lecture 21: rates of diffusion in liquids and gasses
TRANSCRIPT
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Physical Principles in BiologyBiology 3550
Fall 2021
Lecture 21:
Rates of Diffusion in Liquids and Gasses
Wednesday, 20 October 2021
©David P. GoldenbergUniversity of Utah
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A Quick Review from the Last Lecture
Kinetic energy of an object along the x-axis: Ek;x = mv 2x =2
RMS translational kinetic energy of a molecule: Ek;x = kT=2
RMS velocity of molecule: vx = ‹x=fi =pkT=m
The diffusion coefficient:
D =‹2x2fi
=vx‹x2
Calculating ‹x and fi from D and vx :
‹x =2D
vx=
2DpkT=m
fi =‹xvx
=‹2x2D
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Random Walk of a Small Molecule, Bromophenol Blue,in Water
From diffusion experiment: D = 2× 10−10m2=s
Mass = 1.11×10−24 Kg
Velocity:
vx =pkT=m =
p1:38× 10−23 Kg ·m2s−2K−1 × 298K=1:1× 10−24 Kg
= 61m=s
RMS displacement of random-walk step:
‹x =2D
vx= 2× 2× 10−10m2=s÷ 61m=s
= 6:5× 10−12m
RMS time interval for random-walk step:
fi =‹xvx
= 6:5× 10−12m÷ 61m=s = 10−13 s
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Clicker Question #1
A “typical” protein molecule: D = 1× 10−11m2=s and vx = 6m=s
What is the RMS step length, ‹x , as the molecule diffuses via a randomwalk?
A) 1×10−13 m
B) 3×10−13 m
C) 1×10−12 m
D) 3×10−12 m
E) 1×10−11 m
‹x =2D
vx=
2× 1× 10−11m2=s
6m=s= 3:3× 10−12m
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Clicker Question #2
A “typical” protein molecule: D = 1× 10−11m2=s and vx = 6m=s.
What is the RMS average time between direction changes, fi , as themolecule diffuses via a random walk?
A) 5×10−14 s
B) 5×10−13 s
C) 5×10−12 s
D) 5×10−11 s
E) 5×10−10 m
fi =‹xvx
=3:3× 10−12m
6m=s= 5:5× 10−13 s
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A Small Molecule Versus a Big One
Bromophenolblue
Protein
Molar mass 670 g/mol 60,000 g/mol
Velocity 60 m/s 6 m/s
D 2× 10−10m2=s 1× 10−11m2=s
‹x 6×10−12 m 3×10−12 m
fi 1×10−13 s 5×10−13 s
How do we think about the numbers on a molecular scale?
Have to think about liquids.
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The Nature of Liquids
Molecules are very densely packed
Motions of molecules are limited bythe “energy barriers” surroundingthem.
Other molecules in a solution aresimilarly constrained.
Molecules move only very shortdistances before they changedirection.
The larger the molecule is, the lessfreedom it has and the more slowly itmoves.
From a simulation of liquid water.
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RMS Distance of Diffusion
Random walk along one direction: 〈x2〉 = n‹2x
For diffusion: D =‹2x2fi
‹2x = 2Dfi
n = t=fi
〈x2〉 = n‹2x =t
fi2Dfi = 2Dt
RMS(x) =√2Dt
For bromophenol blue (and molecules of similar size):
RMS(x) =pt=s × 2× 10−5m
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RMS Distance of Diffusion
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RMS Distance of Diffusion
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“Velocity” of Diffusion
Apparent velocity decreases with time and distance.
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Time Required for Diffusion Over a Range of Distances
RMS(x) =√2Dt
2Dt = 〈x2〉
t = 〈x2〉=(2D)
Time required is inversely related to the diffusion coefficient.
Diffusion is effective over short distances, but not long.
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Chemical Communication Between Neuronsand Between Neurons and Muscle Cells
Structure of a synapse
Synaptic cleft: ≈ 20 nm wide
Time for diffusion for a small molecule: ≈ 10−6 s = 1—s
Time to diffuse over the length of a sciatic axon (1 m): ≈ 2× 109 s = 80 yr
https://en.wikipedia.org/wiki/Chemical_synapse
Thomas Splettstoesser (www.scistyle.com)
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Calculating Diffusion Coefficients
What determines diffusion coefficient?• Velocity of molecules (temperature)• Size and shape of molecules• How often molecules colide
Stokes-Einstein equation for spherical particles:
D =kT
6ı”r
r = sphere radius
” = viscosity
A key result from the 1905 Einstein paper on Brownian motion.
A testable prediction!
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Viscosity: Property of a Liquid (or Gas)Resistance to Motion of Molecules Past One Another
A molecular property easily observed on the macroscopic scale.
From a 1960s TV ad for Prell Shampoohttps://www.youtube.com/watch?v=9lFsrjoLKq0
Reflects what we commonly think of as the “thickness” of a liquid.
Units are not straight forward! See the notes.
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Some Calculated Diffusion Coefficients
The Stokes-Einstein equation for spherical particles:
D =kT
6ı”r
In water at 25◦C:• Small molecule (1 nm): 2×10−10 m2s−1
• Protein (10 nm): 2×10−11 m2s−1
• Bacterium (1 —m): 2×10−13 m2s−1
• 1 mm sphere: 2×10−16 m2s−1
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Direction Change
Warning!
Diffusion of Gasses
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Clicker Question #3
How Far Apart are Molecules in Air?(on average)
A) ≈ 1 nm
B) ≈ 1—m
C) ≈ 1mm
D) ≈ 1 cm
E) ≈ 1m
All answers count for now.
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Clicker Question #4
Diffusion coefficients of gasses (at atmospheric pressure):≈ 2× 10−5m2=s
From a previous lecture: RMS velocity of N2: ≈ 300m=s.
Calculate the RMS distance an N2 molecule travels before changingdirection.
A) ≈ 1 nm
B) ≈ 10 nm
C) ≈ 100 nm
D) ≈ 1—m
E) ≈ 10—m
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Diffusion in Gasses
Diffusion coefficients of gasses (at atmospheric pressure):≈ 2× 10−5m2=s
From a previous lecture: RMS velocity of N2: ≈ 300m=s.
D = ‹2x=(2fi), and v = ‹x=fi
‹x = 2D=v = 1:3× 10−7m = 130 nm
The “mean-free-path” distance.Is this the distance between molecules?
Time between collisions:
fi = ‹x=v ≈ 4:4× 10−10 s
Much longer random walk steps than in liquids, but still pretty short.