Calculate Exponent in Scientific Notation for Silver Surface Atoms

To find the value of $x$, we determine the number of silver ($\text{Ag}$) atoms present on the given surface area.

Step 1: Calculate the molar volume of silver ($V_m$)
Given:

• Molar mass of $\text{Ag}$, $M = 108 \text{ g mol}^{-1}$
• Density of $\text{Ag}$, $\rho = 10.5 \text{ g cm}^{-3}$

The molar volume is given by:
$V_m = \frac{M}{\rho} = \frac{108 \text{ g mol}^{-1}}{10.5 \text{ g cm}^{-3}} \approx 10.286 \text{ cm}^3 \text{mol}^{-1} = 10.286 \times 10^{-6} \text{ m}^3 \text{mol}^{-1}$

Step 2: Calculate the volume occupied by a single silver atom ($V_{\text{atom}}$)
Using Avogadro's number ($N_A \approx 6.022 \times 10^{23} \text{ mol}^{-1}$):
$V_{\text{atom}} = \frac{V_m}{N_A} = \frac{10.286 \times 10^{-6} \text{ m}^3 \text{mol}^{-1}}{6.022 \times 10^{23} \text{ mol}^{-1}} \approx 1.708 \times 10^{-29} \text{ m}^3$

Step 3: Estimate the effective linear dimension ($d$) of a silver atom
Assuming each atom occupies a cubic volume element of side length $d$:
$d = \left(V_{\text{atom}}\right)^{1/3} = \left(1.708 \times 10^{-29} \text{ m}^3\right)^{1/3} \approx 2.575 \times 10^{-10} \text{ m}$

Step 4: Calculate the surface area occupied by a single atom ($A_{\text{atom}}$)
The cross-sectional area occupied by an atom on the surface is:
$A_{\text{atom}} = d^2 = \left(2.575 \times 10^{-10} \text{ m}\right)^2 \approx 6.63 \times 10^{-20} \text{ m}^2$

Step 5: Calculate the total number of atoms on the surface
Given total surface area $A = 10^{-12} \text{ m}^2$:
$N = \frac{A}{A_{\text{atom}}} = \frac{10^{-12} \text{ m}^2}{6.63 \times 10^{-20} \text{ m}^2} \approx 1.51 \times 10^7$

Comparing $N = 1.51 \times 10^7$ with the standard scientific notation $y \times 10^x$ (where $1 \le y < 10$):
$x = 7$