The field of nanotechnology offers some intriguing possibilities, such as the creation of fibers one atom wide. Suppose you were able to string together 1.00 mol Ag atoms, each of radius 144 pm, into one of these fibers by encapsulating them in carbon nanotubes. How long would the fiber extend?

1 mole of Ag atoms contains 6.02*10^23 atoms (Avogadro's number). Multiply that by twice the radius (244*10^-12 m)to get the length of the nanotube.

1.7x10^14?

1.73x10^11 km

To find the length of the fiber, we can start by determining the total volume occupied by 1.00 mol of silver (Ag) atoms. Then, we can calculate the length of the fiber by dividing this volume by the cross-sectional area of the fiber.

Step 1: Calculate the total volume of Ag atoms
First, let's determine the volume of one Ag atom. Since Ag atoms are assumed to be spherical, we can use the formula for the volume of a sphere:

V = (4/3) * π * r^3

where V is the volume and r is the radius of the atom.

Given:
Radius of Ag atom (r) = 144 pm = 144 × 10^(-12) m

Substituting the values into the equation:

V = (4/3) * π * (144 × 10^(-12))^3

V ≈ 4.190 × 10^(-23) m^3 (rounded to 4 significant figures)

Next, we need to find the total volume of 1.00 mol of Ag atoms. The volume of an individual atom can be multiplied by Avogadro's number (6.022 × 10^23 atoms/mol) to get the total volume:

Total volume = 1.00 mol * [(4.190 × 10^(-23) m^3)/mol] * (6.022 × 10^23 atoms/mol)

Total volume ≈ 25.20 × 10^(-23) m^3 (rounded to 3 significant figures)

Step 2: Calculate the length of the fiber
To determine the length of the fiber, we need to divide the total volume by the cross-sectional area of the fiber. For a one-atom-wide fiber, the cross-sectional area can be calculated using the formula for the area of a circle:

A = π * r^2

Given:
Radius of the fiber (r) = 1 atomic radius = 144 pm = 144 × 10^(-12) m

Substituting the values into the equation:

A = π * (144 × 10^(-12))^2

A ≈ 0.065 × 10^(-23) m^2 (rounded to 3 significant figures)

Now we can calculate the length of the fiber:

Length = Total volume / Cross-sectional area
Length = (25.20 × 10^(-23) m^3) / (0.065 × 10^(-23) m^2)

Length ≈ 387.7 m (rounded to 3 significant figures)

Therefore, if 1.00 mol Ag atoms were strung together in a fiber with a diameter of 1 atom, the fiber would extend approximately 387.7 meters in length.