The consideration of the properties of gases in physics in a first approximation is based on the concept of an ideal gas. In this article, we will study this concept in detail and give an equation that numerically describes the thermodynamic properties of the mentioned fluid substance. This equation is called the Clapeyron-Mendeleev law.
Ideal gas concept
In the school course of physics, the gas state of aggregation of a substance is characterized by arbitrary movement with different speeds of all its atoms and molecules. In the first approximation, these particles are considered absolutely elastic material points. They have mass, but not dimensions. The whole nature of their interaction with each other is in absolutely elastic collisions, as a result of which the momentum and energy are saved. All of the listed properties of particles and their approximations form the concept of an ideal gas.
Any real gas, be it helium, oxygen or air, can be considered ideal with high accuracy if its pressure is on the order of one atmosphere or lower, and the temperature corresponds to room temperature or higher. If these conditions are not met, then the gas is considered real, and the Van der Waals equation should be used to describe it, and not the Clapeyron-Mendeleev law, which will be discussed later in the article.
Prerequisites for the emergence of the ideal gas equation of state
By the ideal gas equation of state it is customary to understand the mathematical formulation of the Mendeleev-Clapeyron gas law. Like any discovery in physics, this equation did not appear from nowhere, but had quite definite historical prerequisites.
In the 60-70s of the XVII century, the Englishman Robert Boyle and the Frenchman Edm Mariotte independently, as a result of many experiments with various gases, established that the product of volume and pressure for a closed system with gas remains constant for any processes that result in temperature does not change. Currently, this gas law bears the names of these scientists.
After almost 1.5 centuries, in the late XVIII - early XIX centuries, the French Charles and Gay Lussac discover two more experimental laws in the behavior of ideal gases. They establish a direct proportional relationship between pressure and temperature at constant volume and between volume and temperature at constant pressure.
Finally, in 1834, Emile Clapeyron deduced, analyzing the gas laws discovered by previous scientists, the Clapeyron equation. Mendeleev's surname appeared in the name of this equation due to his contribution to the transformation of the original expression to its modern form. In particular, Mendeleev introduced the concept of a universal gas constant.
Clapeyron-Mendeleev Law Formula
Above we gave a definition of an ideal gas, talked about the laws that led to the formulation of the universal equation of state. Now it's time to write this equation:
P * V = n * R * T.
Here P, V, n and T are pressure, volume, amount of substance and temperature, respectively. Thus, the product of the volume of the system and the pressure in it is always for an ideal gas in direct proportion to the product of absolute temperature and the amount of substance.
The proportionality coefficient is the already mentioned universal constant R. It is 8.314 J / (mol * K). If 1 mole of gas is heated by 1 kelvin, then in the process of expansion it will do the work of 8.314 Joules. It is interesting to note that the universal value of R is called because it is not determined by the chemical nature of the gas. For all pure gases and their mixtures, it takes on a unique meaning.
Where does the equation being studied come from?
We already said above that Clapeyron received his equation as a result of a banal generalization of the experimental results of various scientists. Nevertheless, the Clapeyron-Mendeleev law can be obtained by purely theoretical methods.
One of them is MKT (molecular kinetic theory). MKT considers the gas system in terms of particle concentration, distribution of their velocities, taking into account their masses and following the concept of ideal gas. The universal gas equation follows unambiguously if we apply Newton’s second law to the process of elastic collision of particles with the walls of a sealed vessel. As a result of the use of MKT, the expression is obtained:
P * V = N * k B * T.
This equality leads to the equation written in the previous paragraph, if we take into account the following expressions:
R = N A * k B ;
n = N / N A.
Using a universal equation to solve a problem
It is known that some gas under a pressure of 2 atmospheres is in the cylinder at a temperature of 25 o C. The volume of the cylinder is 50 liters. How much substance is in the bottle?
Since we know 3 of 4 parameters, we can apply the Clapeyron-Mendeleev law to find the value of n. Before doing this, we transfer all units to the SI system:
P = 2 atm. = 101325 * 2 = 202650 Pa;
T = 25 + 273.15 = 298.15 K;
V = 50 * 10 -3 = 0.05 m 3 .
Now we use the formula, we get:
P * V = n * R * T =>
n = P * V / (R * T) = 202650 * 0.05 / (8.314 * 298.15) = 4.09 mol.
Although the value of 4.09 mol is small, the amount of gas particles will be gigantic. To get it, you should n multiply by N A = 6.02 * 10 23 .