The study of the properties of the gas aggregate state of matter is one of the important areas of modern physics. Considering gases on a microscopic scale, one can obtain all the macroscopic parameters of the system. This article will reveal an important issue of the molecular kinetic theory of gases: what is the Maxwell distribution of molecules in terms of velocities.
Historical background
The idea of gas as a system consisting of microscopic moving particles originated in ancient Greece. It took more than 1700 years for science to develop it.
The founder of the modern molecular-kinetic theory (MKT) of gas is fair to consider Daniil Bernoulli. In 1738 he published a work called "Hydrodynamics". In it, Bernoulli outlined the ideas of the MKT that have been used to this day. So, the scientist believed that gases consist of particles that move randomly in all directions. Numerous collisionsparticles with vessel walls are perceived as the presence of pressure in gases. Particle velocities are closely related to the temperature of the system. The scientific community did not accept Bernoulli's bold ideas because the law of conservation of energy had not yet been established.
Subsequently, many scientists were engaged in building a kinetic model of gases. Among them, Rudolf Clausius should be noted, who in 1857 created a simple gas model. In it, the scientist paid special attention to the presence of translational, rotational and vibrational degrees of freedom in molecules.
In 1859, studying the work of Clausius, James Maxwell formulated the so-called Maxwell distribution over molecular velocities. In fact, Maxwell confirmed the ideas of the MKT, backing them up with a mathematical apparatus. Subsequently, Ludwig Boltzmann (1871) generalized the conclusions of the Maxwell distribution. He postulated a more general statistical distribution of molecules over velocities and energies. It is currently known as the Maxwell-Boltzmann distribution.
Ideal gas. Basic postulates of the ILC
To understand what the Maxwell distribution function is, you need to clearly understand the systems for which this function is applicable. We are talking about an ideal gas. In physics, this concept is understood as a fluid substance, which consists of practically dimensionless particles that do not have potential energy. These particles move at high speeds, so their behavior is completely determined by kinetic energy. Moreover, the distances between the particles are too large forcompared to their sizes, so the latter are neglected.
Ideal gases are described within the MKT. Its main postulates are as follows:
- gas systems are made up of a huge number of free particles;
- particles randomly move at different speeds in different directions along straight trajectories;
- particles collide with vessel walls elastically (the probability of particles colliding with each other is low due to their small size);
- The temperature of the system is uniquely determined by the average kinetic energy of the particles, which is preserved in time if thermodynamic equilibrium is established in the system.
Maxwell's distribution law
If a person had an instrument with which it was possible to measure the speed of a single gas molecule, then, after conducting an appropriate experiment, he would be surprised. The experiment would show that every molecule of any gaseous system moves at a completely arbitrary speed. In this case, within the framework of one system in thermal equilibrium with the environment, both very slow and very fast molecules would be detected.
Maxwell's law of velocity distribution of gas molecules is a tool that allows you to determine the probability of detecting particles with a given velocity v in the system under study. The corresponding function looks like this:
f(v)=(m/(2pikT))3/24piv2 exp(-mv2/(2kT)).
In this expression, m -particle (molecule) mass, k - Boltzmann's constant, T - absolute temperature. Thus, if the chemical nature of the particles (the value of m) is known, then the function f(v) is uniquely determined by the absolute temperature. The function f(v) is called the probability density. If we take the integral from it for some speed limit (v; v+dv), then we get the number of particles Ni, which have speeds in the specified interval. Accordingly, if we take the integral of the probability density f(v) for the velocity limits from 0 to ∞, then we get the total number of molecules N in the system.
Graphic representation of the probability density f(v)
The probability density function has a somewhat complex mathematical form, so it is not easy to represent its behavior at a given temperature. This problem can be solved if you depict it on a two-dimensional graph. A schematic view of the Maxwell distribution graph is shown below in the figure.
