The aggregate state of matter, in which the kinetic energy of particles far exceeds their potential energy of interaction, is called 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 for an ideal gas. We will study it in detail in the article.
Ideal gas and its difference from the real one
As you know, any gas state is characterized by chaotic motion with different speeds of its constituent molecules and atoms. In real gases, such as air, the particles interact with each other in one way or another. Basically, this interaction has a van der Waals character. However, if the temperatures of the gas system are high (room temperature and above) and the pressure is not huge (corresponding to atmospheric), then the van der Waals interactions are so small that notaffect the macroscopic behavior of the entire gas system. In this case, they speak of the ideal.
Combining the above information into 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.
Practically all gases that a person encounters in everyday life (air, natural methane in gas stoves, water vapor) can be considered ideal with accuracy satisfactory for many practical problems.
Prerequisites for the appearance of the ideal gas equation of state in physics
Mankind actively studied the gaseous 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 relation between the volume of the system V and the pressure in it P:
experimentally discovered by Robert Boyle and Edme Mariotte
PV=const, with T=const
Experimenting with various gases in the second half of the 17th century, the mentioned scientists found that the dependence of pressure on volume always has the form of a hyperbola.
Then, at the end of the 18th - at the beginning of the 19th century, French scientists Charles and Gay-Lussac experimentally discovered two more gas laws that mathematically described the isobaric and isochoric processes. Both laws are listed below:
- V / T=const, when P=const;
- P / T=const, with V=const.
Both equalities indicate a direct proportionality between the volume of gas and temperature, as well as between pressure and temperature, while maintaining constant pressure and volume, respectively.
Another prerequisite for compiling the equation of state of an ideal gas was the discovery of the following relation by Amedeo Avagadro in the 1910s:
n / V=const, with T, P=const
The Italian experimentally proved that if you increase the amount of substance n, then at constant temperature and pressure, the volume will increase linearly. The most surprising thing was that gases of different nature at the same pressures and temperatures occupied the same volume if their number coincided.
Clapeyron-Mendeleev law
In the 30s of the 19th century, the Frenchman Emile Clapeyron published a work in which he gave the equation of state for an ideal gas. It was slightly different from the modern form. In particular, Clapeyron used certain constants measured experimentally by his predecessors. A few decades later, our compatriot D. I. Mendeleev replaced the Clapeyron constants with a single one - the universal gas constant R. As a result, the universal equation acquired a modern form:
PV=nRT
It is easy to guess that this is a simple combination of the formulas of gas laws that were written above in the article.
The constant R in this expression has a very specific physical meaning. It shows the work that 1 mole will do.gas if it expands with an increase in temperature by 1 kelvin (R=8.314 J / (molK)).
Other forms of the universal equation
Besides the above form of the universal equation of state for an ideal gas, there are equations of state that use other quantities. Here are them below:
- PV=m / MRT;
- PV=NkB T;
- P=ρRT / M.
In these equalities, m is the mass of an ideal gas, N is the number of particles in the system, ρ is the density of the gas, M is the value of the molar mass.
Recall that the formulas written above are valid only if SI units are used for all physical quantities.
Example problem
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, the volume of which 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 °C.
First, let's write down all units of measure in SI:
1) P=1.5101325=151987.5 Pa;
2) V=7010-3=0.07 m3;
3) T=50 + 273, 15=323, 15 K.
Now we substitute these data into the Clapeyron-Mendeleev equation, we get the value of the amount of substance:
n=PV / (RT)=151987.50.07 / (8.314323.15)=3.96 mol
To determine the mass of nitrogen, you should remember its chemical formula and see the valuemolar mass in the periodic table for this element:
M(N2)=142=0.028 kg/mol.
The mass of gas will be:
m=nM=3.960.028=0.111 kg
Thus, the amount of nitrogen in the balloon is 3.96 mol, its mass is 111 grams.
