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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What is the diffusivity of colloidal Formazine in water?
I have synthesized Foramzine in an aqueous solution (turbidity = 4000 NTU), and will be further diluting it. I would like to know the diffusion coefficient (diffusivity/mass diffusivity) of colloidal ...
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RMS Speed of Gases as Standard Deviation
In kinetic molecular theory, the average velocity of gas particle is zero since the molecule move in different directions, and the overall effect is zero. Howeever, you can calculate different speeds.
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Schottky Anomaly
Simple question about the definitions, but I can't find it explicitly stated anywhere.
If you have a two level system described by $n_i = \frac{N}{Z}e^{-\beta E_i}$ and calculate the heat capacity
$$ ...
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Is it Rigorous to Derive the Arrhenius Exponential Term from the Boltzmann Distribution?
The Boltzmann distribution is a probability density function which expresses the probability of finding a particle in an energy state $\epsilon$ while in thermal equilibrium given a specific ...
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Virial theorem confusion
Virial theorem says that $E_{t} = -\frac{E_{P}}{2}$ however this says that lets say for 2 gas molecules(Ne) the kinetic energy due to temperature a molecule of $Ne$ at 0K will have the same energy ...
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Why do nuclei move considerably slower than electrons
I've been trying to learn quantum chemistry at an introductionary level. While reading I've found out that the Born-Oppenheimer approximation seems to be the reason for the basic and crucial model of ...
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Using a Chi-Square distribution table to calculate fraction of gas molecules with activation energy
I couldn't decide whether to ask this to Chemistry, Physics or Statistics stack exchange. Hopefully I made the right choice.
According to this Wikipedia page, while the speed of particles in an ideal ...
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Can a spontaneous reaction have zero or negative activation energy?
Is it possible for a certain spontaneous reaction to have zero or negative activation energy?
My high school teacher explains that it shouldn't be possible since it breaks the Arrhenius equation, ...
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Cubic nature of van der Waals' gas equation
$V_m^3-\dfrac{RT+bP}{P}V_m^2+\dfrac{a}{P}V_m-\dfrac{ab}{P}=0$
So, At $T<T_c$ the above equation has three real roots say $V_1,V_2,V_3$, my doubt is what does this mean physically because for some ...
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Why probability for an atom is fraction for population?
In the canonical ensemble the probability of finding an atom (or molecule) in its $i$-th energy state is given by the Boltzmann factor.
This probability is interpreted as the number of microstates ...
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According to Maxwell-Boltzmann distribution, what is probability distribution function proportional to?
If, according to the Maxwell-Boltzmann distribution
$${f(v)}\propto\exp\left(-\frac{\varepsilon}{kT}\right),\tag{1}$$
which is the equation from which the whole final equation is derived, then why is ...
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Does Boltzmann distribution refers to microstates or particle's probability?
I am reading about the Boltzmann distribution and I can't grasp the following:
$$p_i = e^{-\frac{E_i}{k_B T}}$$
Is this the probability finding the system at a microstate with Energy $E_i$ or the ...
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Why 3 and not 6 in U = 3/2RT from the equation for the internal energy of an ideal gas?
Internal energy of an ideal monatomic gas is
$$U=\frac{3}{2}RT.\tag{1}$$
While I understand the derivation, I do not understand why we multiply by $3$ rather than by $6.$ According to Khan Academy ...
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Problem with partition function when calculating equlibrium constant?
For example a reaction is given as ($\pu{T = 1000 K}$),
$$\ce{2Na(g) <--> Na2 (g)}$$
The equilibrium constant,
$$\ce{K_{c} = \frac{\frac{q_{Na_{2}}}{V}}{(\frac{q_{Na}}{V})^2}}$$
But when ...
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Microscopically, how does temperature increases in simple exothermic reactions? [duplicate]
Let’s say we have a simple exothermic reaction $\ce{A_{(g)} + B_{(g)} -> C_{(g)}}$
Since it’s exothermic, that means that kinetic energy of C is greater than the kinetic energy of the reagents. ...
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How is relative interaction energy defined and quantified?
I understand the interaction energy is the energy associated or caused by the interaction between the objects. So, for atoms it has to be the sum of van der Waals interaction and Coulomb interaction.
