The kinetic theory of gases plays an important role in understanding the processes occurring in gas substances, which are closely related to the measured macroscopic thermodynamic characteristics of the system. In the article below, we will examine in more detail what Maxwell's law is and how it can be used to describe the behavior of ideal gases.
Ideal gases
We begin the article by revealing what ideal gas is. By this term is meant such a fluid substance that is capable of arbitrarily changing its shape and its volume. The particles that make up this substance are believed to be dimensionless and not interacting with each other. They are considered dimensionless because the distance between them is much larger than their linear dimensions. The absence of interaction between particles is understood to mean their insignificant potential energy in comparison with the energy of kinetic translational and rotational motion.
Ideal gases are characterized by low densities (pressures), high absolute temperatures and chemical inertness of their constituent molecules. A well-known example of ideal gas is air on our planet.
Kinetic theory and velocity of gas molecules
Molecular kinetic theory (MKT) is a powerful tool that, from the analysis of statistical functions, allows predicting the behavior of macroscopic system parameters such as temperature, pressure, and volume. The main provisions of the ILC are as follows:
- particles, whether atoms or molecules, move randomly along direct paths;
- the pressure created is due to constant elastic collisions of particles with the walls of the vessel containing gas;
- the temperature is determined by the average kinetic energy of the system.
From these postulates of MKT, as well as from the answer to the question of what is an ideal gas, an important conclusion can be made: the molecular velocity v is a key microscopic quantity that determines almost all the properties of the system under study.
For completeness, we note that the MKT was developed thanks to the work of Bernoulli, Lomonosov, Krenig and Clausius. In the second half of the 19th century, the success of this theory was due to the advent of Maxwell's statistical law on the distribution of velocities of gas particles. Subsequently, Maxwell's work was generalized by Boltzmann to the case of the particle energies of the system.
The probability density function f (v)
A feature of ideal gases in thermodynamic equilibrium is the variety of velocities of their constituent molecules. Indeed, if you find a way to measure individual speeds, you can find that some particles move very slowly, while others can move faster. This interesting fact was described in 1873 by Maxwell, introducing a special velocity distribution function of the molecules. This function is called the probability of density, it has the following form:
f (v) = (m / (2 * pi * k * T)) 3/2 * 4 * pi * v 2 * exp (-m * v 2 / (2 * k * T)).
In this expression, k, m, and T are the Boltzmann constant, particle mass, and absolute temperature of the system, respectively. The probability density f (v) shows the probability of detecting particles with velocities v ± dv in the system. If we take the integral of this function at all speeds, starting from zero and ending with infinity, then we get the total number of molecules N in the gas system under study.
Probability density function graph
The mathematical formula given in the paragraph above is somewhat cumbersome and contains a power-law and exponential function of temperature and particle mass. To better understand its properties, imagine this function in graphical form.
It can be seen that the Maxwell law leads to an asymmetric distribution of particle velocities. The curve on the graph breaks off relatively sharply at low speeds and gradually decreases for large values of v. The curve has one maximum, which can be interpreted as the most probable speed of molecules in the system.
Maxwell's law obeys Maxwell-Boltzmann statistics, the scope of which is limited only to classical (Newtonian) mechanics. Once particles begin to exhibit quantum properties, other statistical functions should be used to describe their behavior.
Probable mean and quadratic mean velocities
The velocity distribution of molecules, which describes the probability of density f (v), allows us to calculate three important types of velocity. The first of these is the most likely speed. It is it that has the largest number of molecules in the system. It is calculated from the fact that the derivative df (v) / dv is equal to zero. The probable speed v 1 is equal to:
v 1 = √ (2 * R * T / M).
That is, it will be the greater, the higher the temperature of the gas, and the smaller the mass value of its constituent particles. The value of R is here called the universal gas constant.
The next speed is the average. It is calculated by integrating the function v * f (v) at all speeds. The result is the average v 2 :
v 2 = √ (8 * R * T / (pi * M)).
Finally, the third important speed is the mean square value v 3 . It can be calculated by determining the integral of the function v 2 * f (v) over all molecular velocities. It is equal to:
v 3 = √ (3 * R * T / M).
The three formulas recorded show that the largest at any temperature and mass of molecules is the mean square value. It plays a key role in calculating the kinetic energy of a gas.
Velocity v3 and kinetic energy of molecules
Since particles do not interact with each other in ideal gases, their internal energy consists exclusively of kinetic energy. From classical mechanics it is known that the kinetic energy of one molecule, which moves with speed v and has mass m, is calculated by the formula:
E k = m * v 2/2.
Obviously, adding up the energies E k for all N molecules of the system, we obtain the total energy of an ideal gas. If you divide this energy by N, you get the average kinetic energy per molecule. In this case, the corresponding velocity will be the mean square value v 3 . For a monatomic gas having only three translational degrees of freedom, the following formula for the relationship of the quadratic velocity with temperature is valid:
m * v 3 2/2 = 3/2 * k * T.
If we recall the formula for the connection between the constants k and R, then we can obtain the previously written expression for v 3 through the absolute temperature.
Temperature dependence of function f (v)
It follows from Maxwell's law on the distribution of velocities that the probability of density f (v) depends on T. This dependence is most easily traced in the graph below.
Increasing the temperature from 100 K to 700 K for molecular nitrogen, we observe a shift of the peak of the probable velocity to the right. In addition, this peak becomes more blurry and decreases, which indicates the equalization of molecular velocities. Note that the areas under each of the curves remain the same, since the number of nitrogen molecules in the system remains constant.
Task example
Everyone knows colored rubber balls that fill with helium. It is necessary to calculate the probable, mean and quadratic mean velocities for helium atoms at room temperature (298.15 K).
The molar mass of the helium atom is 4 g / mol. Using this value, we obtain the speeds v 1 , v 2 and v 3 :
- v 1 = √ (2 * R * T / M) = √ (2 * 8.314 * 298.15 / 0.004) = 1113 m / s;
- v 2 = √ (8 * R * T / (pi * M)) = √ (8 * 8.314 * 298.15 / (3.14 * 0.004)) = 1257 m / s;
- v 3 = √ (3 * R * T / M) = √ (3 * 8.314 * 298.15 / 0.004) = 1363 m / s.
The calculated velocities indicate that helium atoms even at room temperature move 3.5–4 times faster than the speed of sound (≈340 m / s). Moreover, the average quadratic velocity of atoms is approximately 20% higher than their most probable velocity from the Maxwell distribution.