Hexagonal Packing Factor - Functional Molecules

[VooDoo]

Homework Statement

the sulphur atoms in the self-assembled monolayers are ~ 4.99Å apart, and that they form a hexagonal close-pack structure, estimate the number of functional molecules/cm2 of the substrate

Homework Equations

1 angstrom = 1.0 × 10$^{-10}$ metres

Area of hexagon = 2.598t$^{2}$ where t=length of hexagon side

APF of a hexagonal structure = 0.74

The Attempt at a Solution

I have tried searching google, but the only solution I can come up with is:

Working out the area of the hexagon, by assuming that the 4.99Å is the length of the hexagon side. Then I believe a HCP structure has 7 atoms, therefore I divide the area calculated by 7 to get molecules/Å$^{2}$ which I convert to molecules/cm$^{2}$

No idea if what I am doing is correct!

 

[trendingindia]

To estimate the number of functional molecules per square centimeter of substrate, we first need to calculate the area occupied by each molecule in the hexagonal close packed structures.

In HCP, the sulfur atoms are arranged in a hexagonal lattice with a spacing of 4.99 Å between neighboring atoms. The area of each unit cell in the HCP structure can be calculated as follows:

Area of unit cell = (3√3 / 2) × (4.99 Å)2 = 64.57 Å2

Since there are two sulfur atoms per unit cell, the area occupied by each sulfur atom can be calculated as:

Area per sulfur atom = 64.57 Å2 / 2 = 32.28 Å2

Now, we need to convert the area occupied by each sulfur atom to the area occupied by each functional molecule. Let's assume that each functional molecule contains one sulfur atom and has a cross-sectional area of A.

Then, the number of functional molecules per square centimeter can be calculated as:

Number of functional molecules/cm2 = (1 / A) × (1 / 32.28 Å2) × (10^8 Å2 / cm2)

Solving for the unknown parameter A, we get:

A = (1 / Number of functional molecules/cm2) × (32.28 Å2) × (10^-8 cm2/Å2)

Let's assume that there are N functional molecules per square centimeter. Then, substituting N for Number of functional molecules/cm2, we get:

A = (1 / N) × (32.28 Å2) × (10^-8 cm2/Å2)

Using the given information that the sulfur atoms are spaced 4.99 Å apart, we can calculate the cross-sectional area of each functional molecule as follows:

A = π × (4.99 Å / 2)^2 = 19.57 Å2

Substituting this value into the equation for A, we get:

N = (32.28 Å2) × (10^-8 cm2/Å2) / (19.57 Å2) = 1.65 × 10^13 molecules/cm2

Therefore, the estimated number of functional molecules per square centimeter of substrate is approximately 1.65 × 10^13.

 

[berkeman]

To estimate the number of functional molecules per square centimeter of substrate, we first need to calculate the area occupied by each molecule in the hexagonal close packed structures.

In HCP, the sulfur atoms are arranged in a hexagonal lattice with a spacing of 4.99 Å between neighboring atoms. The area of each unit cell in the HCP structure can be calculated as follows:

Area of unit cell = (3√3 / 2) × (4.99 Å)2 = 64.57 Å2

Since there are two sulfur atoms per unit cell, the area occupied by each sulfur atom can be calculated as:

Area per sulfur atom = 64.57 Å2 / 2 = 32.28 Å2

Now, we need to convert the area occupied by each sulfur atom to the area occupied by each functional molecule. Let's assume that each functional molecule contains one sulfur atom and has a cross-sectional area of A.

Then, the number of functional molecules per square centimeter can be calculated as:

Number of functional molecules/cm2 = (1 / A) × (1 / 32.28 Å2) × (10^8 Å2 / cm2)

Solving for the unknown parameter A, we get:

A = (1 / Number of functional molecules/cm2) × (32.28 Å2) × (10^-8 cm2/Å2)

Let's assume that there are N functional molecules per square centimeter. Then, substituting N for Number of functional molecules/cm2, we get:

A = (1 / N) × (32.28 Å2) × (10^-8 cm2/Å2)

Using the given information that the sulfur atoms are spaced 4.99 Å apart, we can calculate the cross-sectional area of each functional molecule as follows:

A = π × (4.99 Å / 2)^2 = 19.57 Å2

Substituting this value into the equation for A, we get:

N = (32.28 Å2) × (10^-8 cm2/Å2) / (19.57 Å2) = 1.65 × 10^13 molecules/cm2

Therefore, the estimated number of functional molecules per square centimeter of substrate is approximately 1.65 × 10^13.

Welcome to PF.

We generally do not allow solutions of schoolwork threads to be posted, since the student must do the bulk of the work. But since this thread is 12 years old, we can assume the Original Poster (OP) has moved on from this course.

 

[Borek]

Welcome to PF.

We generally do not allow solutions of schoolwork threads to be posted, since the student must do the bulk of the work. But since this thread is 12 years old, we can assume the Original Poster (OP) has moved on from this course.

my bet is it was posted just to post the link (note: they linked to a site that is - what a surprise - called exactly as their account here), so I will edit the link out (from BOTH posts)