# Equipartition Theorem and Microscopic Motion

• ghoultree
In summary: Thus, the correct formula is: frot=\sqrt{{4kT}/{md^2}}/(2\pi)In summary, to find the typical rotational frequency frot for a molecule like N2 at room temperature, we use the formula frot=\sqrt{{4kT}/{md^2}}/(2\pi), where k=1.381x10-23 J/(molecule*K), T=25°C, and mN2=4.65x10-26kg. We must also account for rotations around two axes, so we use 4kT instead of 2kT in the formula.
ghoultree

## Homework Statement

What is the typical rotational frequency frot for a molecule like N2 at room temperature (25°C)? Assume that d for this molecule is 10-10m. Take the atomic mass of N2 to be mN2=4.65x10-26kg. You will need to account for rotations around two axes (not just one) to find the correct frequency.

## Homework Equations

rms angular speed: ω=sqrt{(2kT)/(md2)}
frot=ω/(2$\pi$)
k=1.381x10-23 J/(molecule*K)

## The Attempt at a Solution

For my first attempt, I plugged the given numbers into sqrt{(3kT)/(md2)}. I used 3 instead of 2 because I was accounting for the two different axes. I'm not sure if the was the correct way to go about it. Secondly, I divided the given mass by 2. Then, I divided my answer by 2pi.

The formula you have is derived from equating $\frac 1 2 kT$ to rotational energy, which is $\frac {J \omega^2} {2}$, where $J = \frac {md^2} {2}$ is the moment of inertia of two masses m on a weightless rod d. But you have two degrees of freedom, so you must equate $kT$. That means instead $2kT$ you have in the formula you must take $4kT$, not $3kT$.

As follows from the above, you should not divide m by two, because you are already given the atomic mass.

## 1. What is the Equipartition Theorem?

The Equipartition Theorem is a principle in statistical mechanics that states that in thermal equilibrium, energy is distributed equally among all degrees of freedom of a system. This means that each degree of freedom, such as translational, rotational, and vibrational motion, will have an equal average energy.

## 2. How does the Equipartition Theorem relate to microscopic motion?

The Equipartition Theorem applies to the microscopic motion of particles within a system. It states that the average kinetic energy of each particle is directly proportional to the temperature of the system and the number of degrees of freedom of the particle.

## 3. What is the significance of the Equipartition Theorem in thermodynamics?

The Equipartition Theorem is an important concept in thermodynamics because it helps explain the distribution of energy and the behavior of particles in a system at thermal equilibrium. It also allows us to predict the average energy of a particle based on its degrees of freedom and temperature.

## 4. How does the Equipartition Theorem apply to real-world systems?

The Equipartition Theorem is a fundamental principle that applies to all systems in thermal equilibrium, including real-world systems. It can be used to analyze the behavior of gases, liquids, and solids at different temperatures and understand their energy distribution.

## 5. Are there any limitations to the Equipartition Theorem?

While the Equipartition Theorem is a useful tool in statistical mechanics, it does have some limitations. It assumes that all degrees of freedom have the same average energy, which may not always be the case in complex systems. It also does not account for quantum effects, which become important at low temperatures.

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