Molecular Behavior and Diffusion

Traditional Chemistry vs. Atom-by-Atom Construction

• Traditional Chemistry:

  • Experimenter has limited control over individual atoms.
  • Atoms are dumped together in a container, and thermal energy causes random collisions, potentially leading to reactions.
  • Not all atoms react; many don't, leading to less than 100% percent yield.
  • Percent yield is a common calculation in organic chemistry, and it's rarely 100%.
  • Byproducts may form, or atoms may not react at all.
  • No individual, discrete control over atoms or molecules.

• Atom-by-Atom Construction (Drexlerian concept):

  • Building molecules one atom at a time.
  • Grabbing atoms like Lego bricks and pressing them into a structure.
  • Causing reactions and locking atoms in place to build the desired structure.

Nanoscience and Nanotechnology

• Efforts in nanoscience and nanotechnology aim toward achieving atom-by-atom control in chemistry.
• Current technology is not capable of the Drexlerian vision of atom-by-atom construction.
• Molecular motors powered by ATP and porphyrin rings reshuffled with sharp tips are steps in this direction, but still far from the idea of building molecules from scratch, one atom at a time.

Challenges to Atom-by-Atom Construction

• Many believe atom-by-atom construction is currently impossible and may never be feasible.
• There are fundamental chemical, physical, and biological challenges that these cartoons ignore.
• Need a more detailed understanding of how atoms and molecules behave in real environments.

The Progress of Science and Nobel Prizes

• Science progresses through hypothesis formation and testing.
• Scientists form hypotheses and test them in the lab.
• Nobel Prizes are awarded for universally relevant theories or for proving existing theories with concrete evidence.
• Ideas often come from observations or the work of others, combined in new ways.

Einstein's Nobel Prize-Winning Idea (1905)

• Einstein's work on the behavior of individual building blocks of matter (atoms and molecules) earned him a Nobel Prize.
• In 1905, the existence of atoms and molecules was not yet fully accepted.

Historical Context Leading to Einstein's Idea

• Story starts in 1856 with Eugene Fick:

  • Fick studied the diffusion of food coloring in water.
  • He sought to describe the movement of the coloring mathematically.

• Fick's Law:

  • He developed Fick's law of diffusion, which describes the rate at which material moves across a line in relation to the concentration gradient.
  • $J = -D \frac{dc}{dx}$
  • J is the flux (amount of material passing a point).
  • $\frac{dc}{dx}$ is the change in concentration along the x-axis (concentration gradient).
  • D is the diffusion constant. A negative sign indicates movement away from the high concentration.
  • Fick did not know the underlying cause of this diffusion at the time.

• Back to 1827 with Robert Brown:

  • Botanist who observed pollen grains in water under a microscope.
  • Noticed that pollen grains vibrated randomly; called Brownian motion.
  • Initially thought motion was due to living nature of pollen.
  • Later found non-living material (like Granite chips) also exhibited the same motion.

• Einstein’s Synthesis:

  • By 1905, scientists had a better understanding of atoms and molecules.
  • Einstein connected Brown's observations of Brownian motion with Fick's law of diffusion, proposing a theory to describe the behavior of atoms and molecules.

Einstein's Theory

• Applies Brownian motion to molecules to explain Fick's law of diffusion.
• Describes how molecules diffuse in gases and solutions.
• Molecules starting on one side of a line move across it randomly at the rate described by Fick's law.
• Fick's law works when looking at a large number of molecules. Tracking a single molecule doesn't show Fickian behavior.
• Discrete entities of matter exhibit random behavior, and understanding their collective behavior requires statistical analysis.

Gaussian Distribution

• If all the molecules start out in the center, over time they spread to form a Gaussian (normal, bell-shaped) curve.
  • The center represents the original location, and the standard deviation grows and widens with time.

Molecular Motion

• Computer simulations show molecules moving randomly with trajectories that don't individually look Gaussian.
• Collectively, however, their distribution spreads out in a Gaussian manner.

Dimensional Analysis of Water Molecules

• Calculate the average spacing between water molecules.

• Pure water has a concentration of 55 moles per liter.

