# How Do Particles Leak from a Container and Reach a Spherical Cap?

• J_M_R
In summary, the Maxwell-Boltzmann Spherical Cap is a mathematical model that describes the distribution of velocities of particles in a gas or liquid at a specific temperature. It is calculated using a specific equation and shows the relative number of particles with different velocities. This model is significant in thermodynamics and is related to the ideal gas law through the root-mean-square speed.
J_M_R

## Homework Statement

An ideal gas satisfying the Maxwell-Boltzmann distribution is leaking from a container of the volume V through a circular hole of area A'. The gas is kept in the container under pressure P and temperature T. The initial number density (concentration) is given by n0=N/V.

Find the number of particles leaking from the container per unit of time and reaching a spherical cap with the radius R and the height h, h<R. The centre of the sphere is positioned exactly at the centre of the hole and the base of the cap is in the plane parallel to one of the container's wall.

## Homework Equations

Maxwell-Boltzmann Distribution: f(vi) = √(m/(2πkT))*(e^[(mvi^2)/(2kT)])

where vi stands for either vx, vy, or vz.

k - Boltzmann constant.

## The Attempt at a Solution

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When looking at the hole of the box, if the hole has length vxdt, then the portion of particles with velocity vx getting out of the hole (of area A') in time dt is: n0A'vxdt = dN(vx).
Because the volume covered by particles in a moment of time, dt = A'vxdt. Where no=N/V

I have completed the work with a disk instead of a spherical cap at a distance from the box, removing the y and z velocity components and replacing them with polar coordinates for velocity and think that to solve this problem spherical coordinates are required to replace the x, y and z velocity components associated with the cap.

I have got to this point below but do not know how to derive the limits of the triple integral:

N = no * A' * dt * (m/(2*pi*k*T))^(3/2) ∫ (v * d^3v * e^-(m*v^2/(2*k*T)))

The upper limit of integration is v2 and the lower limit is v1 which are 3-dimensional velocities.

where v^2 = vx^2 + vy^2 + vz^2

and d^3v = dvx dvy dvz = v^2 sinӨdӨdφ

I understand that I need to use a triple integral but I am unsure how to obtain the limits using spherical coordinates, if anybody is able to give me some hints that would be much appreciated.

Last edited:
There is a dv missing in the last equation.

If you plug in your expression for d3 into the integral, you get two integrals that are easy to solve and one where you have to integrate a function of the type of ##v^3 e^{-v^2}##, which can be integrated with standard methods.

## What is the Maxwell-Boltzmann Spherical Cap?

The Maxwell-Boltzmann Spherical Cap is a mathematical model that describes the distribution of velocities of particles in a gas or liquid at a specific temperature. It is named after James Clerk Maxwell and Ludwig Boltzmann, who developed the model in the 19th century.

## How is the Maxwell-Boltzmann Spherical Cap calculated?

The Maxwell-Boltzmann Spherical Cap is calculated using the equation: f(v) = 4πv²(e^(-mv²/2kT))/(2πkT)^(3/2), where f(v) is the fraction of particles with a velocity v, m is the mass of the particle, k is the Boltzmann constant, and T is the temperature.

## What does the Maxwell-Boltzmann Spherical Cap distribution show?

The Maxwell-Boltzmann Spherical Cap distribution shows the relative number of particles with different velocities in a gas or liquid at a specific temperature. It illustrates that the most probable velocity of particles is the average velocity, and the number of particles decreases as the velocity increases.

## What is the significance of the Maxwell-Boltzmann Spherical Cap in thermodynamics?

The Maxwell-Boltzmann Spherical Cap is significant in thermodynamics because it helps explain the relationship between temperature and the average kinetic energy of particles in a gas or liquid. It is also used to calculate various thermodynamic properties, such as the heat capacity and diffusion coefficient.

## How does the Maxwell-Boltzmann Spherical Cap relate to the ideal gas law?

The Maxwell-Boltzmann Spherical Cap is related to the ideal gas law through the root-mean-square speed, which is the most probable speed of particles in a gas. The root-mean-square speed is equal to the square root of (3RT/M), where R is the gas constant, T is the temperature, and M is the molar mass. This is similar to the equation for the most probable speed in the Maxwell-Boltzmann Spherical Cap distribution.