Of the four aggregate states of matter, gas is perhaps the simplest in terms of its physical description. In the article, we consider the approximations that are used for the mathematical description of real gases, and also give the so-called Clapeyron equation.
Ideal gas
All gases that we encounter during life (natural methane, air, oxygen, nitrogen, and so on) can be classified as ideal. Ideal is any gaseous state of matter in which particles move randomly in different directions, their collisions are 100% elastic, particles do not interact with each other, they are material points (they have mass and no volume).
There are two different theories that are often used to describe the gaseous state of matter: molecular kinetic (MKT) and thermodynamics. MKT uses the properties of an ideal gas, the statistical distribution of particle velocities, and the relationship of kinetic energy and momentum to temperature to calculatemacroscopic characteristics of the system. In turn, thermodynamics does not delve into the microscopic structure of gases, it considers the system as a whole, describing it with macroscopic thermodynamic parameters.
Thermodynamic parameters of ideal gases
There are three main parameters for describing ideal gases and one additional macroscopic characteristic. Let's list them:
- Temperature T- reflects the kinetic energy of molecules and atoms in a gas. Expressed in K (Kelvin).
- Volume V - characterizes the spatial properties of the system. Determined in cubic meters.
- Pressure P - due to the impact of gas particles on the walls of the vessel containing it. This value is measured in the SI system in pascals.
- Amount of substance n - a unit that is convenient to use when describing large numbers of particles. In SI, n is expressed in moles.
Further in the article, the Clapeyron equation formula will be given, in which all four described characteristics of an ideal gas are present.
Universal equation of state
The Clapeyron ideal gas equation of state is usually written in the following form:
PV=nRT
Equality shows that the product of pressure and volume must be proportional to the product of temperature and the amount of substance for any ideal gas. The value R is called the universal gas constant and at the same time the coefficient of proportionality between the mainmacroscopic characteristics of the system.
An important feature of this equation should be noted: it does not depend on the chemical nature and composition of the gas. That is why it is often called universal.
For the first time this equality was obtained in 1834 by the French physicist and engineer Emile Clapeyron as a result of the generalization of the experimental laws of Boyle-Mariotte, Charles and Gay-Lussac. However, Clapeyron used a somewhat inconvenient system of constants. Subsequently, all Clapeyron's constants were replaced by one single value R. Dmitry Ivanovich Mendeleev did this, therefore the written expression is also called the formula of the Clapeyron-Mendeleev equation.
Other Equation Forms
In the previous paragraph, the main form of writing the Clapeyron equation was given. Nevertheless, in problems in physics, other quantities can often be given instead of the amount of matter and volume, so it will be useful to give other forms of writing the universal equation for an ideal gas.
The following equality follows from the MKT theory:
PV=NkBT.
This is also an equation of state, only the quantity N (number of particles) less convenient to use than the amount of substance n appears in it. There is also no universal gas constant. Instead, the Boltzmann constant is used. The written equality is easily converted into a universal form if the following expressions are taken into account:
n=N/NA;
R=NAkB.
Here NA- Avogadro's number.
Another useful form of the equation of state is:
PV=m/MRT
Here, the ratio of mass m of gas to molar mass M is, by definition, the amount of substance n.
Finally, another useful expression for an ideal gas is a formula that uses the concept of its density ρ:
P=ρRT/M
Problem Solving
Hydrogen is in a 150-liter cylinder under a pressure of 2 atmospheres. It is necessary to calculate the density of the gas if the temperature of the cylinder is known to be 300 K.
Before we start solving the problem, let's convert pressure and volume units to SI:
P=2 atm.=2101325=202650 Pa;
V=15010-3=0.15m3.
To calculate the density of hydrogen, use the following equation:
P=ρRT/M.
From it we get:
ρ=MP/(RT).
The molar mass of hydrogen can be viewed in the periodic table of Mendeleev. It is equal to 210-3kg/mol. The R value is 8.314 J/(molK). Substituting these values and the values of pressure, temperature and volume from the conditions of the problem, we obtain the following density of hydrogen in the cylinder:
ρ=210-3202650/(8, 314300)=0.162 kg/m3.
For comparison, the air density is approximately 1.225 kg/m3at a pressure of 1 atmosphere. Hydrogen is less dense, since its molar mass is much less than that of air (15 times).
