}^2$

4. Convert the given surface area to consistent units. The given surface area is $A = 10^{-12} \text{ m}^2$. We need to convert this to cm² to be consistent with our calculation for $A_{atom}$. Since $1 \text{ m} = 100 \text{ cm}$, then $1 \text{ m}^2 = (100 \text{ cm})^2 = 10^4 \text{ cm}^2$.

   $A = 10^{-12} \text{ m}^2 \times \frac{10^4 \text{ cm}^2}{1 \text{ m}^2} = 10^{-8} \text{ cm}^2$

5. Calculate the total number of silver atoms on the surface. The total number of atoms ($N$) is the total surface area divided by the area occupied by a single atom.

   $N = \frac{A}{A_{atom}} = \frac{10^{-8} \text{ cm}^2}{6.63 \times 10^{-16} \text{ cm}^2} \approx 0.1508 \times 10^8 \text{ atoms}$

6. Express the result in scientific notation and find x. The number of atoms is expressed in scientific notation as $y \times 10^x$, where $1 \le y < 10$.

   $N = 0.1508 \times 10^8 = 1.508 \times 10^7$ Comparing this with $y \times 10^x$, we can see that $y = 1.508$ and $x = 7$.

Conclusion

The value of x is 7.