Physical Principles in Biology
Biology 3550
Fall 2016
Lecture 20
More on Diffusion at the Molecular Level
Liquids and Gasses
Monday, 17 October
c
David
P. Goldenberg
University of Utah
[email protected]
Quick Review of Diffusion at the Molecular Level
Average (RMS) velocities of molecules reflect thermal (kinetic) energy:
For a single object at a specified velocity:
Ek,x = mvx2 /2
For a population of molecules at thermal equilibrium
RMS(Ek,x ) = mhvx2 i/2 = kT /2
T = temperature (K) and k = Boltzmann’s constant.
The RMS velocity:
q
RMS(vx ) = kT /m
Quick Review of Diffusion at the Molecular Level
From the diffusion coefficient and velocities to random walk steps:
Definition of the diffusion coefficient in terms of random walk steps:
δx2
D=
2τ
2
δx2 = mean-square step length along x-axis (m )
τ = mean step duration (s).
Velocity in terms of random step length and duration:
v = δx /τ
Solving for δx :
δx = 2D/v = p
2D
kT /m
Quick Review of Diffusion at the Molecular Level
From previous slide:
v = δx /τ
δx = 2D/v = p
2D
kT /m
Solving for τ:
τ = δx /v =
2D
kT /m
RMS displacement as a function of time:
t
hx 2 i = nδx2 = × 2Dτ = 2Dt
τ
√
RMS(x) = 2Dt
For a Small Molecule (e.g. a sugar or an amino acid)
D = 2×10−10 m2 /s, T = 298 K
RMS velocity:
v = δx /τ = 61 m/s
RMS step size:
δx = p
2D
= 6.5×10−12 m = 0.065 Å
kT /m
Average step duration:
τ=
2D
= 10−13 s = 0.1 ps
kT /m
How do we think about these very small distances and time intervals?
How Do Gasses Become Liquids?
Potential energy of two molecules as they approach each other:
When the attractive energies between molecules become strong
enough, they start to stick to each other, forming a liquid.
Formation of attractive interactions releases energy as heat.
Temperature and molecular velocities increase.
If temperature is restored, molecules have same kinetic energy and
velocity as in gas, but their motions are much more restrained.
The Nature of Liquids
Molecules are very densely packed
From a simulation of liquid water.
Motions of molecules are limited by
the “energy barriers” surrounding
them.
Other molecules in a solution are
similarly constrained.
What Happens When a Liquid Becomes a Gas?
Molecules in liquids or gasses
have a distribution of velocities
(kinetic energy).
The fastest molecules at the surface of a liquid sometimes break loose
of the attractive interactions with their neighbors.
What happens to the energy?
• The average kinetic energy of the liquid decreases.
(The fastest molecules leave.)
• Breaking attractive interactions absorbs thermal energy.
• Both factors lead to cooling of liquid and gas.
Calculating Diffusion Coefficients
What determines diffusion coefficient?
• Velocity of molecules (temperature)
• Size of molecules
• How often molecules colide
Stokes-Einstein equation for spherical particles:
D=
kT
6πηr
r = sphere radius
η = viscosity
A key result from the 1905 Einstein paper on Brownian motion.
A testable prediction!
Viscosity: Property of a Liquid (or Gas)
Resistance to Motion of Molecules Past One Another
A molecular property easily observed on macroscopic scale.
From a 1960s TV add for Prell Shampoo
https://www.youtube.com/watch?v=9lFsrjoLKq0
Reflects what we commonly refer to as “thickness” of a liquid.
Methods for quantitative measurement:
• Rate of a ball dropping through a liquid!
• Rate of liquid moving through a narrow tube.
• Rate of two closely spaced plates moving past each other.
Units for Viscosity
poise
1 P = 1 g · cm−1 s−1 = 0.1 kg · m−1 s−1
A holdover from the older “centimeter-gram-second” metric system.
Expressed in units of force (1 N = 1 kg · m · s−2 )
1 P = 0.1 N · s · m−2
−2
Expressed in units of pressure (1 Pa = 1 N · m )
1 P = 0.1 Pa · s
1 centipoise = 0.01 poise
Viscosity of water at 25◦ C is 0.9 centipoise
Named for Jean Léonard Marie Poiseuille, 1797–1869. French physicist and physiologist
who studied blood flow.
Some Calculated Diffusion Coefficients
The Stokes-Einstein equation for spherical particles:
D=
kT
6πηr
◦
In water at 25 C:
• Small molecule (1 nm): 2×10−10 m2 s−1
• Protein (10 nm): 2×10−11 m2 s−1
• Bacterium (1 µm): 2×10−13 m2 s−1
• 1 mm sphere: 2×10−16 m2 s−1
Time Required for Diffusion Over a Range of Distances
100,000,000 yr
1012
√
RMS(x) =
2Dt
2Dt = hx 2 i
t = hx 2 i/(2D)
1,000 yr
1 month
106
1
1s
10-6
1 μs
nm
μm
mm
Diffusion is effective over short distances, but not long.
m
Chemical Communication Between Neurons
and Between Neurons and Muscle Cells
Structure of a synapse
Synaptic cleft: ≈ 20 nm wide
−6
Time for diffusion for a small molecule: ≈ 10
s = 1 µs
9
Time to diffuse over the length of a sciatic axon (1 m): ≈ 2×10 s = 80 yr
https://en.wikipedia.org/wiki/Chemical_synapse
Thomas Splettstoesser (www.scistyle.com)
Diffusion in Gasses
Diffusion coefficients of gasses: ≈ 2×10−5 m2 /s
Why are they so much larger than for small molecules in liquids
−10 2
(≈ 1×10 m /s)?
From long ago (before fall break): RMS velocity of N2 ≈ 300 m/s.
D = δx2 /(2τ), and v = δx /τ
δx = 2D/v = 1.3×10−7 m
The “mean-free-path” distance.
and
τ = δx /v ≈ 4.4×10−10 s
Much longer random walk steps than in liquids, but still pretty short.
Diffusion in Liquids vs. Atmosphere
A small molecule in water,
−10 2
D = 2×10 m /s
N2 in atmosphere,
−5 2
D = 2×10 m /s
δx = 6.5×10−12 m
δx = 1.3×10−7 m
τ = 10−13 s
τ = 4.4×10−10 s