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Questions tagged [statistical-mechanics]

Statistical mechanics is used to describe and quantify observed (average) thermodynamic properties as they relate to microscopic states of the system in question.

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Isobaric-isothermal ensemble with degeneracy

I've been studying statistical mechanics for a while, the topic is very far from what I've learned till now so my questions may be superficial. When trying to see what my entropy is in an isobaric-...
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159 views

Lagrange Multipliers for the derivation of Maxwell-Boltzmann distribution

When deriving the MB-Distribution while taking the number of quantum states $g_j$ into account, the final equation I'd have to solve would be: $$\ln(g_j)-\ln(n_j)+\alpha+\beta\cdot E_j=0$$ Where $n_j$ ...
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Why is the Maxwell–Boltzmann formula inapplicable at low temperatures?

This Wikipedia article states that at sufficiently low temperatures (or high pressure), quantum effects lead to a breakdown of Maxwell–Boltzmann statistics. I know that the spacing of energy levels ...
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120 views

Estimating the absolute entropy of water at room temperature

I thought it would be fun to get an idea of how enormously large $W$ in $S = k\ln W$ is. I'd specifically like to estimate this for a mole of water at $\pu{1 atm}$ and $\pu{298 K}$. To do this, I ...
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Hydrogen vs neon in terms of entropy

I know that both molecular size and complexity increase overall entropy, but between the two I'm not sure which increases it more. Would 1 mole of hydrogen have the greater entropy because it is two ...
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Calculating the number of energy states in a particular momentum direction

Given is a 2D scenario of a particle within a container: The circumference shows all the possible 2D spatial directions for a given momentum value of the particle. One of those momentum directions is ...
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Proof of the chemical potential formula

My textbook as well as Wikipedia both state the formula $\mu_A = \mu_A^0 + RT\ln\{A\}$ which gives the chemical potential $\mu_A$ of a chemical species $A$ as function of the standard chemical ...
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Finite probability from discrete distribution and zero probability from continuous distribution in Maxwell-Boltzmann statistics

I have difficulty understanding a certain concept with the derivation of the Maxwell-Boltzmann distribution $f(v)$ function from Boltzmann statistics. The derivation starts with the Boltzmann ...
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Derivation of Maxwell–Boltzmann statistics

I have 2 questions regarding the derivation of the formula which calculates the probability of molecules having a particular amount of kinetic energy $E_x$ in a system of $N$ molecules. It states that:...
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Stat Mech: Maxwell-Boltzmann Distribution of Momenta

I'm trying to show that the MB distribution of scalar momentum that I've found is properly normalized but I'm stuck. My process: I started with the MB distribution of momentum in 2D and found the ...
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Derivation of the Temperature Dependence of the Boltzmann Distribution

In lecture notes, Atkin's Physical Chemistry, etc. the Boltzmann Distribution is derived by first claiming that the system takes on properties of its weightiest configuration. The weight is therefore ...
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Why do we neglect the zero point energy in statistical mechanics?

Quantisation associated with translational modes can be modelled like a particle in a box (PiB). $$E_n = \frac{n^2h^2}{8mL^2} \qquad n \in \mathbb{Z} \, | \, n \ge 1$$ By the equation above, the ...
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Intensity of “surface plasmon resonance “ absorption peaks for a nano-rod

The SPR absorption spectrum of a nano-rod is seen to have two peaks with the longer wavelength peak corresponding to the absorption by the longitudinal plasmons and the other to the transverse ...
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85 views

How to use the calculate velocity correlation from VanHove correlation?

I am trying to derive an expression for the velocity correlation function $C(t) = \langle \mathbf{V}(t) \mathbf{V}(0)\rangle$ from the VanHove correlation: $$ G(\mathbf{r},t) = \frac{1}{N}\left( \...
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How is the number of successful collisions in a time-step distributed?

In a spatial homogeneous reaction system with two different molecules, how is the number of collisions which will lead to a reaction in a fixed time-step distributed? My guess: Poisson distribution, ...
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What dynamical features allow for the existence of a liquid-liquid transition?

Recently it has been shown theoretically[1] and then experimentally[2] that a liquid-liquid transition (LLT) takes place in pure supercooled water. That is to say, two distinct liquid phases have a ...
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Boltzmann Distribution in Molecular Dynamics Simulation?

The Boltzmann distribution is often used to describe the population of energy states at different temperatures, i.e. something like $$\dfrac{g_i}{g_j} = \mathrm{exp}\left(-(E_i-E_j)\times \frac{1}{...
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Is the Arrhenius Equation a solution to a differential equation?

I just learned about the Arrhenius Equation, $$k(T)=A\exp{\left(\frac{-E_\mathrm a}{RT}\right)}$$ and noticed that it appears to be in the same form as the solution to a linear homogeneous ...
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What is the difference between statistical mechanics and quantum mechanics in terms of accuracy?

It is my impression that everything relevant to 'quantum' is the truth and others is just a simplification. Is this the case in those two kinds of mechanics?
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119 views

How do repulsive solute interactions affect the van't Hoff factor?

Context. Consider an ideal solution, one in which the intermolecular solvent-solvent, solute-solute, and solute-solvent interactions are all equal. The presence of a solute therefore produces (solely) ...
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What is the physical significance of molecular partition function?

What is the physical significance of molecular partition function? I reason, that the molecular partition function \begin{align} q &= \sum_i \exp(-\beta\varepsilon_i),& \beta &= \frac{1}...
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Should the normal mode of vibration leading to the transition state be removed?

