Chapter 7: Diffusion- Random Motion At The Molecular Level

Question 1

Define Brownian motion

Answer 1

The random process by which molecules move in a solution. It is driven by the thermal energy of the system.

Question 2

Define diffusion

Answer 2

The net movement of ions or molecules from a region of high concentration to on of lower concentration until evenly distributed.

Question 3

Diffusion is an example of _______ transport.

Answer 3

Passive

Question 4

State Fick’s first law

Answer 4

The molecular flux due to diffusion is proportional to the concentration gradient, given an inhomogeneous initial concentration field.

Question 5

Give the equation for Fick’s first law

Answer 5

D = diffusion coefficient
J = diffusive flux
dc/dx = concentration gradient

Question 6

What is the Stoke-Einstein formula?

Answer 6

An expression of the diffusion constant, D.

D = diffusion constant
T = temperature
ξ = frictional constant = 6πηa (η = viscosity; a = radius of sphere)
µ = mobility

Question 7

Give the equation for Fick’s first law in three dimensions

Answer 7

∆c = change in concentration

Question 8

State Fick’s second law

Answer 8

There is a fundamental connection between the passage of time and the square of the period that diffusion occurs over, given an initial inhomogeneous concentration field.

Question 9

Give the equation for Fick’s second law

Answer 9

D = diffusion coefficient
t = time

Question 10

How can Fick’s second law be solved?

Answer 10

The initial conditions and the boundary conditions must be known for the diffusion pathway.

Question 11

Give the equation for Fick’s second law in three dimensions

Answer 11

Question 12

Convert Cartesian coordinates to spherical coordinates

Answer 12

Question 13

Convert Cartesian unit vectors to spherical unit vectors

Answer 13

Question 14

What is the Langevin model?

Answer 14

A model of molecular motion that assumes that particles in a solution can be represented as having two forces acting on them, a dissipative force due to friction and a random force representing Brownian motion that fluctuates randomly as a function of time. Collisions with other particles due to Brownian motion cause the particle to move in one direction and the drag force acts in the opposite direction to slow the particle down.

Question 15

Give the equation for the Langevin model in 1D

Answer 15

m = mass of the particle
dv/dt = acceleration of the particle
f(t) = fluctuating random force
v = velocity of the particle

Question 16

The random force in the Langevin model fluctuates randomly so there is the ____ probability of it acting from the left or right. Hence the average force is ___.

Answer 16

Same
Zero

Question 17

Define time autocorrelation function

Answer 17

A concept that is used to analyse and describe the dynamics of a system.

Question 18

What is the velocity autocorrelation function for a system?

Answer 18

The average autocorrelation function for all particles, found by taking the product at times t = 0 and later times.

Question 19

Give the equipartition theorem for ideal gas

Answer 19

v_rms = root mean square velocity
C(0) = autocorrelation function

Question 20

What is the purpose of the equipartition theorem for an ideal gas?

Answer 20

To describe the autocorrelation function for velocity at t = 0.

Question 21

How does the rms velocity change in 3 dimensions?

Answer 21

A factor of 3 is added to the is added to the numerator (hence, there is a factor of 3 in the autocorrelation function).

Question 22

What is the velocity autocorrelation function of the Langevin model in 1D?

Answer 22

m = mass of the particle
C(t) = velocity autocorrelation function
ξ = frictional constant

Question 23

What is the Green-Kubo relationship?

Answer 23

The relationship between the autocorrelation function (integrated from 0 to infinity) and the diffusion constant.

Question 24

Give the equation for the Green-Kubo relationship

Answer 24

D = diffusion constant
ξ = frictional constant
m = mass of the particle

Question 25

According to the Green-Kubo relationship, the faster the movement of molecules, the _____ collisions and the ______ the rate of diffusion.

Answer 25

Fewer
Faster

Question 26

What is the Green-Kubo relationship also known as?

Answer 26

The fluctuation-dissipation theorem

Question 27

For three dimensional models of the Green-Kubo relationship, the autocorrelation function equals ____ times the diffusion constant, _D.

Answer 27

Three
3D