The aggregate state of matter in which the kinetic energy of particles far exceeds their potential interaction energy is called a gas. The physics of such substances is beginning to be considered in high school. The key issue in the mathematical description of this fluid substance is the equation of state of an ideal gas. We will study it in detail in the article.
The ideal gas and its difference from the real
As is known, chaotic motion with different velocities of its constituent molecules and atoms is characteristic of any gas state. In real gases, for example, air, particles interact in one way or another with each other. Basically, this interaction is of a van der Waals character. Nevertheless, if the temperature of the gas system is high (room temperature and higher), and the pressure is not huge (corresponds to atmospheric), then the van der Waals interactions are so small that they do not affect the macroscopic behavior of the entire gas system. In this case, they talk about the ideal.
Collecting the above information in one definition, we can say that an ideal gas is a system in which there are no interactions between particles. The particles themselves are dimensionless, but have a certain mass, and the collisions of particles with the walls of the vessel are elastic.
Almost all gases that a person encounters in everyday life (air, natural methane in gas stoves, water vapor) can, with satisfactory accuracy for many practical problems, be considered ideal.
Prerequisites for the emergence of the equation of state of an ideal gas in physics
Mankind has actively studied the gas state of matter from a scientific point of view during the XVII-XIX centuries. The first law that described the isothermal process was the following relationship between the volume of the system V and the pressure P in it, experimentally discovered by Robert Boyle and Edm Mariott:
- P * V = const, at T = const.
Carrying out experiments with various gases in the second half of the XVII century, the scientists mentioned that the dependence of pressure on volume always has the form of a hyperbola.
Then, at the end of the XVIII - at the beginning of the XIX century, French scientists Charles and Gay-Lussac discovered experimentally two more gas laws that described mathematically isobaric and isochoric processes. Both laws are listed below:
- V / T = const, at P = const;
- P / T = const, with V = const.
Both equalities indicate direct proportionality between the gas volume and temperature, as well as between pressure and temperature while maintaining constant pressure and volume, respectively.
Another prerequisite for the preparation of the equation of state of an ideal gas was the discovery by Amedeo Avagadro in the 10s of the XIX century of the following relationship:
- n / V = ββconst, at T, P = const.
The Italian experimentally proved that if the amount of substance n is increased, then at constant temperature and pressure the volume will increase linearly. The most surprising thing was that gases of different nature at the same pressure and temperature occupied the same volume, if their number coincided.
Clapeyron-Mendeleev Law
In the 30s of the XIX century, the Frenchman Emile Clapeyron published a work in which he presented the equation of state of an ideal gas. It was slightly different from the modern form. In particular, Clapeyron used certain constants measured experimentally by his predecessors. After several decades, our compatriot D.I. Mendeleev replaced the Clapeyron constants with one single one - the universal gas constant R. As a result, the universal equation acquired a modern form:
It is not difficult to guess that this is a simple combination of gas law formulas, which were written above in the article.
The constant R in this expression has a very specific physical meaning. It shows the work that 1 mol of gas will do if it expands with increasing temperature by 1 kelvin (R = 8.314 J / (mol * K)).
Other forms of writing the universal equation
In addition to the above form of the universal equation of state for an ideal gas, there are equations of state that use other quantities. We give them below:
- P * V = m / M * R * T;
- P * V = N * k B * T;
- P = Ο * R * T / M.
In these equalities, m is the ideal gas mass, N is the number of particles in the system, Ο is the gas density, and M is the molar mass value.
Recall that the formulas written above are valid only if SI units are used for all physical quantities.
Task example
Having received the necessary theoretical information, we will solve the following problem. Pure nitrogen is at a pressure of 1.5 atm. in a cylinder whose volume is 70 liters. It is necessary to determine the number of moles of an ideal gas and its mass, if it is known that it is at a temperature of 50 Β° C.
First, write all the units in SI:
1) P = 1.5 * 101325 = 151987.5 Pa;
2) V = 70 * 10 -3 = 0.07 m 3 ;
3) T = 50 + 273.15 = 323.15 K.
We substitute these data into the Clapeyron-Mendeleev equation, we obtain the value of the amount of substance:
- n = P * V / (R * T) = 151987.5 * 0.07 / (8.314 * 323.15) = 3.96 mol.
To determine the mass of nitrogen, one should recall its chemical formula and look at the value of the molar mass in the periodic table for this element:
- M (N 2 ) = 14 * 2 = 0.028 kg / mol.
The mass of gas will be equal to:
- m = n * M = 3.96 * 0.028 = 0.111 kg.
Thus, the amount of nitrogen in the cylinder is 3.96 mol, its mass is 111 grams.