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Maxwell-Boltzmann Distribution Derivation
Introduction
In thermodynamics and statistical mechanics, we often study gas molecules inside a container. In such a case, it is impossible to study each molecule separately. Indeed, even in a single mL of gas, there are almost an uncountable number of gas molecules. Thus, instead of working with individual molecules, we focus on them as a system.
When we work with systems, we describe them by picking a range of velocities and trying to figure out the possible number of molecules that are moving with that velocity. This is where Maxwell-Boltzmann Distribution comes in handy.
What Is Macrostate?
To understand this term, you should understand what macroscopic properties are. Macroscopic properties are those properties of a system that can be seen and observed by the naked eye. For instance, the temperature, pressure, and volume of a system are macroscopic variables.
Thus, macrostate refers to the state of the system in the context of these properties. For example, to say that a system is at a temperature of 300K is a statement about its macrostate. It doesn’t matter how the system attained this temperature, or what the temperatures of the individual molecules inside are. The system itself is at 300K, and that is it.
In statistical mechanics, a macrostate is a set of values of energy, number of particles, and volume that characterizes a system.
Another example you can take is that of a coin being tossed twice. You can have either of the following outcomes:
Head-Head
Head-Tail
Tail-Head
Tail-Tail
One could then say that one head and one tail is a macrostate. It doesn’t matter which time the coin landed on the head and which time on the tail; just that there was one head and one tail.
What Is Microstate?
In contrast to a macrostate, the microstate refers to how the system is organized in terms of its microscopic properties. For example, to achieve a bottle of water at 300K, you could mix equal amounts of hot and cold water at 290K and 310K, respectively. Or you could mix different amounts of water at different temperatures to get the same outcome. Each of the different ways of achieving a particular macrostate would then be termed a microstate.
In the example of the double coin toss, one head and one tail is a macrostate, characterized by two possible microstates: either H-T or T-H. In statistical mechanics, a microstate is one of the many possible ways in which, a particular system can achieve a particular macrostate.
Maxwell Boltzmann Distribution Law
When you take a container filled with any gas, the gas molecules aren’t stationary inside it. Instead, they are randomly moving about in different directions with varying speeds. Indeed, it is this movement that gives rise to the pressure of a gas. Since the exact speed of each particle is impossible to describe, we use what is known as the Maxwell- Boltzmann Distribution Law.
Simply put, this law describes the fraction of particles that have velocities in the range of v to v + dv. Mathematically, this law is given by −
$$\mathrm{F\:(v)\:d^{3}v\:=\:(\frac{m}{2\Pi\:KT})^{3/2}\:e^{-\:\frac{mv^{2}}{2KT}}\:d^{3}\:v}$$
Here, 𝒇(𝒗) 𝒅𝟑𝒗 represents the fraction of particles whose velocities are in the range of v to v + dv.
Derivation of Maxwell Distribution Law
Consider a system of n particles in a container of volume V. Let the possible energy levels be represented by 𝜖1, 𝜖2, … , 𝜖𝑟 . The number of particles with energy 𝜖𝑖 is represented by 𝑛𝑖.
