A gas is contained in a harmonic potential well V(r)=αr^2, where αis a constant and r is the dis

step 1 Hello and welcome to this video solution of numerate. Here the equation of an ideal gas is given

A gas is contained in a harmonic potential well V(r)=αr^2,...

A gas is contained in a harmonic potential well $V(r)=\alpha r^2$, where $\alpha$ is a constant and $r$ is the distance from the centre of the well. Show that under isothermal conditions, the number density profile is $n=n_0$ $\exp \left(-\alpha r^2 / k_{\mathrm{B}} T\right)$

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Key Concepts

Isothermal Equilibrium

Isothermal conditions imply that the system is maintained at a constant temperature throughout its volume. This uniform temperature ensures that the thermal energy k_B T is constant, allowing the use of the Boltzmann factor uniformly across the system to describe how the external potential affects the local particle density.

Boltzmann Distribution

The Boltzmann distribution describes how particles in a system at thermal equilibrium are distributed over various energy states. In an external potential, the probability of finding a particle at a certain position is proportional to exp(-V/(k_B T)), where V is the potential energy, k_B is the Boltzmann constant, and T is the temperature.

Harmonic Potential Well

A harmonic potential is a quadratic potential of the form V(r) = ?r², which is characteristic of systems where restoring forces are proportional to the displacement. This type of potential well is fundamental in physics as it approximates the behavior of many systems near their equilibrium state, and it simplifies the analysis of particle distribution in external fields.

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