ch 23 pages 573-580

Ch 23pages 573-580

Lecture 15 – Molecular interactions

In the second half of the course, we will discuss properties of biological molecules from the point of view of their atomic and molecular structure and interactions. This subject will be introduced by analogy with the discussion of ideal gases, and how molecular properties such as interactions and internal vibrations and rotations for multiatomic molecules make real gases deviate from their ideal PV=NRT behavior. In the next two weeks, we will relate thermodynamic properties of gases to the microscopic features that define their molecular structure and interactions and, by showing how observables cannot be explained on the basis of the classical description of molecular properties, will introduce quantum mechanics.

The equation of state PV=NRT is only valid for ideal gases, which includes the fact that gas molecules do not interact

Real gases deviate from ideal behavior because there are molecular forces between gas molecules and within multiatomic gas molecules (even simple diatomic gases such as O2) and thus do not behave like ideal gases

The main deviations result from intermolecular interactions; these are negligible at very low gas densities where the gas molecules are separated on average by large distances, but at higher densities these interactions are no longer negligible

Intermolecular Interactions: The Virial Expansion

Deviations from ideal gas behavior can generally be expressed as a infinite power series in the density =N/V:


...)()( 32 TCTBkTP

The power expansion in density is called a virial expansion.

The coefficients B(T), C(T), etc are dependent on temperature and express the deviation of the behavior of a gas from ideal; they are called virial coefficients. B(T) is called the second virial coefficient, etc.


...)()( 32 TCTBkTP

Virial coefficients reflect the presence of intermolecular interactions. If these interactions are zero, or at very low gas density, the virial expansion reduces to the ideal gas law. As the density increases, progressively higher terms become important.

The second viral coefficient reflects strictly two-body interactions. Intuitively, you can already see how this would be the case, since two-body interactions are dominant at low densities where three, four, etc. molecular encounters are rare.


...)()( 32 TCTBkTP

At very low densities, the viral expansion may be truncated after the second term:

2)( TBkTP

This equation can be rearranged to the following form:

)(TBkTP


...)()( 32 TCTBkTP

Measuring the pressure of a gas recorded as a function of density at a given T may be used to measure the second virial coefficient B(T); A plot of P/ has the following shape:

)(TBkTP


...)()( 32 TCTBkTP

Deviations in the graph of P/ versus =N/V from linearity at higher pressures indicate that the approximation of truncating the expansion to the first non-ideal term is no longer valid and contributions from higher order terms, reflecting interactions involving more than two particles, are becoming important. The truncation would then assume the form:

)(TBkTP

2)()(

TCTBkTP


...)()( 32 TCTBkTP

In order to relate the macroscopic properties we measure (P, V, T) to microscopic properties of the gas (the potential energy function that describes interaction between gas molecules) we have to return to the statistical mechanical description of matter introduced in the first week of the course.

)(TBkTP

Statistical Interpretation of Pressure : no interactions

From classical statistical mechanics (see chapter 11 and lectures 1-3), we have established that the pressure P was related to the molecular partition function by the equation:

TVqkTP

ln

q e E k T

i

i B /

All thermodynamic quantities can be obtained once q is calculated. However, we first have to introduce the correct partition function.


The molecular partition function relates to the energy levels of individual molecules, but if a system is composed of N molecules, then the partition function for the system of N molecules is:

If the molecules are all identical and do not interact with each other, so that energy levels of one molecules are not affected by those of another molecule, then:

l

kTE

j

kTE

i

kTE iii eeeQ /// ......

NqQ


However, if the particles are non-distinguishable (they are all equivalent), then we must introduce a correction reflecting the fact that having molecule a in state 1 and molecule b in state 2 is the same as having molecule a in state 2 and molecule b in state 1, etc (indistinguishable particles). The correct partition function for the case of N indistinguishable particles is:

Q qN

N

!


We will now derive the ideal gas law from the expression relating pressure and partition function

We have already discussed the partition function for a system composed of non-interacting particle moving in three dimensions

If the molecules do not interact, the total energy is the kinetic energy. Furthermore, we have already discussed how the sum over energy states can be converted into an integral because the energy levels of a classical system are continuous

Statistical Interpretation of Pressure: no interactions

Let us then estimate again the molecular partition function:

By substituting

2

2

/ / 2

0

/ 2

0

2

2

E kT mc kTx y z x y z

p mkTx y z

q e dp dp dp dxdydz e dp dp dp dxdydz

e dp dp dp dxdydz

p p p px y z2 2 2 2

2 2 2

2

( ) / 2

0

33 / 2/ 2

0

2

2 2

x y zp p p mkTx y z

p mkTx

q e dp dp dp dxdydz

e dp dx V mkT

Statistical Interpretation of Pressure: no interactions

Therefore, the partition function for N non-interacting, indistinguishable particles is

We can now calculate the pressure as follows:

3 / 22! !