We see that it starts from zero, since the velocity v of molecules cannot have negative values. The graph ends somewhere in the region of high speeds, smoothly falling to zero (f(∞)->0). The following feature is also striking: the smooth curve is asymmetric, it decreases more sharply for small speeds.
An important feature of the behavior of the probability density function f(v) is the presence of one pronounced maximum on it. According to the physical meaning of the function, this maximum corresponds to the most probable value of the velocities of molecules in the gassystem.
Important speeds for the function f(v)
The probability density function f(v) and its graphic representation allow us to define three important types of speed.
The first kind of speed that is obvious and that was mentioned above is the most likely speed v1. On the graph, its value corresponds to the maximum of the function f(v). It is this speed and values close to it that will have most of the particles of the system. It is not difficult to calculate it, for this it is enough to take the first derivative with respect to the speed of the function f(v) and equate it to zero. As a result of these mathematical operations, we get the final result:
v1=√(2RT/M).
Here R is the universal gas constant, M is the molar mass of molecules.
The second kind of speed is its average value for all N particles. Let's denote it v2. It can be calculated by integrating the function vf(v) over all velocities. The result of the noted integration will be the following formula:
v2=√(8RT/(piM)).
Because the ratio is 8/pi>2, the average speed is always slightly higher than the most probable one.
Every person who knows a little about physics understands that the average speed v2 of molecules must be of great importance in a gas system. However, this is an erroneous assumption. Much more important is the RMS speed. Let's denote itv3.
According to the definition, root-mean-square velocity is the sum of the squares of the individual velocities of all particles, divided by the number of these particles, and taken under the square root. It can be calculated for the Maxwell distribution if we define the integral over all velocities of the function v2f(v). The formula for the average quadratic speed will take the form:
v3=√(3RT/M).
Equality shows that this speed is greater than v2 and v1 for any gas system.
Thus, all considered types of velocities on the Maxwell distribution graph lie either on the extremum or to the right of it.
Importance of v3
It was noted above that the mean square velocity is more important for understanding the physical processes and properties of the gas system than the simple average velocity v2. This is true, since the kinetic energy of an ideal gas depends precisely on v3, and not on v2.
If we consider a monatomic ideal gas, then the following expression is valid for it:
mv32/2=3/2kT.
Here, each part of the equation represents the kinetic energy of one particle of mass m. Why does the expression contain exactly the value v3, and not the average speed v2? Very simple: when determining the kinetic energy of each particle, its individual velocity v is squared, then all velocitiesare added and divided by the number of particles N. That is, the procedure for determining the kinetic energy itself leads to the value of the mean square velocity.
Dependence of function f(v) on temperature
We have established above that the probability density of molecular velocities uniquely depends on temperature. How will the function change if T is increased or decreased? The chart below will help answer this question.
It can be seen that the heating of the closed system leads to smearing of the peak and its shift towards higher speeds. An increase in temperature leads to an increase in all types of velocities and to a decrease in the probability density of each of them. The peak value decreases due to the conservation of the number of particles N in a closed system.
Next, we will solve a couple of problems to consolidate the received theoretical material.
Problem with nitrogen molecules in the air
It is necessary to calculate the speeds v1, v2 and v3 for air nitrogen at temperature of 300 K (about 27 oC).
The molar mass of nitrogen N2 is 28 g/mol. Using the above formulas, we get:
v1=√(2RT/M)=√(28, 314300/0, 028)=422 m/s;
v2=√(8RT/(piM))=√(88, 314300/(3, 140, 028))=476 m/s;
v3=√(3RT/M)=√(38, 314300/0, 028)=517 m/s.
Oxygen tank problem
The oxygen in the cylinder was at a certain temperature T1. Then the balloon was placed in a colder room. How will the Maxwell velocity distribution plot for oxygen molecules change when the system comes to thermodynamic equilibrium?
Remembering the theory, we can answer the question of the problem in this way: the values of all types of velocities of molecules will decrease, the peak of the function f(v) will shift to the left, become narrower and higher.