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Pressure dependence of the equilibrium constant
The equilibrium constant of a chemical reaction can be calculated if the $\Delta_\mathrm rG^\circ$ is known which is a function of only temperature. So if a reaction happens inside a container ...
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Interpretation of azeotropes from statistical mechanics derivation
In an earlier posted question titled Reason for the formation of azeotropes, @porphyrin presented an elegant derivation from statistical mechanics to interpret partial pressures for the two components ...
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Saturated vapor and liquid densities (VLC curve) from Helmholtz free energy equation of state
I have been reading about the statistical associating fluid theory (SAFT) which computes the Helmholtz free energy ($A$) from the SAFT EoS (a hell lot of equations) for a molecule of interest, and is ...
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Is this a valid interpretation of the Mean relative speed between two (Maxwell-Boltzmann) species?
In textbooks, the collision frequency $Z_{ab}$ is always given as something like:
$$Z_{ab} = n_an_b\sigma_{ab} \Bigg(\frac{8 k T}{\pi \mu}\Bigg)^{1/2}= n_an_b\sigma_{ab} \bar{C}_{rel}$$
where the ...
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Do solids in a solution also apply partial pressure? If yes, can this be used as an intuitive explanation for osmosis?
I know that in a mixture of different gases, we can assign every type of gas molecules a partial pressure so that the total pressure is the sum of all partial pressures. This can be extended to gas ...
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Why are MD simulations necessary for obtaining Boltzmann populations?
Given that MD simulations converge to the Boltzmann distribution $\rho \sim \exp(-\beta \epsilon)$ after sufficiently long times, and all the macroscopic quantities can be computed from the Boltzmann ...
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Practical use of the partition function in molecular simulations
In the second chapter of Understanding Molecular Simulation, Frenkel and Smit derive an equation for the partition function and the thermal average of the generic observable A, stating that these ...
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Statistical entropy of perfect gas and indistinguishability
The weight, $W$ (the same $W$ as in Boltzmann formula $S=k\text{ln}W$) of a configuration {$N_0, N_1, N_2$, ...} is given by
$$W=\frac{N!}{N_A!N_B!N_C!...} \tag{1}$$
where $N$ is the total number of ...
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How can I determine the spin of a molecule?
I was reading statistical mechanics today and came across a line
For $\ce{O2}$ and $\ce{N2},$ we get half-spin, and that is why we get $σ = 2$ in their rotational partition equation.
I didn't ...
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Number of possible microscopic states and their probabilities
I'm having trouble with the following problem:
Boltzmann defined entropy as a measure of the number of possible microscopic states of a system, $W$,
as:
${S = k_BlnW}$
Here, $k_B$ is the Boltzmann ...
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Why is the spectroscopic entropy of carbon monooxide higher than that of nitrogen?
While solving a question set, I found a question with this data table (at $\pu{298K}$):
$$
\begin{array}{lrrr}
\hline
\text{Substance} & T_\mathrm b/\pu{K} & S^⦵_\mathrm{cal}/R & S^⦵_\...
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Practical meaning of omega potential
In statistical thermodynamics, the quantity omega potential is defined and is different for fermions, photons, phonons and other particles .
I have been understanding the derivations of omega ...
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Can someone help me understand the motivation behind and visualize the dihedral potential and the Urey-Bradley potential?
I am studying statistical mechanics and force-fields, and I see a lot of this formula being thrown around with no explanation:
$$U_{\mathrm{dihedral}} = \begin{cases}
k(1+\cos (n\psi + \phi)),&...
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Does population of vibrational states depend on population of electronic states?
We know that molecules have different energy states (vibrational, rotational, electronic). Using Boltzmann distribution we can find the popoulation of each energy state. If we have only electronic ...
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Getting close to zero percent of particles with energy greater than an activation energy of 101kJ/mol using the Maxwell-Boltzmann energy distribution
I experimentally obtained the activation energy of a reaction to be approximately 101kJ/mol. I was then going to use the value for the activation energy to see how the number of particles with ...
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How to model CRISPR/Cas9 binding and cleavage using a generalized Kittel's zipper model?
I am thinking about how to model the binding of a CRISPR/Cas9+guide-RNA complex to a double-stranded DNA that cause its cleavage.
This post describes four steps at which a CRISPR/Cas9 system binds and ...
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Is the average kinetic energy of evaporating water molecules (at room temperature) equivalent to the average kinetic energy of boiling water?
Purpose:
On new year's eve, after a splendid red and an assortment of sumptuous repasts, I made a bold remark which, on further consideration, may turn out to be incorrect. Unless! Unless I can ...
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What is the implication of having the internal energy equal the Gibbs free energy?
I came up with a hypothetical equation of state, where one of the consequences is that $U=G$, where $U$ is the internal energy and $G$ is the Gibbs free energy. I wanted to ask the chemists, is there ...
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Is there an error on Wikipedia's page on characteristic rotational temperature of HF?
I am referencing this article: https://en.wikipedia.org/wiki/Rotational_temperature
So, I see that $\theta _R = \frac{(h/2\pi)^2}{2Ik_b}$.
I took the value of $I$ from https://cccbdb.nist.gov/...
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What is b in the Gompertz model?
I am wondering how to use the Gompertz model in modelling drug diffusion:
$$x(t) = x_\mathrm{max}\exp(-a\exp(b\log t)).$$
$x_\mathrm{max}$ is to be taken as $100,$ $a$ is the undissolved proportion at ...
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Understanding transition state theory
Let's say we have standard chemical reaction given by
$$A-B + C \leftrightharpoons [A\cdot\cdot B\cdot\cdot C]^{\dagger} \rightarrow A +B-C$$
I want to estimate the pre-exponential factor of this ...
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Thermodynamics- How are various property tables made? [closed]
I've always been curious about how different thermodynamic property tables are made. It's really interesting to me. I want to know how different things such as critical properties and variations of ...
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My Misconception of Entropy
It was recently brought to my attention that my understanding of entropy is wonky at best.
In my experience, entropy was introduced (superficially at best) during general chemistry/foundations of ...
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How do we get g and E values for various levels to calculate electronic partition function
When we calculate electronic partition function using the formula:
$$q_\mathrm{elec}=\sum^\infty_{n=1}g_ne^{-E_n/k_BT}$$
How can I get the $g_n$ value and $E_n$ values for $n=0,1,2,3...$ levels?
I ...
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Determining the torsion energy profile of butadiene by molecular dynamics
Disclaimer: this question is related to my previous question, although it is much more detailed
Consider the following rotation of butadiene:
I want to determine the relative energy of each dihedral ...
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How to calculate the half time of a unimolecular reaction given the Arrhenius coefficient and the activation energy?
Consider the following unimolecular reaction:
$$\ce{H2O2 + M -> OH + OH + M}$$
The high pressure limit unimolecular rate coefficient for this reaction is $k_\mathrm{uni} = A\cdot \mathrm{e}^{...
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Constructing a partition function for a gas-solid equilibrium in canonical ensemeble
For a closed system at fixed N,V,T there is a solid in equilibrium with its vapor. I'm told that I can minimize the helmholtz free energy to find that the equilibrium condition is given by
$N_g = \...
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Pressure of a system of particles
I'll derive the pressure of a system of particles, which would be used in a MD simulation for example, according to Allen's Computer Simulation of Liquids and show where my issue is.
Consider a ...
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What exactly is temperature?
I've read at many places that temperature is the average kinetic energy of particles present in an object. I just don't intuitively get how kinetic energy is connected with temperature. And how is ...
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Does the Boltzmann distribution hold for supercritical fluids?
Above the critical temperature of a real gas, if we compress it, to whatever extent, it doesn't get liquefied (no gas liquid interface). However, it does behave like liquids , esp. in case of very ...
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Balance Repeatability Sample or Population for SD
A question for the masses about standard deviation (SD), selection of population vs sample in balance measurements.
Analytical balances are typically checked for repeatability by taking 10 weight ...
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Do "random" "2D" polymers constrained to a box actually form "space filling" curves?
Background: The self avoiding walk model is (allegedly) a reasonable model for polymers, at least with respect to certain "universal" (i.e. lattice independent) constants -- see section 1.1. in this ...
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Statistical thermodynamics [closed]
Entropy of indistinguishable particles $S$ is
$$S = k\ln W,$$
but in some places notation is different:
$$S = k\ln Q + \frac{U}{T}$$
Can you explain, please?
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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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