380
9 Bioenergetics
Einstein, in 1905, showed 53 how the rate of diffusion of particles in a fluid
depends on the temperature of the fluid, its viscosity, and the size of the particle.
Suppose an external force f , such as from gravity or an electric field, acts on the
particles in the fluid, causing a concentration gradient in the x direction. Then
particles in the fluid will tend to diffuse toward lower concentration, following
Fick’s Law, while the external force will drag them in the opposite direction. At
equilibrium, according to Boltzmann, the distribution of particles is determined by
the stored energy of each particle, i.e.
C = C o exp (−f x/ (k B T )) .
(9.153)
It follows that the concentration gradient at equilibrium is given by
∂C/∂x = − (f/ (k B T )) C
(9.154)
so that, from Fick’s law,
J D = −D (f/ (k B T )) C
(9.155)
The particles which feel an external force will, through collisions, reach a
terminal drift speed v f . Newton’s law for each particle takes the form m¨ r+(1/μ)˙ r =
f, where μ is called the mobility of the particle in the surrounding medium
and characterizes the drag on the particle as it moves. After the particle stops
accelerating because of drag, it reaches the terminal speed is v f = μf . The net flow
of particles in the direction of the force will then be J f = v f C. Now, at equilibrium,
the flow of particles due to diffusion toward lower concentrations will be balanced
by the flow due to the external force: J D + J f = 0. With Fick’s law, this balance of
flow becomes
− D(f/(k B T ))C + v f C = 0.
(9.156)
Einstein realized that the same relation applies to particles within the fluid diffusing
through collisions and being impeded by viscous drag. By observing diffusion of
small particles in a fluid, the diffusion constant becomes
D =
v f
f
k B T = μk B T .
(9.157)
For a large spherical molecule of radius a diffusing in the fluid, f can be
approximated by Stokes’ law: f = 6πηav f , where η is the viscosity of the fluid.
The Einstein relation becomes
53 A. Einstein, Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von
in ruhenden Flüssigkeiten suspendierten Teilchen, Annalen der Physik 322:8, 549–560 (1905).
9 Bioenergetics
Einstein, in 1905, showed 53 how the rate of diffusion of particles in a fluid
depends on the temperature of the fluid, its viscosity, and the size of the particle.
Suppose an external force f , such as from gravity or an electric field, acts on the
particles in the fluid, causing a concentration gradient in the x direction. Then
particles in the fluid will tend to diffuse toward lower concentration, following
Fick’s Law, while the external force will drag them in the opposite direction. At
equilibrium, according to Boltzmann, the distribution of particles is determined by
the stored energy of each particle, i.e.
C = C o exp (−f x/ (k B T )) .
(9.153)
It follows that the concentration gradient at equilibrium is given by
∂C/∂x = − (f/ (k B T )) C
(9.154)
so that, from Fick’s law,
J D = −D (f/ (k B T )) C
(9.155)
The particles which feel an external force will, through collisions, reach a
terminal drift speed v f . Newton’s law for each particle takes the form m¨ r+(1/μ)˙ r =
f, where μ is called the mobility of the particle in the surrounding medium
and characterizes the drag on the particle as it moves. After the particle stops
accelerating because of drag, it reaches the terminal speed is v f = μf . The net flow
of particles in the direction of the force will then be J f = v f C. Now, at equilibrium,
the flow of particles due to diffusion toward lower concentrations will be balanced
by the flow due to the external force: J D + J f = 0. With Fick’s law, this balance of
flow becomes
− D(f/(k B T ))C + v f C = 0.
(9.156)
Einstein realized that the same relation applies to particles within the fluid diffusing
through collisions and being impeded by viscous drag. By observing diffusion of
small particles in a fluid, the diffusion constant becomes
D =
v f
f
k B T = μk B T .
(9.157)
For a large spherical molecule of radius a diffusing in the fluid, f can be
approximated by Stokes’ law: f = 6πηav f , where η is the viscosity of the fluid.
The Einstein relation becomes
53 A. Einstein, Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von
in ruhenden Flüssigkeiten suspendierten Teilchen, Annalen der Physik 322:8, 549–560 (1905).