We just discussed statistical mechanical principles which allow us to calculate the properties of a...

We just discussed statistical mechanical principles which allow us to calculate the properties of a complex macroscopic system from its microscopic characteristics (energy levels)• A fundamental tool of statistical mechanics is the Boltzmann distribution e-E/(kT) that describes how energy levels are occupied for a system at thermodynamic equilibrium From the partition function q we can derive thermodynamic properties such as pressure, energy or entropy By being able to relate macroscopic properties to the microscopic molecular properties of a system, we can use macroscopic measurements to obtain microscopic parameters such as energy differences between states

E

N E

N

N

NE PE

ii

ii

ii i i

i

The Ideal Gas

• Now, we will reexamine ideal gases from the point of view of their microscopic nature

• The ideal gas is the simplest macroscopic system and therefore an ideal introduction to more complex macroscopic collections of molecules

• By doing so, we will be able to introduce molecular motion and transport, and therefore apply the statistical mechanical tools we have been describing to characterize transport properties of biological molecules (diffusion, separation, etc.)

•The macroscopic description of a substance consists of an equation of state, which provides a relationship between the state variables P, V, and T

• The ideal gas equation PV=nRT is an example of an equation of state (n is the number of moles of gas)

• Does statistical mechanics allow us to reinterpret the ideal gas equation in terms of the molecular properties of the gas?

The ideal gas: statistical mechanical description

Does statistical mechanics allow us to reinterpret the ideal gas equation in terms of the molecular properties of the gas?

Of course it does; remember that, for an isolated system composed of N non-interacting particle:

The ideal gas: statistical mechanical description

• We will use Boltzmann distribution to derive the equation of state for an ideal gas

kNT

E

N

qkNS

V

qNkTP

T

qNkTE

T

V

ln

ln

ln2

• What is an ideal gas from the microscopic perspective?

• It is a system composed of N non interacting particles of mass m confined within a certain volume V=abc, where a, b and c are the dimensions of the container

• In the first lecture, I have provided the expression for the energy levels for such a system:

The ideal gas: partition function

2

2

2

2

2

22

8),,(

c

n

b

n

a

n

m

hnnnE zyxzyx

• We have also calculated the partition function for the 1-dimensional case under the assumption that the energy levels are spaced very close together (corresponding to a system of large mass, or a macroscopic, classical system)


22 2/

20

2exp

8i B BE k T

i B

ma k Tn hq e dn

hma k T

• Because the expression for the energy level is a sum, we can do the same calculation independently for each dimension:

Vh

Tmkcba

h

Tmkqqqq bbzyx

3222

322

• We can now use the above expressions


• To obtain, for one mole of gas:

kNT

E

N

qkNS

V

qNkTP

T

qNkTE

T

V

ln

ln

ln2

V

RTP

RTE

2

3

• Here we will described an ideal gas from in its mechanical properties

• We aim to provide insight into the microscopic meaning of temperature and on transport properties

• It will also introduce the subsequent analysis of transport properties of biomolecules

• We will do so by relating the pressure of a gas with the collisions of the gas molecules against the walls of the container

The ideal gas: microscopic interpretation of temperature

• An ideal gas is a collection of a very large number of particles (molecules or atoms). At a given instance in time, each particle (mass m) in the gas has a position described by (x,y,z) and a velocity (u,v,w), where u, v, and w are the x, y, and z components of the velocity, respectively. The history of a particle’s position/velocity is called a trajectory. Each point on a trajectory of a particles is described by 6 parameters (x,y,z,u,v,w). The speed of a particle c is related to the components of its x, y, and z velocity components (i.e. u, v, and w, respectively):


c u v w2 2 2 2


c u v w2 2 2 2

• A trajectory is obtained by applying the laws of classical mechanics, also called Newton’s laws of motion


• Pressure is is the force F exerted per unit area A of the container wall by gas molecules as they collide with the walls

• In principle, one should be able to obtain an expression for the pressure P by applying Newtons’ laws of motion

• This mechanical view of pressure is called the Kinetic Theory of Gases


• Newton’s First Law of Motion (Law of Inertia): Every body persists in its state of rest or of uniform motion in a straight line unless it is compelled to change that state by forces impressed on it

Newton’s Second Law of Motion: The sum of all forces (F) acting on a body with mass (m) is related to its vector acceleration (a) by the equation F=ma

Newton’s Third Law of Motion: To every action there is always opposed an equal reaction; or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts

• The kinetic (i.e. mechanical) theory of gases is based on the following assumptions:


A gas is composed of molecules in random motion obeying Newton’s laws of motion

The volume of each molecule is a negligibly small fraction of the volume V occupied by the gas

No appreciable forces act on the molecules except during collisions between molecules or between molecules and the container walls

Collisions between molecules and with the container walls conserve momentum and kinetic energy (elastic collisions)


• Consider a particle with mass m moving with velocity u (i.e. in the +x direction). It collides with a wall of unit area A

• Momentum p of a particle is mass times velocity. Just prior to a wall collision the momentum of a gas particle is p=mu. Just following an elastic collision the momentum is p=-mu (the particle has the opposite direction, i.e. it is moving in the –x direction). The change in momentum p is:


• Suppose in an instant of time t a particle with velocity u covers a distance d, collides with the wall and subsequently covers a distance d; then ut=2d or t=2d/u; the change in momentum during the time period t is given by

p

t

mu

d u

mu

d

2

2

2

/

mumumup 2)(

•From Newton’s Second Law (see above) this is the force exerted by the gas particle as it collides with the wall:


• To get the total force F resulting from the collisions of N particles against an area A of the container wall, add up all the particle forces fi:

f map

t

mu

d

2

F f f f f fmu

d

N

N

m

duN i

i

i

N

i

N

ii

N

1 2 3

2

11

2

1

...

• This equation can be rearranged to read:


Now just divide both sides by A to get the pressure:

F f f f f fmu

d

N

N

m

duN i

i

i

N

i

N

ii

N

1 2 3

2

11

2

1

...

PF

A

Nm

A du

Nm

Vu

2 2

2

1

21u

d

Nmu

Nd

NmF

N

ii

This mechanical calculation states that the product of pressure P and volume V is proportional to the average squared velocity

Temperature does not explicitly appear in this equation, and it cannot: temperature is not a mechanical concept, it is a thermodynamic, macroscopic concept


PF

A

Nm

A du

Nm

Vu

2 2

•Let us reconcile the macroscopic view of the pressure of an ideal gas PV=nRT and the microscopic (mechanical) view of the gas; let us being back the statistical mechanical formulation we have give earlier:


PV=nRT

mechanical description

statistical description

2uNmPV

•We can interpret the temperature, a thermodynamic concept, in terms of the average velocity or kinetic energy of the molecules that compose the gas


PV=nRT

mechanical description

statistical description

2uNmPV

Kum

Tk

um

NPV

b

A

2

2

22

3

23

2

• Pressure is related to the average kinetic energy per molecule

• Temperature is a measure of the average kinetic energy of each molecule in the gas

• The proportionality constant that relates T and average molecular kinetic energy is Boltzmann’s constant


• Every atom or molecule in every molecule (protein, nucleic acid or oxygen) in any environment has an average kinetic energy proportional to its T

• Average speed depends on molecular mass in addition to T


• The average kinetic energy of molecules or atoms depends only on the temperature of the system; this is true for the translational energy of solids and liquids as well

• The average speed is an important quantity; for example, the rate at which molecules collide (an important determinants of chemical reactivity) depends on it

• Of course not all gas molecules in a container have the same speed: molecular speeds are distributed (like exam grades)

• What is the distribution like?

The microscopic interpretation of temperature

• Of course not all gas molecules in a container have the same speed: molecular speeds are distributed (like exam grades); what is the distribution like?

The Maxwell-Boltzmann distribution of speed

cN

n cn

Nc f ci i

i

Mi

i

M

i ii

M

i2 2

1 1

2

1

21

c

Nn c

n

Nc f ci i

i

Mi

i

M

i ii

M

i2 2

1 1

2

1

21

cN

n cn

Nc f ci i

i

Mi

i

M

i ii

M

i2 2

1 1

2

1

21

• We should now be used to the idea of averaging microscopic properties using Boltzmann distribution to calculate macroscopic properties of a system

• For the average (or mean) squared speed:

• In the weighted average equation we are averaging over the groups of molecules with equal speeds, where ni is the

number of molecules with speed ci, and fi=ni/N is the fraction

of molecules with velocity ci (distribution)

• We also know that energies associated with the motions of microscopic particles are quantized. However, the spacing between energy levels is very small for large amplitude motions such as molecular translation; thus, quantization is not an important effect in molecular speed distributions