N NNq VQ mkT

N N

P kT QdV

kT VV

NkT VV

NkTV

N

ln ln ln

This is of course the ideal gas law, derived from statistical mechanical principles. If intermolecular interactions are not present, all virial coefficients are zero.

Statistical Interpretation of Pressure: interactions

Under these conditions, the energy of the system that can be written as follows:

E K Upm

UTi

i

N

iji j

N

2

1 2

Where the first term is the kinetic energy and the second term is the potential energy describing all pair-wise interactions by summing over all pairs of molecules in the system

The partition function has of course changed as a result of the presence of the potential energy term; however, the exponential nature of the partition function, allows different energy terms to be ‘partitioned’.


E K Upm

UTi

i

N

iji j

N

2

1 2

We saw in Lecture 2 that, since to a high degree of approximation, the energy of a molecule in a particular state is the sums of various types of energy (translational, rotational, vibrational, electronic, etc.)

... vibrpttr EEEE

......///vibrottr

kTEvibkTErotkTE qqqeeeq tr


E K Upm

UTi

i

N

iji j

N

2

1 2

Similarly, we can partition Q by separating kinetic and potential energy as follows:

......///vibrottr

kTEvibkTErotkTE qqqeeeq tr

2 / 2 /1 1 1

0

1 2 ...!

N

p mkT U kTN N NQ e dp e dx dy dz dx dy dz

N

3 / 21 2 ( , )!

NmkT Z T VN


E K Upm

UTi

i

N

iji j

N

2

1 2

The interaction potential energy

3 / 21 2 ( , )!

NmkT Z T VN

U U iji j

N

can often be assumed to be a function of the positions or coordinates of the molecules. If the molecules are monatomic, then the pair-wise interactions are a function only of the inter-molecular distance r and not of any direction (this is in general not true of diatomic and more complex molecules):

N

jiijij rUU )(


E K Upm

UTi

i

N

iji j

N

2

1 2

The integral

3 / 21 2 ( , )!

NmkT Z T VN

is called the configuration integral because it is a function of the potential energy which is in turn a function of the molecular coordinates.

N

jiijij rUU )(

rdeTVZ NrU ijij )(...),(


E K Upm

UTi

i

N

iji j

N

2

1 2

Let us now calculate the pressure for a system of N identical particles interacting with each other through a pair-wise potential:

3 / 21 2 ( , )!

NmkT Z T VN

Z(V,T) directly relates intermolecular interactions to pressure!

N

jiijij rUU )(

3 / 2 ln ,ln 1ln 2 ( , )!

N Z V TQP kT kT mkT Z T V kTV V N V


E K Upm

UTi

i

N

iji j

N

2

1 2

In general, Z(V,T) can be very complex, but for monoatomic gases at low density, the following approximation is valid:

3 / 21 2 ( , )!

NmkT Z T VN

where

N

jiijij rUU )(

VNb

VTVZ N2

21),(

drreb kTrU 2

0

)/)(2 )1(2


U(r) is the functional form for the pair-wise interaction between molecules and is only dependent on the inter-molecular distance r. Clearly, if U(r)=0, then b2=0 and Z(V,T)=VN. We can then re-obtain the ideal gas law from the expression:

VNkT

VVkT

VTVZkTP

N

ln),(ln

If intermolecular interactions are present, then b2 is non-zero, and the pressure will have an additional term that is dependent on the nature of the molecular interactions:

VNb

VkT

VNkT

VNb

VV

kTVTVZkTP N

22

221ln),(ln


The integral:

is independent of volume so we can write:

VNb

VkT

VNkT

VNb

VV

kTVTVZkTP N

22

221ln),(ln

drreb kTrU 2

0

)/)(2 )1(2

2

2

22

VNb

VN

VNb

VVN

kTP


Recalling the virial expansion of pressure at low density:

We can reinterpret the second virial coefficient B(T)=-b2.

Evaluating b2 as a function of U(r) is therefore of great interest. We will study various forms of intermolecular interactions in the next lecture.

2

2

22

VNb

VN

VNb

VVN

kTP

)(TBkTP


...)()( 32 TCTBkTP

Measuring the pressure of a gas recorded as a function of density at a given T may be used to measure the second virial coefficient B(T); A plot of P/ has the following shape:

)(TBkTP

ch 23 pages 573-580

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