8.1: The Boltzmann Factor is used to Approximate the Fraction of Particles in a Large System The proportionality constant \(k\) (or \(k_B\)) is named after Ludwig Boltzmann. It plays a central role in all statistical thermodynamics. The Boltzmann factor is used to approximate the fraction of particles in a large system. The Boltzmann factor is given by: \(e^{-\beta E_i}\). 8.2: The Boltzmann Distribution represents a Thermally Equilibrated Distribution The Boltzmann distribution represents a thermally equilibrated most probable distribution over all energy levels. There is always a higher population in a state of lower energy than in one of higher energy. 8.3: Translational Partition Functions of Monotonic Gases The energy levels of translation are very closely spaced, so a large number of translational states are accessible and available for occupation by the molecules of a gas. This result is very similar to the result of the classical kinetic gas theory. 8.4: Most Atoms are in the Ground Electronic State The energy difference between the ground electronic state of a system and its first excited state are typically much larger the thermal energy, \(kT\). This means that most atoms are in their ground electronic state, unless the temperature of the system is very high. 8.5: Pressure can be Expressed in Terms of the Canonical Partition Function The canonical partition function can be used to derive an equation of state for pressure, \(P\). 8.6: Most Molecules are Rotationally Excited at Ordinary Temperatures At room temperature, many rotational states will be populated. This is due to the smaller rotational energies compared to vibrational or electronic energies. 8.7: Rotational Partition Functions of Diatomic Gases Contain a Symmetry Number Homonuclear diatomic molecules have a high degree of symmetry and rotating the molecule by 180° brings the molecule into a configuration which is indistinguishable from the original configuration. This leads to an overcounting of the accessible states. To correct for these symmetry factors, we divide the partition function by \(σ\), which is called the symmetry number. 8.8: Rotational Partition Functions of Polyatomic Molecules Depend on the Sphar of the Molecule For a polyatomic molecule containing NNN atoms, the total number of degrees of freedom is 3N3N3N. Out of these, three degrees of freedom are taken up for the translational motion of the molecule as a whole. The translational partition function was discussed previously and now we have to consider the three rotational degrees of freedom and the 3N–63N–63N–6 vibrational degrees. 8.9: Most Molecules are in the Ground Vibrational State At room temperature, most molecules are in the ground vibrational state. This is because the vibrational energies of molecules are larger than the average thermal energy available. 8.10: Vibrational Partition Functions of Polyatomic Molecules Include the Partition Function for Each Normal Coordinate The partition function for polyatomic molecules include the partition functions for translational, electronic, vibrational, and rotational states. For translational states, the number of states available is far greater than the number of molecules. For electronic states, we only consider the ground electronic state due to the large gap between electronic states. For vibrational states, we include all the normal modes of vibration. 8.11: The Partition Function of Distinguishable, Independent Molecules is the Product of the Molecular Partition Functions A system, such as a gas can, consist of a large number of subsystem. How is the partition function of the system built up from those of the subsystems depends on whether the subsystems are distinguishable or indistinguishable. For distinguishable systems, the partition function is the product of the molecular partition functions. 8.12: Partition Functions can be Decomposed into Partition Functions of Each Degree of Freedom Just as the partition function of a system of \(N\) particles can be decomposed into the product of partition functions for each molecule, a molecular partition function can be decomposed into the product of the partition functions for each degree of freedom. 8.13: The Average Ensemble Energy is Equal to the Observed Energy of a System The probability of finding a molecule with energy \(E_i\) is equal to the fraction of the molecules with energy \(E_i\). The average energy is obtaining by multiplying \(E_i\) with its probability and summing over all \(i\): \[\langle E \rangle=\sum_i{E_iP_i} \nonumber\]. Using the Boltzmann distribution for \(P_i\) allows us to show that the average ensemble energy is equal to the observed energy of the system. 8.14: The Energy of a Diatomic/Polyatomic Molecule Can Be Approximated as a Sum of Separate Terms A reasonable partition function of a diatomic/polyatomic molecule is the product of the partition function for the translational, vibrational, rotational, and electronic degrees of freedom. The total energy of the molecule then becomes the sum of the translational, vibrational, rotational, and electronic energies. 8.15: Heat Capacity at Constant Volume is the Change in Internal Energy with Temperature The heat capacity at constant volume, denoted \(C_V\), is defined to be the change in thermodynamic energy with respect to temperature. 8.16: Molar Heat Capacities The heat capacity of a substance is a measure of how much heat is required to raise the temperature of that substance by one degree Kelvin. For a simple molecular gas, the molecules can simultaneously store kinetic energy in the translational, vibrational, and rotational motions associated with the individual molecules. In this case, the heat capacity of the substance can be broken down into translational, vibrational, and rotational contributions. 8.17: The Equipartition Principle The equipartition theorem states that every degree of freedom that appears only quadratically in the total energy has an average energy of ½kT in thermal equilibrium and contributes ½k to the system's heat capacity. Here, k is the Boltzmann constant, and T is the temperature in Kelvin. The law of equipartition of energy states that each quadratic term in the classical expression for the energy contributes ½kBT to the average energy. 8.18: The Famous Equation of Statistical Thermodynamics is S=k ln W Entropy of can be calculated from the molecular viewpoint when considering the number of microstates that exist in a corresponding macrostate. 8.19: Entropy Can Be Expressed in Terms of a Partition Function We have seen that the partition function of a system gives us the key to calculate thermodynamic functions like energy or pressure as a moment of the energy distribution. We can extend this formulism to calculate the entropy of a system once its Q is known. The derivation is shown on page 840 and involves the use of the Stirling approximation. The end result is 8.20: The Statistical Definition of Entropy is Analogous to the Thermodynamic Definition The molecular formula for calculating entropy is directly related to the macroscopic, thermodynamic formula for calculating changes in entropy. 8.21: Practical Absolute Entropies of Gases Can Be Calculated from Partition Functions If the partition function, \(Q\) is known for a system, the practical absolute entropy of the system can calculated. 8.22: Standard Entropies Depend Upon Molecular Mass and Structure Entropy is related to the number of microstates a collection of particles can occupy. As both the molecular mass and molecular structure of the particles will affect the number of available microstates, they also affect the entropy of the collection of particles. In general, the entropy of a system increases with molecular mass and the number of atoms in a molecule. 8.23: Spectroscopic Entropies sometimes disgree with Calorimetric Entropies Some substances have residual entropy that arises when multiple configurations exist for the structure of the substance at zero kelvin.