• Because the energy level spacing is small for the translational motion, the sum


cN

n cn

Nc f ci i

i

Mi

i

M

i ii

M

i2 2

1 1

2

1

21

c

Nn c

n

Nc f ci i

i

Mi

i

M

i ii

M

i2 2

1 1

2

1

21

c f cii

M

i2

1

2 can be replaced by:

0

2 )( dccfc

• In general, the average of any property can be calculated from the distribution as follows:


cN

n cn

Nc f ci i

i

Mi

i

M

i ii

M

i2 2

1 1

2

1

21

c

Nn c

n

Nc f ci i

i

Mi

i

M

i ii

M

i2 2

1 1

2

1

21

0

)( dxxxfx

• The function f(c) represents the probability of the particle having a certain speed c and is called Maxwell-Boltzmann speed distribution function

• What is the form of the speed distribution function?

• In lecture 2 we considered the general form for the Boltzmann distribution function, which in quantized systems gives the population of particles in the energy

level is proportional to


cN

n cn

Nc f ci i

i

Mi

i

M

i ii

M

i2 2

1 1

2

1

21

c

Nn c

n

Nc f ci i

i

Mi

i

M

i ii

M

i2 2

1 1

2

1

21

kTEie /

•Let us consider that the only energy of the system under consideration is kinetic energy


cN

n cn

Nc f ci i

i

Mi

i

M

i ii

M

i2 2

1 1

2

1

21

c

Nn c

n

Nc f ci i

i

Mi

i

M

i ii

M

i2 2

1 1

2

1

21

• the probability of finding a molecule between c and c+dc is:

• since:

• then:

2

2

1mcE

0

2/2

2/2

2

2

)()(

dcec

dcecdccfcdP

kTmc

kTmc

2/3

0

2

4

12

adxex ax

dceckT

mdccfcdP kTmc 2/2

2/32

24)()(

•This is the Maxwell-Boltzmann speed distribution function that can be used to calculate mechanical averages:


cN

n cn

Nc f ci i

i

Mi

i

M

i ii

M

i2 2

1 1

2

1

21

c

Nn c

n

Nc f ci i

i

Mi

i

M

i ii

M

i2 2

1 1

2

1

21

In the argument of the exponential, the energy is the molecular kinetic energy

• The constant term

• The term

ensures that

4 2 c skews the distribution function toward higher speeds

0

2 )( dccfc

2/3

2

kT

m

0

1)(cdP

• How does the probability depends on the mass of the molecules and the absolute temperature?


cN

n cn

Nc f ci i

i

Mi

i

M

i ii

M

i2 2

1 1

2

1

21

c

Nn c

n

Nc f ci i

i

Mi

i

M

i ii

M

i2 2

1 1

2

1

21

3D Velocity Distribution Functions:

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

0.0018

0

150

300

450

600

750

900

1050

1200

1350

1500

1650

1800

1950

• How does the probability depends on the mass of the molecules and the absolute temperature?


cN

n cn

Nc f ci i

i

Mi

i

M

i ii

M

i2 2

1 1

2

1

21

c

Nn c

n

Nc f ci i

i

Mi

i

M

i ii

M

i2 2

1 1

2

1

21


0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

0.0018

0

150

300

450

600

750

900

1050

1200

1350

1500

1650

1800

1950

•The graph shows the results of experimental measurements of the distribution of molecular speeds for nitrogen gas N2 at

T=0oC, 1000oC, and 2000oC; each curve also corresponds very closely to the Maxwell-Boltzman distribution. The y axis is the fraction of molecules with a given speed (x axis)


cN

n cn

Nc f ci i

i

Mi

i

M

i ii

M

i2 2

1 1

2

1

21

c

Nn c

n

Nc f ci i

i

Mi

i

M

i ii

M

i2 2

1 1

2

1

21


0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

0.0018

0

150

300

450

600

750

900

1050

1200

1350

1500

1650

1800

1950

• As the temperature is increased, the distribution spreads out and the peak of the distribution is shifted to higher speed; that is why T has such an effect, for example, on reaction rates (for a given mass, molecules move faster, encounter each other more frequently)

• We can use the Maxwell-Boltmann distribution to calculate the mean speed <c> and the mean square speed <c2>:


cN

n cn

Nc f ci i

i

Mi

i

M

i ii

M

i2 2

1 1

2

1

21

c

Nn c

n

Nc f ci i

i

Mi

i

M

i ii

M

i2 2

1 1

2

1

21

2/1

0

8)(

m

kTdcccfc

m

kTdccfcc

3)(

0

22

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