# Microstates for a gas of hard spheres in a box

• happyparticle
In summary, according to this model, the number of microstates of a gas of N spheres of radius r and volume v is N/X, where X is a small number.
happyparticle
Homework Statement
Find the number of microstates for a gas of N hard spheres of radius r and volume v in box taking into account the reduced volume after each sphere. V sphere << V box.
Find the state equation for the gas.
Relevant Equations
##\Omega = \frac{N!}{(N-n)!n!}##
##dE = Tds - Pdv##
Hi,

I have to found the number of microstates for a gas of N spheres of radius r and volume v in box taking into account the reduced volume after each sphere. V sphere << V box.

I'm struggling to find the microstates in general.
I don't see how to find the number of microstates without knowing the volume.After each hard sphere the free space decreases, so the pression increase and if the pression increase the temperature should increase as well and so the energy.

If this statement is right I don't know know how to write it mathematically to find ##\Omega##

Last edited by a moderator:
EpselonZero said:
I don't see how to find the number of microstates without knowing the volume.
You are given the volume of the box, ##V_{box}## and the radius ##r## of a hard sphere. What volume do you need that you cannot find?

kuruman said:
You are given the volume of the box, ##V_{box}## and the radius ##r## of a hard sphere. What volume do you need that you cannot find?
The volume is not given. The statement just say that the volume of the sphere is really really small compare to the volume of the box.

EpselonZero said:
The volume is not given. The statement just say that the volume of the sphere is really really small compare to the volume of the box.
Assume that the volume of the sphere is ##v_{\text{sphere}}## and the volume of the box is ##V_{\text{box}}##. These are implicitly given in the inequality ##v_{\text{sphere}}<<V_{\text{box}}.##

So in this case is N ##V_{box}## and n ##v_{spheres}##?

I confess I do not understand the question. There's no mention of energy, so I presume these are microstates distinguished by positions, not velocities. But isn't the number of positions for even a single sphere astronomical, and related to the Planck length?
Maybe we are supposed to consider the box as a collection of disjoint microboxes each able to take one sphere, but such would not pack into a simple connected volume.
Likely there is some standard model of which I am ignorant, but I could not find anything helpful on the net.

As regards given volumes, I am inclined to agree with the OP. The wording does not imply that ##V_{box}, V_{sphere}## are to be taken as given values. Only v and r are expressed that way. OTOH, it does seem that ##V_{box}## ought to be given.

Delta2
The idea behind this question is to find the equation of state for a gas modeled as hard spheres. Equations of state include the volume occupied by the gas so if it is not given, it is to be assumed as given. As for ##V_{\text{sphere}}##, it is essentially given because we are told that the radius of a hard sphere is ##r##.

I think it is reasonable then to say that the number of microstates ##\Omega## is the number of ways ##N## spheres can be placed in ##n=\dfrac{fV_{\text{box}}}{V_{\text{sphere}}}## available slots, where ##f## is the sphere reduction factor that one can look up (see post #7). Of course, these spheres are "hard" which means that a slot can have an occupancy of 1 or 0.

Tom.G
kuruman said:
The idea behind this question is to find the equation of state for a gas modeled as hard spheres. Equations of state include the volume occupied by the gas so if it is not given, it is to be assumed as given. As for ##V_{\text{sphere}}##, it is essentially given because we are told that the radius of a hard sphere is ##r##.

I think it is reasonable then to say that the number of microstates ##\Omega## is the number of ways ##N## spheres can be placed in ##n=\dfrac{fV_{\text{box}}}{V_{\text{sphere}}}## available slots, where ##f## is the sphere reduction factor that one can look up (see post #7). Of course, these spheres are "hard" which means that a slot can have an occupancy of 1 or 0.
I remain unconvinced that, in effect, treating the spheres as that amount larger fairly represents the behaviour. The added corners would disproportionately reduce the number of ways the spheres could be packed not-quite-optimally.
A test would be, how is the estimate of microstates translated into a prediction of some observable statistic?

Edit:
e.g., consider a simplified model, n discs radius r=4 ("Planck lengths") in a square box side L+2r=256+8. ##L^2=Xn(4r^2)##, where I suggest X is about 4; we want the total area of discs to be a significant fraction of the area available.
A program randomly positions disc centres within the LxL region, retrying one if it overlaps an existing disc. (Might need to increase X so that this does not take forever.) After placing all discs it counts how many are centred in the left half and starts again. It builds up statistics on these counts.
This can then be compared with the expected pattern from picking n of ##k\frac{L^2}{4r^2}## sites to find the right value of k.

4 might be a bit small for r to represent circles well.

An alternative starts with evenly spaced circles then let's them drift, but avoiding overlaps. Not sure how to detect when the drifting has continued long enough that the initial positions don't matter.

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Delta2

## 1. What is a microstate in the context of a gas of hard spheres in a box?

A microstate refers to the specific arrangement and velocities of the individual particles in a gas of hard spheres within a given box. It includes information about the positions and momenta of each particle and is used to describe the state of the system at a specific point in time.

## 2. How does the number of microstates relate to the entropy of the system?

The entropy of a system is directly proportional to the number of microstates it can occupy. This means that as the number of microstates increases, the entropy also increases, indicating a greater degree of disorder and randomness in the system.

## 3. Can the number of microstates for a gas of hard spheres in a box be calculated?

Yes, the number of microstates for a gas of hard spheres in a box can be calculated using the formula N! / (n1! * n2! * ... * nk!), where N is the total number of particles and n1, n2, etc. are the number of particles in each energy level.

## 4. How does the volume and number of particles affect the number of microstates in a gas of hard spheres in a box?

The number of microstates increases as the volume of the box and the number of particles in the gas increase. This is because a larger volume and more particles allow for more possible arrangements and velocities of the particles, resulting in a greater number of microstates.

## 5. How does the concept of microstates relate to the kinetic theory of gases?

The kinetic theory of gases explains the macroscopic properties of gases, such as pressure and temperature, based on the movement and collisions of individual particles. Microstates are used to describe the specific arrangements and velocities of these particles, providing a microscopic understanding of the behavior of gases in accordance with the kinetic theory.

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