The study of the properties of the gas state of aggregation of matter is one of the important areas of modern physics. Examining gases on a microscopic scale, all macroscopic parameters of the system can be obtained. In this article, an important question of the molecular-kinetic theory of gases will be revealed: what is the Maxwell distribution of molecules in velocity.
Historical essay
The idea of gas as a system consisting of microscopic moving particles originated in ancient Greece. It took science more than 1700 years to get its development.
It is fair to consider Daniel Bernoulli the founder of the modern molecular kinetic theory (MKT) of gas. In 1738, he published a work called Hydrodynamics. In it, Bernoulli outlined the ideas of the ILC used to date. So, the scientist believed that gases consist of particles that randomly move in all directions. Numerous collisions of particles with the walls of blood vessels are perceived as the presence of pressure in gases. Particle velocities are closely related to the temperature of the system. The community of scientists did not accept Bernoulli's bold ideas, since the law of conservation of energy had not yet been established.
Subsequently, many scientists were engaged in the construction of a kinetic model of gases. Among them, Rudolf Clausius, who in 1857 created a simple gas model, should be noted. In it, the scientist paid special attention to the presence of translational, rotational and vibrational degrees of freedom for molecules.
In 1859, studying the work of Clausius, James Maxwell formulated the so-called Maxwell distribution of molecular velocities. In fact, Maxwell confirmed the ideas of the MKT, reinforcing them with a mathematical apparatus. Subsequently, Ludwig Boltzmann (1871) summarized the conclusions of the Maxwell distribution. He postulated a more general statistical distribution of molecules in speed and energy. It is currently known as the Maxwell-Boltzmann distribution.
Perfect gas. The main postulates of MKT
To understand what the Maxwell distribution function is, it is necessary to clearly understand the systems for which this function is applicable. It's about perfect 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 the kinetic energy. Moreover, the distances between the particles are too large in comparison with their sizes; therefore, the latter are neglected.
Ideal gases are described in the framework of the MKT. Its main postulates are the following:
- gas systems consist of a huge number of free particles;
- particles randomly move at different speeds in different directions along straight paths;
- particles collide with the walls of the vessels elastically (the probability of collision of particles with each other is small, due to their small size);
- the temperature of the system is uniquely determined by the average kinetic energy of the particles, which is stored in time in the event of the establishment of thermodynamic equilibrium in the system.
Maxwell's distribution law
If a person possessed a device with which it was possible to measure the speed of an individual gas molecule, then, having conducted the corresponding experiment, he would be surprised. An experiment would show that every molecule of any gas system moves with a completely arbitrary speed. In this case, within the framework of a single system in thermal equilibrium with the environment, both very slow and very fast molecules would be detected.
The Maxwell distribution law for the velocities of gas molecules is a tool that allows you to determine the probability of detecting particles with a given speed v in the system under study. The corresponding function is as follows:
f (v) = (m / (2 * pi * k * T)) 3/2 * 4 * pi * v 2 * exp (-m * v 2 / (2 * k * T)).
In this expression, m is the mass of the particle (molecule), k is the Boltzmann constant, T is the 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 a certain speed limit (v; v + dv), then we get the number of particles N i that have velocities in the indicated interval. Accordingly, if we take the integral of the probability density f (v) for the speed limits from 0 to ∞, then we get the total number of molecules N in the system.
Graphic representation of probability density f (v)
The probability density function has a somewhat complex mathematical form, so it is not easy to imagine its behavior at a given temperature. This problem can be solved by depicting it on a two-dimensional graph. A schematic diagram of the Maxwell distribution graph is shown in the figure below.
We see that it starts from zero, since the velocity v of the molecules cannot have negative values. The graph ends somewhere in the region of high speeds, gradually falling to zero (f (∞) -> 0). The following feature is also striking: a smooth curve is asymmetric, it decreases more sharply for small values of speeds.
An important feature of the behavior of the probability density function f (v) is the presence of one expressed maximum on it. According to the physical meaning of the function, this maximum corresponds to the most probable value of the velocities of the molecules in the gas system.
Important types of velocity for the function f (v)
The probability density function f (v) and its graphic image allow us to define three important types of velocity.
The first type of speed, which is obvious, and which was mentioned above, is the most probable speed v 1 . On the graph, its value corresponds to the maximum of the function f (v). It is this velocity and the values close to it that the majority of the particles of the system will have. It is not difficult to calculate it, for this it is enough to take the first derivative with respect to speed of the function f (v) and equate it to zero. As a result of these mathematical operations, we obtain the final result:
v 1 = √ (2 * R * T / M).
Here R is the universal gas constant, M is the molar mass of molecules.
The second type of velocity is its average value for all N particles. Denote it by v 2 . It can be calculated if the function v * f (v) is integrated over all velocities. The result of the noted integration will be the following formula:
v 2 = √ (8 * R * T / (pi * M)).
Since the ratio is 8 / pi> 2, the average speed is always somewhat higher than the most probable.
Every person who is a little familiar with physics understands that the average speed of v 2 molecules should be of great importance in the gas system. Nevertheless, this is an erroneous judgment. Much more important is the root mean square velocity. Denote it by v 3 .
By definition, the 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 as the square root. It can be calculated for the Maxwell distribution if the integral over all velocities of the function v 2 * f (v) is determined. The formula for the average quadratic velocity will take the form:
v 3 = √ (3 * R * T / M).
Equality shows that this speed is greater than the values of v 2 and v 1 for any gas system.
Thus, all the considered types of velocities on the Maxwell distribution graph lie either at the extremum or to its right.
The importance of v3
It was noted above that the mean square velocity is more important for understanding the physical processes and properties of a gas system than the simple average velocity v 2 . This is indeed so, since the kinetic energy of an ideal gas depends precisely on the value of v 3 , and not on v 2 .
If we consider a monatomic ideal gas, then the following expression holds for it:
m * v 3 2/2 = 3/2 * k * T.
Here, each part of the equality represents the kinetic energy of one particle of mass m. Why is the value v 3 rather than the average speed v 2 in the expression? It is very simple: when determining the kinetic energy of each particle, its individual velocity v is squared, then all speeds are added up and divided by the number of particles N. That is, the procedure for determining the kinetic energy itself leads to the mean square velocity.
Dependence of the function f (v) on temperature
We have established above that the probability density of the velocities of molecules uniquely depends on temperature. How will the function change if we increase or decrease the value of T? The chart below will help answer this question.
It can be seen that heating 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 solve a couple of problems to consolidate the obtained theoretical material.
The problem with nitrogen molecules in the air
It is necessary to calculate the speeds v 1 , v 2 and v 3 for air nitrogen at a temperature of 300 K (about 27 o C).
The molar mass of nitrogen N 2 is 28 g / mol. Using the above formulas, we obtain:
v 1 = √ (2 * R * T / M) = √ (2 * 8.314 * 300 / 0.028) = 422 m / s;
v 2 = √ (8 * R * T / (pi * M)) = √ (8 * 8.314 * 300 / (3.14 * 0.028)) = 476 m / s;
v 3 = √ (3 * R * T / M) = √ (3 * 8.314 * 300 / 0.028) = 517 m / s.
The task with oxygen in the tank
Oxygen in the cylinder was at a certain temperature T 1 . Then the balloon was placed in a cooler room. How will the graph of the Maxwell velocity distribution for oxygen molecules change when the system comes into thermodynamic equilibrium?
Remembering the theory, one can answer the question of the problem: the values of all kinds of molecular velocities will decrease, the peak of the function f (v) will shift to the left, become narrower and higher.