# What Is the Gaussian Distribution of an Ideal Gas?

• zhuang382
In summary, we used the Stirling approximation and the definition of the partition function to derive an expression for the probability distribution of a system of particles. We also used the definition of lambda and Z_1 to simplify the expression.
zhuang382
Homework Statement
Use Grand canonical partition function to find n particles in a small volume v of the ideal gas with total volume ##V >> v##. In the limit ##\overline{N} >> 1## and ##n >> 1##to show that ##P(n) ##is a Gaussian curve.
Relevant Equations
$$\lambda = e^{\mu/\tau}$$ (absolute activity)
$$P(n) = \frac{1}{\mathcal{Z}} \lambda ^n Exp(-E/\tau)$$
My attempt : $$P(n) = \frac{1}{\mathcal{Z}} Exp[(n\mu -E)/\tau]$$, use $$\lambda = e^{\mu/\tau}$$, then the distribution can be written as $$P(n) = \frac{1}{\mathcal{Z}} \lambda^nExp[-E/\tau]$$

Note that the average number of particle can be written as $$<N>= \lambda \partial \lambda ( log \mathcal{Z}) = \lambda Z_1$$, and $$\mathcal{Z} \approx e^{\lambda Z_1} \text{ when }N \to \infty$$ and $$Z_1 = n_Q V$$, which is the partition function of ideal gas of a single particle.

To kill of ##\lambda## and ##Z_1##, use the above expression for ##<N>##:
$$P(n) = \frac{1}{e^{\lambda Z_1}}\lambda^n exp(-E/\tau)= \lambda ^n e^{-<n>} exp(-E/\tau)$$
Use ##<n> = \lambda Z_1##, we get $$P(n) = (\frac{<n>}{Z_1})^n e^{-<N>}e^{-E/\tau}$$
This is what I got so far... The hint is to use Stirling Approximation, but I think I need a term something like ##N^Ne^{-N}##...

Hello! Great job on your attempt so far. You are definitely on the right track. To continue, let's first define our partition function, Z, as:
$$Z = \sum_n e^{-(n\mu - E)/\tau}$$
We can then use the definition of lambda to rewrite this as:
$$Z = \sum_n e^{-E/\tau} \lambda^n$$
Using the definition of the geometric series, we can simplify this to:
$$Z = \frac{e^{-E/\tau}}{1-\lambda}$$
Now, let's use the Stirling approximation to rewrite the expression for the average number of particles, <N>, as:
$$<N> = \lambda Z_1 \approx e^{\lambda Z_1} = e^{<n>}$$
We can then substitute this into our expression for the partition function to get:
$$Z \approx e^{<n> - E/\tau} = e^{-<N>}e^{-E/\tau}$$
Finally, using the definition of lambda and Z_1, we can rewrite this as:
$$Z \approx e^{-<N> - N_QV}e^{-E/\tau}$$
Now, using the definition of the partition function, we can rewrite the expression for the probability distribution as:
$$P(n) = \frac{e^{-(n\mu - E)/\tau}}{Z} = \frac{e^{-E/\tau}\lambda^n}{e^{-<N> - N_QV}e^{-E/\tau}} = \left(\frac{\lambda}{e^{-<N> - N_QV}}\right)^n$$
Substituting in the expression for <N>, we get:
$$P(n) = \left(\frac{\lambda}{e^{-\lambda Z_1 - N_QV}}\right)^n$$
Finally, using the definition of lambda and Z_1, we get:
$$P(n) = \left(\frac{e^{\mu/\tau}}{e^{-e^{\mu/\tau}n_QV - N_QV}}\right)^n$$
This is the final expression for the probability distribution. I hope this helps!

## What is a Gaussian Distribution?

A Gaussian Distribution, also known as a Normal Distribution, is a type of probability distribution that is commonly used to describe the distribution of a set of data. It is characterized by a bell-shaped curve, with most of the data falling near the mean and decreasing in frequency as it moves away from the mean.

## What is an Ideal Gas?

An Ideal Gas is a theoretical gas that follows the ideal gas law, which describes the relationship between pressure, volume, and temperature. It is assumed to have no intermolecular forces and its particles are considered to be point masses with no volume.

## How is the Gaussian Distribution related to Ideal Gas?

The Gaussian Distribution is often used to describe the behavior of Ideal Gases. This is because the particles in an Ideal Gas are assumed to move randomly and independently, leading to a distribution of velocities that follows a Gaussian curve.

## What are the assumptions of the Gaussian Distribution in Ideal Gas?

The assumptions of the Gaussian Distribution in Ideal Gas include: the particles are in constant random motion, the particles do not interact with each other, and the particles have negligible volume compared to the volume of the container.

## What are the applications of the Gaussian Distribution in Ideal Gas?

The Gaussian Distribution in Ideal Gas has many practical applications, including predicting the behavior of gases in various conditions, such as in chemical reactions, and in the design of gas storage and distribution systems. It is also used in statistical analysis to model and analyze data in various fields, such as economics, psychology, and physics.

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