# 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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### 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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### 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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### Why does the plot of volume and temperature have a kink in a second order phase transition?

The $(b)$ part of the figure shows how thermodynamic variables change in a second order phase transformation. We observe there is a kink in the plot of volume with temperature. Why is that so? The ...
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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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### 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 ...
1 vote
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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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### 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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### 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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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 ...