Download Lecture 18 More on Diffusion and Kinetic Energy

Physical Principles in Biology
Biology 3550
Fall 2016
Lecture 18
More on Diffusion and Kinetic Energy
Monday, 3 October
c
David
P. Goldenberg
University of Utah
[email protected]
Random Walk Distances
Step length: δ
(or RMS step length)
Number of steps: n
Total distance walked: nδ
Mean-square end-to-end
2
2
distance: hr i = nδ
Root-mean-square end-to-end
p
distance: RMS(r ) = hr 2 i
r
Fick’s First and Second Laws of Diffusion
C(x)
x
First law:
J = −D
x
Second law:
dC
dx
Flux, J, at position x is
proportional to the concentration
gradient at that position.
2
d C
dC
=D 2
dt
dx
Rate of change in concentration
at position x is proportional to
the derivative of the
concentration gradient.
Clicker Question #1
Where will the flux, J, be greatest?
C(x)
1
2
3
1
0
x
2
3
0
x
At point 2, where the concentration gradient is greatest.
Clicker Question #2
Where will where will the concentration change most rapidly? (After time 0)
C(x)
1
2
3
1
0
x
2
0
x
NOT at point 2, where the concentration gradient is constant.
3
Diffusion from a Sharp Boundary
Relative Concentration
Distribution of molecules from all starting points, a ≥ 0.
0 a
x
At position x, concentration is the sum of molecules that have diffused
from a ≥ 0
Z∞
2
1
C (x, t) =
e −(x−a) /(4Dt) da
√
0
4πDt
Diffusion from a Sharp Boundary
0
0
x
Diffusion from a Sharp Boundary
0
0
0
x
A Simple Diffusion Experiment
48 hours after forming a (pretty)
sharp boundary.
We can use this to estimate the
diffusion coefficient, D.
Estimating D from Diffusion from a Sharp Boundary
Zx
√
0
1
δx2
D=
2τ
2
/(4Dt)
dx
4πDt
t = 48 hr = 1.7×105 s
D ≈ 2×10−10 m2 /s
e −x
Relative Concentration
1
C (x, t) = +
2
1
0.5
0
-4
-2
0
2
x (cm)
What are the average length (δx ) and duration (τ) of the random walk
steps?
4
Before Moving On
C(x)
0
x
0
x
This is a particularly simple case!
• Single concentration gradient along the x-axis.
• Initial state is defined by a simple concentration difference.
• Volume has a simple shape.
Same mathematics applies to the flow of heat.
Solutions pioneered by Joseph Fourier, 1768–1830.
Warning!
Direction Change
Molecular Motion and Kinetic Energy
What is energy?
Capacity to do work.
What is work?
Mechanical work: The application of force over distance:
Zb
Fdx
w=
a
The units of work and energy.
• Force: Units defined by Newton’s second law: F = mass × acceleration
SI unit of mass: Kg
2
Acceleration: change in velocity (m/s) with time. SI units: m/s
SI units of Force: Kg · m/s2
• Work or energy: Kg · m2 /s2
1 N = 1 Kg · m/s2
1 J = 1 N · m = 1 Kg · m2 /s2
Kinetic Energy
A object of mass, m, moving with velocity, v , in the x-direction has
kinetic energy in that direction of:
Ek,x = mv 2 /2
Check the units: Kg × (m/s)2 = Kg · m2 /s2
It’s OK!
What does this mean?
• The energy required to accelerate the mass, m, from rest to velocity, v .
• Also the energy released during the deceleration of the mass from
velocity, v , to rest.
• Kinetic energy does not depend on the rate of acceleration, only the final
velocity.
• But, amount of wasted energy likely does depend on rate of acceleration!
2
Ek,x is proportional to v . What are the implications?