Ch 23 pages 573-580

1. Lecture 15 – Molecular interactions Ch 23 pages 573-580

2. 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.

3. Intermolecular Interactions: The Virial Expansion 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

4. Intermolecular Interactions: The Virial Expansion Deviations from ideal gas behavior can generally be expressed as a infinite power series in the density r =N/V: 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.

5. Intermolecular Interactions: The Virial Expansion 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.

6. Intermolecular Interactions: The Virial Expansion At very low densities, the viral expansion may be truncated after the second term: This equation can be rearranged to the following form:

7. Intermolecular Interactions: The Virial Expansion 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/r has the following shape:

8. Intermolecular Interactions: The Virial Expansion Deviations in the graph of P/r versus r=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:

9. Intermolecular Interactions: The Virial Expansion 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.

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

11. Statistical Interpretation of Pressure : no interactions 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:

12. Statistical Interpretation of Pressure : no interactions 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:

13. Statistical Interpretation of Pressure : no interactions 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

14. Statistical Interpretation of Pressure: no interactions Let us then estimate again the molecular partition function: By substituting

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

16. Statistical Interpretation of Pressure: interactions Under these conditions, the energy of the system that can be written as follows: 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’.

17. Statistical Interpretation of Pressure: interactions 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.)

18. Statistical Interpretation of Pressure: interactions Similarly, we can partition Q by separating kinetic and potential energy as follows:

19. Statistical Interpretation of Pressure: interactions The interaction potential energy 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):

20. Statistical Interpretation of Pressure: interactions The integral is called the configuration integral because it is a function of the potential energy which is in turn a function of the molecular coordinates.

21. Statistical Interpretation of Pressure: interactions Let us now calculate the pressure for a system of N identical particles interacting with each other through a pair-wise potential: Z(V,T) directly relates intermolecular interactions to pressure!

22. Statistical Interpretation of Pressure: interactions In general, Z(V,T) can be very complex, but for monoatomic gases at low density, the following approximation is valid: where

23. Statistical Interpretation of Pressure: interactions 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: 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:

24. Statistical Interpretation of Pressure: interactions The integral: is independent of volume so we can write:

25. Statistical Interpretation of Pressure: interactions 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.

26. Intermolecular Interactions: The Virial Expansion 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/r has the following shape: