CHEM 127-Chapter 15: Statistical Thermodynamics

CHEM 127-Chapter 15: Statistical Thermodynamics Chapter 15: Statistical Thermodynamics Problem numbers in italics indicate that the solution is included in the Student’s Solutions Manual. Questions on Concepts Q15.1) What is the relationship between ensemble energy and the thermodynamic concept of internal energy? The ensemble energy is equal to the difference in internal energy at some finite temperature to that present at 0 K. Q15.2) List the energetic degrees of freedom expected to contribute to the internal energy at 298 K for a diatomic molecule. Given this list, what spectroscopic information do you need to numerically determine the internal energy? Translational, rotational, and vibrational energetic degrees of freedom are expected to contribute to the internal energy at 298 K. Translational degrees of freedom will be in the high-temperature limit; therefore, a molar contribution of 1/2RT per degree of freedom is expected. Rotations are also expected to be in the high-temperature limit. Since a diatomic has two non-vanishing moments of intertia, a molar contribution of RT is expected. Finally, one needs to know the vibrational frequency to determine if the high-temperature limit is appropriate, and if not to determine the vibrational contribution to the internal energy. Q15.3) List the energetic degrees of freedom for which the contribution to the internal energy determined by statistical mechanics is equal to the prediction of the equipartition theorem at 298 K. The high-temperature approximation will generally be applicable to translations and rotations at 298 K; therefore, these degrees of freedom will make contributions to the internal energy in accord with the equipartition theorem. Q15.4)Write down the contribution to the constant volume heat capacity from translations and rotations for an ideal monatomic, diatomic, and nonlinear polyatomic gas, assuming that the high-temperature limit is appropriate for the rotational degrees of freedom. monatomic diatomic Non-linear polyatomic translations 3/2 R 3/2 R 3/2 R rotations 0 R 3/2 R Chapter 15/Statistical Thermodynamics 15-2 Q15.5) When are rotational degrees of freedom expected to contribute R or 3/2R (linear and nonlinear, respectively) to the molar constant volume heat capacity? When will a vibrational degree of freedom contribute R to the molar heat capacity? The rotational degrees of freedom will contribute R or 3/2R to the molar constantvolume heat capacity when the high-temperature limit is valid, defined as when T > 10ΘR where ΘR is the rotational temperature, defined as B/k. A vibrational degree of freedom will contribute R to the molar heat capacity when the hightemperature limit is applicable, defined as when T > 10ΘV where ΘV is the vibrational temperature, defined as ν /k. Q15.6) Why do electronic degrees of freedom generally not contribute to the constant volume heat capacity? The electronic degrees of freedom do not generally contribute to the constant volume heat capacity since the electronic-energy level spacings are generally quite large compared to kT. Therefore, the higher electronic energy levels are not readily accessible so that the contribution to the heat capacity is minimal. Q15.7) What is the Boltzmann formula, and how can it be used to predict residual entropy? S = kln(W) where S is entropy, W is weight, and k is Boltzmann’s constant. This equation can be used to determine the residual entropy in a crystal at 0 K associated with the number of spatial arrangements available to the system. Q15.8) How does the Boltzmann formula provide an understanding of the third law of thermodynamics? For a perfect crystal at low temperature, only one spatial arrangement of the atoms or molecules will be present so that W = 1 and S = 0. Q15.9) Which thermodynamic quantity is used to derive the ideal gas law for a monatomic gas? What molecular partition function is employed in this derivation? Why? The Helmholtz energy can be used to derive the ideal gas law for a monatomic gas. The translational partition function is employed in the derivation since one is dealing with a monatomic gas for which rotations and vibrations are not present. It should be noted that the derivation can be performed for a molecular system, and will also result in the ideal gas law (see Problem P15.24 of this chapter). Q15.10) What is the definition of “zero” energy employed in constructing the statistical mechanical expression for the equilibrium constant? Why was this definition necessary? Chapter 15/Statistical Thermodynamics 15-3 The zero of energy is the dissociation energy for each molecule. It is employed to establish a common energetic reference state for all species involved in the reaction of interest. Q15.11) Assume you have an equilibrium expression that involves monatomic species only. What difference in energy between reactants and products would you use in the expression for KP? The difference in energy would be zero; therefore, only the ratio of translational and electronic partition functions would be relevant when calculating the equilibrium constant. Problems P15.1) Consider two separate ensembles of particles characterized by the energy-level diagram provided in the text. Derive expressions for the internal energy for each ensemble. At 298 K, which ensemble is expected to have the greatest internal energy? 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