• 1 mole = $6.02 \times 10^{23}$ molecules (Avogadro's Number)

• Use Dimensional Analysis to determine spacing between molecules:

  • 1 Liter = 1 cubic decimeter = $(0.1 \text{ meter})^3$
  • 1 decimeter = $10^8$ nanometers
  • 1 Liter = $(10^8 \text{ nm})^3 = 10^{24} \text{ nm}^3$

• Therefore:

  • $55 \frac{\text{moles}}{\text{Liter}} \times 6.02 \times 10^{23} \frac{\text{molecules}}{\text{mole}} = 3.31 \times 10^{25} \frac{\text{molecules}}{\text{Liter}}$
  • $\frac{3.31 \times 10^{25} \frac{\text{molecules}}{\text{Liter}}}{10^{24} \frac{\text{nm}^3}{\text{Liter}}} = 33.1 \frac{\text{molecules}}{\text{nm}^3}$
  • Invert: $0.03 \frac{\text{nm}^3}{\text{molecule}}$

• Each water molecule occupies 0.03 cubic nanometers of space.

• Dimensions of the cube: $(\sqrt[3]{0.03}) = 0.31$ nm on each side

• Water molecules are not touching; there is empty space between them.

Molecular Velocity

• Water molecules have linear velocity, can be calculated using Kinetic Energy calculations:
  • Kinetic Energy Equation = $\frac{1}{2}mv^2$
• Using Einstein's connections:
  • Kinetic Energy = $\frac{1}{2} k_B T$
  • kB = Boltzmann Constant ($1.38 \times 10^{-23} kg \cdot m^2 / K \cdot s^2$)
  • T = Temperature (in Kelvin)
• Velocity $(v)$: Derived from combining and rearranging the KE equations such that
  • $v = \sqrt{\frac{k_B T}{m}}$
• Calculate mass of water molecule:
  • Hydrogen: 1 gram/mole; Oxygen: 16 grams/mole; Water: 18 grams/mole
  • 18 grams/mole = 0.018 kg/mole
  • \therefore mass of ONE h2o molecule= $\frac{(0.018 \frac{kg}{mole})}{6.02 \times 10^{23} \frac{molecules}{mole}} = 2.99 \times 10^{-26} kg$
• $Velocity= \sqrt{\frac{1.38 \times 10^{-23} \times 300}{2.99 \times 10^{-26}}} = 372 \frac{meters}{second}$
• This is almost 4 soccer fields covered every second (very very fast, like a bullet).
• However, molecules are constantly colliding, due their tiny mass and high speeds.

Mean Squared Displacement (MSD)

• Due to collisions (10^14 collisions per second), direction is always changing, velocity is not so reliable.

• Need statistical information to describe molecular motion.

• MSD defined:

  • Describe it over 1 dimension, X coordinate:
    • $<[x(t)-x(0)]^2>$
    • MSD is the average of all squared distances a molecule travels over set timeframe.
    • Formula:
    • $MSD = \frac{1}{N} \sum<em>{i=1}^{N} (x</em>i (t) - x_i (0))^2$

• Einstein showed connection between MSD and Fick's diffusion constant:

  • $MSD = 2 \cdot n \cdot D \cdot t$
    • Where n = number of dimensions, D = diffusion constant, t = time.
    • In 1 dimension: MSD = 2Dt
    • In 2 dimensions: MSD = 4Dt
    • In 3 dimensions: MSD = 6Dt

• Bigger D implies faster diffusion; smaller D, slower diffusion.

E. Coli Example

• E. Coli is a cylindrical bacteria, surrounded by a cell membrane (lipid planar bilayer).
• Diffusion constant of a lipid molecule in cell membrane: $1 \times 10^{-7} \frac{cm^2}{s}$
• Question: How long for a lipid molecule to move from one end of the bacteria to the other?
• E. Coli dimensions: 1000nm wide, 3000nm long = 3 microns long.
• Answer: Using 2D approximation
  • Convert all units into consistent units, ie: meters
    • $3 microns = 3 \times 10^{-6} meters$, is movement, get (MSD), square it.
    • Diffusion Coefficient is in $cm^2 \rightarrow ((1 \times 10^{-7}) \frac{cm^2}{s}) = 1 \times 10^{-11} \frac{m^2}{s}$
• Calculate the relationship:
  • Rearrange and get:
    • $t= \frac{(3 \times 10^{-6})^2}{4 \cdot (1 \times 10^{-11})} = .225 seconds$

Universality of Diffusion Constants

• Diffusion constants apply to gases, liquids, and solids.
• Though the values differ, the concept is universal.
• Even molecules in solids (like glass) have diffusion constants and exhibit Brownian motion, albeit very slowly. The glass won't exist in 100 years, due to diffusion.