The study of processes occurring in statistical systems is complicated by the minimum particle size and their huge number. It is practically impossible to consider each particle separately, therefore statistical quantities are introduced: average particle velocity, particle concentration, particle mass. The formula characterizing the state of the system, taking into account microscopic parameters, is called the basic equation of the molecular kinetic theory of gases (MKT).
A bit about the average particle velocity
The determination of the particle velocity was first experimentally carried out. The experience known from the school curriculum conducted by Otto Stern allowed us to create an idea of particle velocities. During the experiment, the motion of silver atoms in rotating cylinders was studied: first, in the stationary state of the apparatus, then when it was rotated at a certain angular velocity.
As a result, it was found that the speed of silver molecules exceeds the value of the speed of sound and is 500 m / s. The fact is quite interesting, since it is difficult for a person to feel such speeds of particles in substances.
Perfect gas
It seems possible to continue the study only in a system whose parameters can be determined by direct measurements using physical instruments. Speed is measured with a speedometer, but the idea of attaching a speedometer to a separate particle is absurd. Only a macroscopic parameter related to particle motion can be directly measured.
Consider the gas pressure. Pressure on the walls of the vessel is created by the impact of the molecules of the gas in the vessel. A peculiarity of the gaseous state of matter is in sufficiently large distances between particles and their small interaction with each other. This allows you to directly measure its pressure.
Any system of interacting bodies is characterized by potential energy and kinetic energy of motion. Real gas is a complex system. The variability of potential energy cannot be systematized. The problem can be solved by introducing a model that carries the characteristic properties of a gas, which notes the complexity of the interaction.
An ideal gas is a state of matter in which the interaction of particles is negligible, the potential interaction energy tends to zero. Significant can be considered only the energy of motion, depending on the speed of the particles.
Ideal gas pressure
To reveal the relationship between gas pressure and the velocity of its particles allows the basic equation of the MKT ideal gas. A particle moving in a vessel, upon impact with a wall, gives it an impulse, the value of which can be determined on the basis of Newton’s Law II:
A change in the particle momentum during elastic impact is associated with a change in the horizontal component of its velocity. F is the force acting from the side of the particle on the wall for a short time t; m 0 Is the mass of the particle.
A surface of area S over time Δt is collided by all gas particles moving in the direction of the surface at a speed v x and located in a cylinder of volume Sυ x Δt. At a particle concentration of n, exactly half of the molecules move toward the wall, the other half in the opposite direction.
Having considered the collision of all particles, we can write down Newton’s law for the force acting on the site:
Since the gas pressure is defined as the ratio of the force acting perpendicular to the surface to the area of the latter, we can write:
The resulting relation, as the basic equation of the MKT, cannot describe the entire system, since the movement in only one direction is considered.
Maxwell distribution
The incessant frequent collisions of gas particles with the walls and with each other lead to the establishment of a certain statistical distribution of particles by velocities (energies). The directions of all velocity vectors are equally probable. This distribution is called the Maxwell distribution. In 1860, this pattern was deduced by J. Maxwell on the basis of the MKT. The main parameters of the distribution law are called velocities: the probable, corresponding to the maximum value of the curve, and the rms vq = √ ‹v 2 › is the mean square of the particle velocity.
An increase in gas temperature corresponds to an increase in velocity.
Based on the fact that all speeds are equal, and their modules have the same value, we can assume:
- ‹V 2 › = ‹v x 2 › + ‹v y 2 › + ‹v z 2 ›, whence: ‹v x 2 › = ‹v 2 ›: 3
The basic MKT equation, taking into account the average value of the gas pressure, has the form:
This ratio is unique in that it determines the relationship between microscopic parameters: speed, particle mass, particle concentration and gas pressure in general.
Using the concept of kinetic energy of particles, the basic equation of MKT can be rewritten differently:
- p = 2nm 0 ‹v 2 ›: 6 = 2n ‹E to ›: 3
The gas pressure is proportional to the average value of the kinetic energy of its particles.
Temperature
Interestingly, for a constant amount of gas in a closed vessel, the gas pressure and the average value of the particle motion energy can be related. The pressure measurement can be done by measuring the energy of the particles.
How to proceed? What value can be compared with kinetic energy? This value is the temperature.
Temperature is a measure of the thermal state of substances. To measure it, a thermometer is used, the basis of which is the
thermal expansion of the working fluid (alcohol, mercury) when heated. The thermometer scale is created experimentally. Typically, labels are placed on it that correspond to the position of the working fluid during a certain physical process that occurs when the thermal state remains constant (water boiling, ice melting). Different thermometers have different scales. For example, Celsius, Fahrenheit.
Universal temperature scale
From the point of view of independence from the properties of the working fluid, gas thermometers can be considered more interesting. Their scale does not depend on the type of gas used. In such a device, it is possible to hypothetically isolate the temperature at which the gas pressure tends to zero. Calculations show that this value corresponds to -273.15 . The temperature scale (absolute temperature scale or Kelvin scale) was introduced in 1848. The possible point of zero gas pressure was taken as the main point of this scale. The unit of the scale is equal to the unit value of the Celsius scale. Writing the basic MKT equation using temperature seems more convenient when studying gas processes.
The relationship of pressure and temperature
Empirically, one can verify the proportionality of the gas pressure to its temperature. At the same time, it was found that the pressure is directly proportional to the concentration of particles:
where T is the absolute temperature, k-constant value equal to 1.38 • 10 -23 J / K.
The fundamental value, which has the same value for all gases, is called the Boltzmann constant.
Comparing the dependence of pressure on temperature and the basic equation of MKT gases, we can write:
The average value of the kinetic energy of motion of gas molecules is proportional to its temperature. That is, temperature can serve as a measure of the kinetic energy of particle motion.