12 votes
Accepted

Why is the Maxwell–Boltzmann formula inapplicable at low temperatures?

Since the explanation was a little more complicated than I initially thought, I figured it would be worth it to combine my comments (and info from Physics SE) into an answer. Quantum particles satisfy ...
  • 11.4k
12 votes

What exactly is temperature?

Heat is the transfer of energy to or from the body in forms other than matter flow or work (organized energy transfer, such as pushing). Temperature is only a well-defined property for a collective ...
  • 1,510
12 votes

What exactly is temperature?

Temperature vs kinetic energy [OP:] I've read at many places that temperature is the average kinetic energy of particles present in an object. Temperature has to do with the average kinetic energy ...
  • 32.6k
11 votes

Details of Boltzmann distribution derivation

What you say is a good idea, but is not quite correct because we must maximize the distribute subject to two constraints. I will reiterate the derivation following along with McQuarrie's Statistical ...
  • 12.7k
9 votes
Accepted

What is the physical significance of molecular partition function?

The partition function $q=\sum_i\exp(-E_i/k_BT)$ in your question can be regarded as the effective number of levels accessible to the molecule at a given temperature. It also means that in the ...
  • 27.6k
8 votes
Accepted

Entropy change when indistinguishable particles suddenly become distinguishable

Very interesting question. The issue is that your formula for $\ln Q_\mathrm{indis}$ does not hold for $N = 1$. Since the rotational, vibrational and electronic degrees of freedom do not come into ...
  • 66.8k
8 votes
Accepted

Derivation of mean kinetic energy

A simple validation The result you quoted is the average translational kinetic energy for an ideal gas. First, let's sketch out a rough derivation for the average kinetic energy of a particles of an ...
  • 8,335
8 votes
Accepted

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

In the solution there are two types of molecules $N_1$ and $N_2$. Assume that they do not interact with one another but simply occupy particular 'lattice' sites by blocking them. The total number ...
  • 27.6k
7 votes
Accepted

What is the classical partition function for a system of anharmonic oscillators?

Here follows a complete mathematical derivation of the expressions for the internal energy and isochoric heat capacity. I am not sure why taking the high-temperature limit is unphysical, but maybe ...
  • 4,653
7 votes
Accepted

Deriving the partition function for a harmonic oscillator

I'm confused why you're interpreting the partition function as a count of states. It can't be a count; it's continuous. The zero point energy doesn't actually matter because you can just shift the ...
  • 16.9k
7 votes
Accepted

How is the volume kept constant in an NVE MD simulation?

The answer to this is linked to how you define the “volume” of your system, and the boundary conditions. The simplest case is simply to consider an isolated systems, like a molecule in gas phase. This ...
  • 23.3k
7 votes

What is the difference between statistical mechanics and quantum mechanics in terms of accuracy?

Quantum mechanics is about the physics of very small things, molecules and smaller. Classical mechanics is about macroscopic things. Quantum mechanics covers the whole of classical mechanics as well, ...
  • 4,539
7 votes

Boltzmann Distribution in Molecular Dynamics Simulation?

Within statistical mechanics (SM) a molecular property $X$ is computed by $$\left<{X}\right>_{SM}=\sum^{states}_i X_i p_i$$where $X_i$ is the value of $X$ for energy state $i$ and $p_i$ is the ...
  • 4,779
7 votes

What exactly is temperature?

Temperature is related to kinetic energy, but it can't be simply equated to the average kinetic energy of the system. As I wrote in response to another answer, different systems can have different ...
  • 11.6k
7 votes
Accepted

Why are MD simulations necessary for obtaining Boltzmann populations?

In principle you can, assuming you are given $V(\{\bf{r_i}\})$ where $\{\bf {r_i} \}$ is the set of variables that define a configuration in the system - this is typically the coordinates of the atoms,...
  • 2,530
6 votes

What is the difference between statistical mechanics and quantum mechanics in terms of accuracy?

