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Thermo_ISM_ch14 - Chapter 14 Ensemble and Molecular...

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14-1 Chapter 14: Ensemble and Molecular Partition Functions Problem numbers in italics indicate that the solution is included in the Student’s Solutions Manual. Questions on Concepts Q14.1) What is the canonical ensemble? What properties are held constant in this ensemble? An ensemble is a collection of identical units or replicas of the system of interest. In the canonical ensemble, V , N , and T are held constant. Q14.2) What is the relationship between Q and q ? How does this relationship differ if the particles of interest are distinguishable versus indistinguishable? Q is the canonical partition function, while q is the molecular partition function, or the partition function that describes an individual member of the ensemble. The relationship between Q and q depends on if ensemble members are distinguishable are indistinguishable. For N members or units in the ensemble: Q = q N (distinguishable) Q = q N N ! (indistinguishable) Q14.3) List the atomic and/or molecular energetic degrees of freedom discussed in this chapter. For each energetic degree of freedom, briefly summarize the corresponding quantum mechanical model. Translational: Particle in a box Vibrational: Harmonic oscillator Rotational: Rigid rotor Electronic: Hydrogen-atom model, MO theory Q14.4) For which energetic degrees of freedom are the spacings between energy levels small relative to kT at room temperature? At room temperature, the spacings between translational energy levels are extremely small relative to kT . The rotational energy level spacings are also generally small relative to kT , although some species involving light atoms (such as H 2 ) may have energy levels spacings that are comparable to kT . Vibrational and electronic energy levels usually have spacings that are greater than kT .
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Chapter 14/Ensemble and Molecular Partition Functions 14-2 Q14.5) For the translational and rotational degrees of freedom, evaluation of the partition function involved replacement of the summation by integration. Why could integration be performed? How does this relate back to the discussion of probability distributions of discrete variables treated as continuous? Integration can be performed in evaluating q for the translational and rotational degrees of freedom because the energy level spacings along these degrees of freedom are small relative to kT . Since there are numerous states within the energy range of interest (given by kT , it is reasonable to treat the variable of interest (i.e., the number of states) as continuous. Q14.6) When is the high-T approximation for rotations and vibrations? For which degree of freedom do you expect this approximation to be generally valid at room temperature? The high-temperature approximation is when the product kT is significantly greater than the energy level spacings. This will be true for translations at all but lowest temperatures. For rotations, the high-temperature limit is appropriate with T 10 Θ R where Θ R is the rotational temperature, equal to B/k where B is the rotational constant.
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