Gas is one of the four aggregate states of matter around us. Humanity began to study this state of matter using a scientific approach, starting from the 17th century. In the article below, we will study what an ideal gas is and which equation describes its behavior under various external conditions.
The concept of an ideal gas
Everyone knows that the air we breathe, or the natural methane we use to heat our homes and cook our food, is a prime example of the gaseous state of matter. In physics, to study the properties of this state, the concept of an ideal gas was introduced. This concept involves the use of a number of assumptions and simplifications that are not essential in describing the basic physical characteristics of a substance: temperature, volume and pressure.
So, an ideal gas is a fluid substance that satisfies the following conditions:
- Particles (molecules and atoms)moving randomly in different directions. Thanks to this property, in 1648, Jan Baptista van Helmont introduced the concept of "gas" ("chaos" from ancient Greek).
- Particles do not interact with each other, that is, intermolecular and interatomic interactions can be neglected.
- Collisions between particles and with vessel walls are absolutely elastic. As a result of such collisions, kinetic energy and momentum (momentum) are conserved.
- Each particle is a material point, that is, it has some finite mass, but its volume is zero.
The set of the above conditions corresponds to the concept of an ideal gas. All known real substances correspond with high accuracy to the introduced concept at high temperatures (room and above) and low pressures (atmospheric and below).
Boyle-Mariotte Law
Before writing down the equation of state for an ideal gas, let us present a number of particular laws and principles, the experimental discovery of which led to the derivation of this equation.
Let's start with the Boyle-Mariotte law. In 1662, the British physical chemist Robert Boyle and in 1676 the French physical botanist Edm Mariotte independently established the following law: if the temperature in a gas system remains constant, then the pressure created by the gas during any thermodynamic process is inversely proportional to its volume. Mathematically, this formulation can be written as follows:
PV=k1 for T=const,where
- P, V - pressure and volume of an ideal gas;
- k1 - some constant.
Experimenting with chemically different gases, scientists have found that the value of k1 does not depend on the chemical nature, but depends on the mass of the gas.
The transition between states with a change in pressure and volume while maintaining the temperature of the system is called an isothermal process. Thus, the isotherms of an ideal gas on the graph are hyperbolas of the dependence of pressure on volume.
Charles and Gay-Lussac's Law
In 1787, the French scientist Charles and in 1803 another Frenchman Gay-Lussac empirically established another law that described the behavior of an ideal gas. It can be formulated as follows: in a closed system at constant gas pressure, an increase in temperature leads to a proportional increase in volume and, conversely, a decrease in temperature leads to a proportional compression of the gas. The mathematical formulation of the law of Charles and Gay-Lussac is written as follows:
V / T=k2 when P=const.
The transition between the states of a gas with a change in temperature and volume and while maintaining pressure in the system is called an isobaric process. The constant k2 is determined by the pressure in the system and the mass of the gas, but not by its chemical nature.
On the graph, the function V (T) is a straight line with slope tangent k2.
You can understand this law if you draw on the provisions of molecular kinetic theory (MKT). Thus, an increase in temperature leads to an increasekinetic energy of gas particles. The latter contributes to an increase in the intensity of their collisions with the walls of the vessel, which increases the pressure in the system. To keep this pressure constant, the volumetric expansion of the system is necessary.
Gay-Lussac's Law
The already mentioned French scientist at the beginning of the 19th century established another law related to the thermodynamic processes of an ideal gas. This law states: if a constant volume is maintained in a gas system, then an increase in temperature affects a proportional increase in pressure, and vice versa. The Gay-Lussac formula looks like this:
P / T=k3 with V=const.
Again we have the constant k3, which depends on the mass of the gas and its volume. A thermodynamic process at constant volume is called isochoric. Isochores on a P(T) graph look the same as isobars, that is, they are straight lines.
Avogadro Principle
When considering the equation of state of an ideal gas, they often characterize only three laws that are presented above and which are special cases of this equation. Nevertheless, there is another law, which is commonly called the principle of Amedeo Avogadro. It is also a special case of the ideal gas equation.
In 1811, the Italian Amedeo Avogadro, as a result of numerous experiments with different gases, came to the following conclusion: if the pressure and temperature in the gas system is maintained, then its volume V is in direct proportion to the amountsubstances n. It does not matter what chemical nature the substance is. Avogadro established the following ratio:
n / V=k4,
where the constant k4 is determined by the pressure and temperature in the system.
Avogadro's principle is sometimes formulated as follows: the volume occupied by 1 mole of an ideal gas at a given temperature and pressure is always the same, regardless of its nature. Recall that 1 mole of a substance is the number NA, reflecting the number of elementary units (atoms, molecules) that make up the substance (NA=6.021023).
Mendeleev-Clapeyron law
Now it's time to return to the main topic of the article. Any ideal gas in equilibrium can be described by the following equation:
PV=nRT.
This expression is called the Mendeleev-Clapeyron law - after the names of scientists who have made a huge contribution to its formulation. The law states that the product of pressure times the volume of a gas is directly proportional to the product of the amount of substance in that gas and its temperature.
Clapeyron first obtained this law, summarizing the results of the studies of Boyle-Mariotte, Charles, Gay-Lussac and Avogadro. The merit of Mendeleev is that he gave the basic equation of an ideal gas a modern form by introducing the constant R. Clapeyron used a set of constants in his mathematical formulation, which made it inconvenient to use this law for solving practical problems.
The value R introduced by Mendeleevis called the universal gas constant. It shows how much work is done by 1 mole of a gas of any chemical nature as a result of isobaric expansion with an increase in temperature by 1 kelvin. Through the Avogadro constant NA and the Boltzmann constant kB this value is calculated as follows:
R=NA kB=8, 314 J/(molK).
Derivation of the equation
The current state of thermodynamics and statistical physics allows us to obtain the ideal gas equation written in the previous paragraph in several different ways.
The first way is to generalize only two empirical laws: Boyle-Mariotte and Charles. From this generalization follows the form:
PV / T=const.
This is exactly what Clapeyron did in the 30s of the XIX century.
The second way is to invoke the provisions of the ICB. If we consider the momentum that each particle transfers when colliding with the wall of the vessel, take into account the relationship of this momentum with temperature, and also take into account the number of particles N in the system, then we can write the ideal gas equation from the kinetic theory in the following form:
PV=NkB T.
By multiplying and dividing the right side of the equation by the number NA, we get the equation in the form in which it is written in the paragraph above.
There is a third more complicated way to obtain the ideal gas equation of state - from statistical mechanics using the concept of Helmholtz free energy.
Writing the equation in terms of gas mass and density
The figure above shows the ideal gas equation. It contains the amount of substance n. However, in practice, the variable or constant mass of an ideal gas m is often known. In this case, the equation will be written in the following form:
PV=m / MRT.
M - molar mass for a given gas. For example, for oxygen O2 it is 32 g/mol.
Finally, transforming the last expression, we can rewrite it like this:
P=ρ / MRT
Where ρ is the density of the substance.
Mixture of gases
A mixture of ideal gases is described by the so-called D alton's law. This law follows from the ideal gas equation, which is applicable for each component of the mixture. Indeed, each component occupies the entire volume and has the same temperature as the other components of the mixture, which allows us to write:
P=∑iPi=RT / V∑i i.
That is, the total pressure in the mixture P is equal to the sum of the partial pressures Pi of all components.