From plain Wikipedia, Statistical mechanics is a branch of theoretical physics that uses probability theory to study the average behaviour of a mechanical system, where the state of the system is ...
6 votes

Equation to calculate the triple point

In principle, you can use the integrated Clapeyron and Clausius–Clapeyron equations to calculate $P$ vs. $T$ co-existence lines and find the crossing point which is at the triple point. You need the $\...
  • 27.6k
6 votes
Accepted

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

Given the accuracy it sounds like you're looking for, and the systems you're studying, this indeed appears to be a Very Hard Problem. I think your intuitions are well founded. Briefly, to your ...
  • 16.9k
6 votes
Accepted

Is the Arrhenius Equation a solution to a differential equation?

To answer the question of the title. In Adam-Gibbs theory the activation enegy is defined as [1]: $$E_\mathrm a=-R\left[ \frac{\partial \ln k}{\partial (1/T)} \right]$$ Now clearly this is a first ...
6 votes
Accepted

According to Maxwell-Boltzmann distribution, what is probability distribution function proportional to?

The $v$ in your first expression (that $f(v) \propto \exp(-\varepsilon/kT)$) most likely refers to the true velocity which is a vector $$\vec{v} = (v_x, v_y, v_z),$$ whereas the $v$ in the Maxwell–...
  • 66.8k
5 votes
Accepted

Probability of finding a molecule in the ground vibrational level

You're on the right track. Also, using $i$ as an index can be confusing some times because it can be confused with the imaginary number; however, here it should not present a problem. As a matter of ...
  • 8,335
5 votes

How to find the number of atomic microstates for a given electronic configuration?

You really have to calculate out things exactly, and particularly carefully if electrons are in the same orbital, e.g. $p^2$, when the Pauli principle has to be checked. Its a bit tedious but it can ...
  • 27.6k
5 votes

Details of Boltzmann distribution derivation

(1) You have to maximize $\Omega$ subject to the constraints that the total number of particles and the total energy is constant, which is more complicated than just taking the derivative and setting ...
5 votes

Lagrange Multipliers for the derivation of Maxwell-Boltzmann distribution

Ignoring multiplicative constants, the required integral is (in general form) $$ \int \sqrt{x} e^{ax} \mathrm{d} x = \frac{1}{a}\sqrt{x}e^{ax}+\frac{i\sqrt{\pi}}{2a^{3/2}} \mathrm{erf}(i\sqrt{ax})$$ ...
  • 19k
5 votes

How do we get g and E values for various levels to calculate electronic partition function

$E_n$ represents the energy of the $n$th electronic state relative to the ground state (i.e. the ground-state electronic energy $E_0 = 0$). Often, electronic states are very high in energy and the ...
  • 66.8k
5 votes
Accepted

Practical use of the partition function in molecular simulations

The actual partition function is unimaginably formidable. For just $N$ point particles in a 3D box, it's already got $3N$ dimensions. If the box is length $L$, GROMACS would probably divide the box ...
4 votes
Accepted

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

I think that your first probabalistic argument is correct, i.e that as the solute interaction increases the effective number of them is reduced. In the second case, as the interaction become repulsive,...
  • 27.6k
4 votes

What is the physical significance of molecular partition function?

We first come across the canonical partition function $Q(\beta) := \sum_i e^{-\beta E_i}$ as a normalization constant for the probability that a given microstate is occupied, as given by the Boltzmann ...
4 votes

What is the physical significance of molecular partition function?

In the simplest terms and most convenient definitions, it represents the total amount of states that the energy can be in. The probability expression you wrote emphasizes this. A probability is a ...
4 votes
Accepted

Populations of a two level system

I'm getting basically the same answer when I work out the math. I got $T=3398.45\ K$, but we probably just rounded the constants differently. This is a good lesson in how much thermal energy is ...
  • 12.7k

Only top scored, non community-wiki answers of a minimum length are eligible