Silver (atomic weight = 108 g mol-1) has a density of 10.5 g cm-3. The number of silver atoms on a surface of area 10-12 m2 can be expressed in scientific notation as y × 10x. The value of x is?

To find the value of x in the expression for the number of silver atoms on a surface area of \(10^{-12} \, \text{m}^2\), we need to follow a few logical steps involving the density and atomic weight of silver. Let's break this down step by step.

Understanding the Problem

We know the following about silver:

• Atomic weight of silver (Ag) = 108 g/mol
• Density of silver = 10.5 g/cm³

We need to calculate how many silver atoms fit into a surface area of \(10^{-12} \, \text{m}^2\).

Step 1: Convert Density to Appropriate Units

First, we should convert the density from g/cm³ to kg/m³ for consistency with the area we are using:

• 1 g/cm³ = 1000 kg/m³
• Thus, 10.5 g/cm³ = 10.5 × 1000 kg/m³ = 10500 kg/m³

Step 2: Calculate the Volume of Silver

Next, we need to determine how much mass of silver corresponds to the given surface area. Assuming a thin layer of silver, we can express the volume (V) of silver in terms of its density (ρ) and mass (m):

V = m / ρ

However, we need to find the mass of silver that would cover the surface area of \(10^{-12} \, \text{m}^2\). Let's assume a thickness (t) of silver. The volume can also be expressed as:

V = Area × Thickness = \(10^{-12} \, \text{m}^2 \times t\)

Step 3: Relating Mass to Volume

Now, we can relate the mass of silver to the volume using the density:

m = ρ × V = 10500 \, \text{kg/m}³ × \(10^{-12} \, \text{m}^2 \times t\)

To find the number of atoms, we need to convert mass to moles using the atomic weight:

Number of moles (n) = m / (atomic weight) = \( \frac{10500 \times 10^{-12} \times t}{108 \times 10^3} \, \text{kg/mol}\)

Note that we converted the atomic weight from grams to kilograms by multiplying by \(10^{-3}\).

Step 4: Calculate the Number of Atoms

The number of atoms (N) can be calculated using Avogadro's number (approximately \(6.022 \times 10^{23} \, \text{atoms/mol}\)):

N = n × Avogadro's number

Substituting for n, we get:

N = \( \left( \frac{10500 \times 10^{-12} \times t}{108 \times 10^{-3}} \right) \times 6.022 \times 10^{23}\)

Step 5: Simplifying the Expression

Now, let's simplify this expression to find the number of atoms per unit area:

N = \( \frac{10500 \times 6.022}{108} \times 10^{-12} \times 10^{3} \times t\)

Calculating the constant:

\( \frac{10500 \times 6.022}{108} \approx 586.5\)

Thus, we have:

N ≈ \(586.5 \times 10^{-9} \times t\)

Final Calculation

For a very thin layer of silver, let's assume a thickness of \(t = 1 \, \text{nm} = 1 \times 10^{-9} \, \text{m}\). Substituting this value:

N ≈ \(586.5 \times 10^{-9} \times 10^{-9} = 586.5 \times 10^{-18}\)

In scientific notation, this can be expressed as:

N ≈ \(5.865 \times 10^{-16}\)

Identifying x

In the expression \(y \times 10^x\), we see that \(x = -16\). Therefore, the value of x is:

x = -16