I would like to know whether the normal mode of vibration of the reactant should be deleted when calculating $\Delta ZPE$ and $\delta G_{\text{corr}}(T)$, where $\Delta ZPE$ is the zero-point energy ...
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Brownian motion of colloids at high density

Colloidal particles in dilute suspensions tend to perform a universal type of Brownian motion at finite temperature, according to the Stokes Einstein relation $D\approx\frac{k_BT}{6\pi \eta r}$ in the ...
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How does physical adsorption of mixed gas occur on solid? (eg I mean how the gas molecules of compete each other for a slot on solid surface)

I am working on sorption phenomenon of gases (including inert gas) on coal seam surface. For a mixture of 2 gases at a certain pressure, carbon dioxide and methane, for instance, $\ce{CO2}$ is ...
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353 views

Populations of a two level system

Take the two level system in atomic iodine where the ground state is four fold degenerate and the excited state at $7603 \,\mathrm{cm}^{-1}$ above the ground state (which is taken to have zero energy) ...
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Good sources to understand Stat Mech with concepts

I never had a proper course on Statistical Mechanics during my undergrad. I've tried picking up the subject by reading books, but I find that many focus on the mathematical derivations and I get ...
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What is the classical partition function for a system of anharmonic oscillators?

This is an exam question that I'm trying to solve. Here's the text: A system is composed of N localised, but independent one-dimensional classical oscillators. Assume that the potential energy ...
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Possible to find expression for the energy of mixing with the Helmholtz free Energy?

I am aware that the energy of mixing can be found using the thermodynamic identity for the Gibbs free Energy and plugging in the expression for the entropy of mixing which finds;-∆G_Mixing≈-TNk (xlnx+(...
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732 views

Deriving the entropy of mixing of a non-ideal solution

In the earlier posted question named Derive expression for internal energy of mixing and entropy of mixing using statistical thermodynamics the entropy of mixing was found by assuming an ideal ...
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1answer
996 views

Derive expression for internal energy of mixing and entropy of mixing using statistical thermodynamics

I want to derive an expression for the internal energy of mixing, $\Delta_\mathrm{mix}U$, and entropy of mixing, $\Delta_\mathrm{mix}S$. The framework for this is the Lattice Theory of Ideal Solutions....
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4k views

Derivation of mean kinetic energy

I read from a book that average kinetic energy is equal to $3kT/2$ where $k$ is Boltzmann's constant and $T$ is the kelvin temperature. I don't know how the formula was derived. Any help to gain ...
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461 views

Deriving the partition function for a harmonic oscillator

This question is in reference to an answer I posted here yesterday. In it I derived the partition function for a harmonic oscillator as follows $$q = \sum_{j}e^{-\frac{\epsilon_j}{kT}}$$ For the ...
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Probability of finding a molecule in the ground vibrational level

I wanted to estimate the probability of finding a molecule in the ground vibrational level using the Boltzmann distribution: $$ p_i = \frac{e^{-\epsilon_i/kT}}{\sum_{i=0}^{N}e^{-\epsilon_i/kT}} $$ ...
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267 views

Estimate Partition Function from a simulation

I have a system and I've carried out a long molecular dynamics simulation over it. I would like to estimate the partition function $Z.$ Theoretically, one would compute: $$Z=\dfrac{1}{N!h^{3N}}\int \...
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Entropy change when indistinguishable particles suddenly become distinguishable

We have that $$ S = \frac{U}{T} + k\ln Q $$ Using that for indistinguishable particles in the canonical ensemble $$ \ln Q_{\mathrm{indis}} = N\ln \frac{q_{\mathrm{tr}}e}{N} + N\ln q_{\mathrm{rot}} ...
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How is the volume kept constant in an NVE MD simulation?

The simplest way to conduct a molecular dynamics simulation is to set up some initial conditions and propagate the system in time by solving Newton's equations of motion. When one does not interact ...
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Derivation of the form of the time correlation function in NMR spectroscopy

For a given protein, I know that the NMR spectrometer magnet generates a field $B_0$ and that the interactions with the spins in the local environment generates a much smaller field $B_\mathrm{loc}$ (...
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8k views

Equation to calculate the triple point

Is there any equation that can calculate the triple point of a substance? Because a method for calculating the critical temperature and the Boyle temperature of a substance that I posted in another ...
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Difference between microcanonical and canonical ensemble

I feel a little unclarity about this. Canonical ensemble is pictured as many systems in heat reservoir of infinite capacity having $N$ (number of particles), $V$ (volume) and $T$ (temperature) ...
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1answer
452 views

What is the relation between chemical thermodynamics and chemical kinetics? [closed]

Can one be understood on the basis of the other or are they not interrelated at all? The first thing my kinetics textbook demonstrated was how thermodynamics ignores time taken for a process whereas ...
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1answer
635 views

Effect on the partition function upon the mixing of two ideal gases

Consider a box that is separated into two compartments by a thin wall. Each compartment has a volume V and temperature T. The first compartment contains N atoms of ideal monatomic gas A and the ...
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1answer
968 views

Computing accurate vibrational and rotational contributions to the free energy of transition states and loosely bound complexes

Recently I have been dealing with a lot of transition states and relatively loosely bound ion-dipole complexes and I have some trouble figuring out how to make sure that the rotational and vibrational ...
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How to find the number of atomic microstates for a given electronic configuration?

Given some electron configuration, I know the following formula can be used to determine the number of microstates: $$\text{# mircostates} = \frac{(\text{# electron positions})!}{((\text{# electrons})...
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124 views

What is the mathematical foundation for the probabilistic nature of chemistry?

Is this best answered through quantum theory, Boltzmann statistics, or what? For example, in a reaction energy diagram, only some of the reactants go over to the side of the products, and then get ...
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What units do I use for energy in the Boltzmann distribution?

I am a mathematics student doing a course in thermodynamics and am working on the basics now. The question I am asked is: Suppose two states differ in energy by one electronvolt. What can be said ...