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- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Advanced_Statistical_Mechanics_(Tuckerman)/02%3A_Foundations_of_classical_statistical_mechanics/2.01%3A_The_Ensemble_Concept_(Heuristic_Definition)Since, from the point of view of macroscopic properties, precise microscopic details are largely unimportant, we might imagine employing a construct known as the ensemble concept in which a large numb...Since, from the point of view of macroscopic properties, precise microscopic details are largely unimportant, we might imagine employing a construct known as the ensemble concept in which a large number of systems with different microscopic characteristics but similar macroscopic characteristics is used to "wash out'' the microscopic details via an averaging procedure. This is an idea developed by individuals such as Gibbs, Maxwell, and Boltzmann.
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Advanced_Statistical_Mechanics_(Tuckerman)/01%3A_Classical_mechanics/1.05%3A_Classical_microscopic_states_or_microstates_and_ensemblesA microscopic state or microstate of a classical system is a specification of the complete set of positions and momenta of the system at any given time. In the language of phase space vectors, it is a...A microscopic state or microstate of a classical system is a specification of the complete set of positions and momenta of the system at any given time. In the language of phase space vectors, it is a specification of the complete phase space vector of a system at any instant in time. A given macroscopic property, \(A\), and its microscopic function \(a = a (x) \), which is a function of the positions and momenta of a system, i.e.
- https://chem.libretexts.org/Courses/Pacific_Union_College/Kinetics/05%3A_The_Ensemble_Treatment/5.01%3A_Ensembles_of_N-molecule_SystemsImagine collecting n N-molecule, constant-volume, constant-temperature systems. An aggregate of many multi-molecule systems is called an ensemble. Just as we assume that no forces act among the non-in...Imagine collecting n N-molecule, constant-volume, constant-temperature systems. An aggregate of many multi-molecule systems is called an ensemble. Just as we assume that no forces act among the non-interacting molecules we consider earlier, we assume that no forces act among the systems of the ensemble. However, as we emphasize above, our model for the systems of an ensemble recognizes that intermolecular forces among the molecules of an individual system can be important
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/23%3A_The_Ensemble_Treatment/23.01%3A_Ensembles_of_N-molecule_SystemsImagine collecting n N-molecule, constant-volume, constant-temperature systems. An aggregate of many multi-molecule systems is called an ensemble. Just as we assume that no forces act among the non-in...Imagine collecting n N-molecule, constant-volume, constant-temperature systems. An aggregate of many multi-molecule systems is called an ensemble. Just as we assume that no forces act among the non-interacting molecules we consider earlier, we assume that no forces act among the systems of the ensemble. However, as we emphasize above, our model for the systems of an ensemble recognizes that intermolecular forces among the molecules of an individual system can be important
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_(LibreTexts)/17%3A_Boltzmann_Factor_and_Partition_Functions/17.05%3A_Pressure_can_be_Expressed_in_Terms_of_the_Canonical_Partition_FunctionThis page explains the derivation of pressure from the canonical partition function in statistical mechanics, linking it to thermodynamic principles and the ideal gas law. It includes equations that r...This page explains the derivation of pressure from the canonical partition function in statistical mechanics, linking it to thermodynamic principles and the ideal gas law. It includes equations that relate average pressure to energy and the partition function, along with a thought experiment illustrating gas compression with a piston.
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_(LibreTexts)/17%3A_Boltzmann_Factor_and_Partition_Functions/17.07%3A_Partition_Functions_of_Indistinguishable_Molecules_Must_Avoid_Over_Counting_StatesThis page discusses the partition function in statistical mechanics, comparing calculations for distinguishable and indistinguishable particles. Using a two-particle model, it shows that distinguishab...This page discusses the partition function in statistical mechanics, comparing calculations for distinguishable and indistinguishable particles. Using a two-particle model, it shows that distinguishable particles have four states, leading to a squared partition function. In contrast, indistinguishable particles have three states, requiring a modification to the partition function through a factor of \(N!\) to prevent overcounting.
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Statistical_Thermodynamics_(Jeschke)/03%3A_Classical_Ensembles/3.01%3A_Statistical_EnsemblesInstead of considering a large ensemble of systems at the same time (ensemble average), we could also consider a long trajectory of a single system in phase space. The single system will go through di...Instead of considering a large ensemble of systems at the same time (ensemble average), we could also consider a long trajectory of a single system in phase space. The single system will go through different microstates and if we observe it for a sufficiently long time, we might expect that it visits all accessible points in phase space with a frequency that corresponds to the associated probability density.
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_(LibreTexts)/17%3A_Boltzmann_Factor_and_Partition_Functions/17.03%3A_The_Average_Ensemble_Energy_is_Equal_to_the_Observed_Energy_of_a_SystemThis page explains the canonical ensemble in statistical mechanics, highlighting the probability of molecules at specific energy levels based on the Boltzmann distribution. It details how to compute a...This page explains the canonical ensemble in statistical mechanics, highlighting the probability of molecules at specific energy levels based on the Boltzmann distribution. It details how to compute average energy using the partition function and introduces the variable \(\beta\) for simplification. The notation may vary, for instance, using \(Z\) instead of \(Q\). The focus is on foundational concepts relating to energy distributions.
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_(LibreTexts)/20%3A_Entropy_and_The_Second_Law_of_Thermodynamics/20.08%3A_Entropy_Can_Be_Expressed_in_Terms_of_a_Partition_FunctionThis page explains how to calculate the entropy of a system using Boltzmann's definition, deriving the ensemble entropy from the partition function and the state probabilities. It presents the formula...This page explains how to calculate the entropy of a system using Boltzmann's definition, deriving the ensemble entropy from the partition function and the state probabilities. It presents the formula for entropy, \(S_{system} = - k \sum_j p_j \ln p_j\), and arrives at the final expression \(S = \frac{U}{T} + k \ln Q\), where \(U\) represents internal energy and \(T\) is temperature.
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_(LibreTexts)/20%3A_Entropy_and_The_Second_Law_of_Thermodynamics/20.09%3A_The_Statistical_Definition_of_Entropy_is_Analogous_to_the_Thermodynamic_DefinitionThis page discusses the relationship between entropy (\(S\)) and microstates (\(W\)) in an ensemble, deriving ensemble entropy using Stirling's approximation. It simplifies the entropy derived from th...This page discusses the relationship between entropy (\(S\)) and microstates (\(W\)) in an ensemble, deriving ensemble entropy using Stirling's approximation. It simplifies the entropy derived from the ensemble and examines the impact of microstate probabilities on entropy. The text concludes by relating changes in entropy to reversible heat transfer, culminating in the expression \(dS = \frac{\delta q_{rev}}{T}\).
- https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_(LibreTexts)/17%3A_Boltzmann_Factor_and_Partition_Functions/17.06%3A_The_Partition_Function_of_Distinguishable_Independent_Molecules_is_the_Product_of_the_Molecular_Partition_FunctionsThis page discusses the derivation of the partition function for a system of distinguishable subsystems, such as gas molecules. It highlights that energy is additive, allowing the total energy to be e...This page discusses the derivation of the partition function for a system of distinguishable subsystems, such as gas molecules. It highlights that energy is additive, allowing the total energy to be expressed as the sum of individual molecule energies. For distinguishable, independent molecules, the overall partition function is the product of individual partition functions.