Then, the number of ways to attain a particular microstate is given using the following formula −
$$\mathrm{W\:=\:\frac{n!}{n_{1}!\times\:n_{2}!\times\:n_{3}!.......\:n_{r}!}}$$
This formula can prove tedious to work with and we take its natural logarithm to remove the fraction. Further, we apply Stirling’s approximation. Then,
$$\mathrm{ln\:ln\:W\:=\:ln\:ln\:n!\:-\:ln\:ln\:({n_{1}!\times\:n_{2}!\times\:n_{3}!.......\:n_{r}!})\:ln\:ln\:(n!)\:=\:n\:ln\:ln\:(n)-\:n\:ln\:ln\:W\:=\:n\:ln\:ln\:(n)\:-\:n\:-\:\displaystyle\sum\limits_{i=0}^r (n_{1}\:ln\:ln\:(n)\:-\:n_{i}}$$
We need to maximize the value of W. To do that, we differentiate the above equation and equate it to zero. Since n is a constant, we get −
$\mathrm{\partial\:ln\:ln\:W\:=\:-\:\displaystyle\sum\limits_{i=0}^r (\partial\:n_{i}\:ln\:ln\:(n_{i}))+n_{i}\times\:\frac{1}{n_{i}}\:-\:\partial\:n_{i}\:=\:0}$ Thus,$\mathrm{\displaystyle\sum\limits_{i=0}^r (\partial\:n_{i}\:ln\:ln\:(n_{i}))\:=\:0}$
Now, with n = constant, the sum $\mathrm{\displaystyle\sum\limits^r\:n_{i}\:=\:n}$ Thus,
$$\mathrm{\displaystyle\sum\limits_{i}^r\partial\:n_{i}\:=\:0}$$
Also, since the total energy is constant,
$$\mathrm{\displaystyle\sum\limits_{i}^r\varepsilon\:_{i}\partial\:n_{i}\:=\:0}$$
We now use Lagrange’s method of undetermined multipliers to get
$$\mathrm{\displaystyle\sum\limits_{i}^r(Ln\:ln\:n_{i}\:+\:\alpha\:+\:\beta\:\varepsilon\:_{i})\:\partial\:n_{i}\:=\:0}$$
The term in the bracket must be individually zero for the above equation to be valid. This only holds when
$$\mathrm{n_{i}\:=\:e^{-\alpha}\:\times\:e^{-\beta\:\varepsilon_{i}}}$$
This is similar in form to Maxwell-Boltzmann Distribution given above since energy is directly proportional to velocity squared. The value of the constants is a lengthier derivation that cannot fit inside a single article.
Applications
Calculating average, most probable, and root mean squared speeds.
Using Maxwell-Boltzmann Distribution, it is possible to derive the values for the average speed of particles, the speed that is most likely to be possessed by the molecules, and the root mean squared speed of the molecules. These are important parameters in describing a system.
Calculating Energies
The mean energy associated with the system as well as the average energy per particle can be calculated from the distribution function we have derived. Indeed, the results turn out to be in excellent agreement with derivations performed via other means.
Conclusion
The macrostate of a system refers to its configuration in terms of its macroscopic properties like temperature and pressure etc. Statistically, it is a set of energies, number of particles, and the volume of the system.
he Maxwell-Boltzmann Distribution can give us the fraction of molecules that have energies in the range of v to v + dv. It was given by James Clerk Maxwell on heuristic grounds, while Boltzmann later worked on its physical interpretations. The distribution function is given as follows −
$$\mathrm{F\:(v)\:d^{3}v\:=\:(\frac{m}{2\Pi\:KT})^{3/2}\:e^{-\:\frac{mv^{2}}{2KT}}\:d^{3}\:v}$$
FAQs
1. I have seen multiple names like Maxwell distribution law, Maxwell-Boltzmann Distribution, Maxwellian Distribution, etc. Are they the same or is one particular name correct?
All these terms refer to the same law: the Maxwell-Boltzmann Distribution Law. There is no penalty for using one particular name over another. Since Boltzmann also contributed significantly to this law, it is most commonly referred to with the complete name
2. Can Maxwell Distribution be given in terms of parameters other than velocity?
Yes. A gas molecule’s velocity is responsible for its energy and pressure. Sometimes, the distribution function is given in terms of energy instead of velocity. It does not change the nature of the law.
3. Is Maxwell Distribution quantum mechanically valid?
Maxwell-Boltzmann Distribution holds only when quantum effects aren’t significant
4. Are there multiple ways to derive Maxwell Distribution?
Yes. There is no rigorous proof for deriving this law. There are multiple ways to derive it, and each has its own merits and demerits. One can even derive it without talking about microstates and macrostates.
5. What does the graph of Maxwell Distribution look like?
The graph has a peak at one particular value, while it decays off on either side of this value. The graph changes with temperature.
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