Gas is one of the four aggregate states of matter surrounding 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 what equation describes its behavior under various external conditions.
The concept of perfect gas
Everyone knows that the air we breathe, or the natural methane that we use to heat homes and cook food, are vivid representatives of the gas state of matter. In physics, the concept of an ideal gas was introduced to study the properties of this state. This concept involves the use of a number of assumptions and simplifications that are not essential when describing the basic physical characteristics of a substance: temperature, volume, and pressure.
So, a fluid substance is called an ideal gas, which satisfies the following conditions:
- Particles (molecules and atoms) move randomly in different directions. Due to this property, in 1648, Jan Baptista van Helmont introduced the concept of "gas" ("chaos" from the ancient Greek).
- Particles do not interact with each other, i.e. intermolecular and interatomic interactions can be neglected.
- Collisions between particles and with the walls of the vessel are absolutely elastic. As a result of such collisions, kinetic energy and momentum (momentum) are conserved.
- Each particle represents a material point, that is, it has some finite mass, but its volume is zero.
The set of conditions described above corresponds to the concept of an ideal gas. All known real substances with high accuracy correspond to the introduced concept at high temperatures (room and above) and low pressures (atmospheric and below).
Boyle-Marriott Law
Before writing the equation of state of an ideal gas, we present a number of particular laws and principles whose experimental discovery led to the conclusion of this equation.
Let's start with the Boyle-Marriott law. In 1662, the British physicist-chemist Robert Boyle and in 1676 the French physicist-botanist Edm Marriott independently established the following law: if the temperature in the 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:
P * V = k 1 at T = const, where
- P, V - pressure and volume of an ideal gas;
- k 1 is some constant.
Carrying out experiments with chemically different gases, scientists found that the value of k 1 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 the ideal gas on the graph are hyperbolas of the dependence of pressure on volume.
Charles and Gay-Lussac 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 a 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 = k 2 at P = const.
The transition between gas states with a change in temperature and volume and while maintaining pressure in the system is called an isobaric process. The constant k 2 is determined by the pressure in the system and the mass of the gas, but not by its chemical nature.
In the graph, the function V (T) is a straight line with the slope k 2 .
One can understand this law if one draws on the provisions of the molecular kinetic theory (MKT). Thus, an increase in temperature leads to an increase in the kinetic energy of gas particles. The latter helps to increase the intensity of their collisions with the walls of the vessel, which increases the pressure in the system. In order to keep this pressure constant, a volumetric expansion of the system is necessary.
Gay Lussac Law
The already mentioned French scientist at the beginning of the XIX century established another law related to the thermodynamic processes of an ideal gas. This law states: if a constant volume is maintained in the gas system, then an increase in temperature affects a proportional increase in pressure, and vice versa. The formula of the Gay-Lussac law is as follows:
P / T = k 3 at V = const.
Again, we have a constant k 3 depending on the mass of the gas and its volume. The thermodynamic process with a constant volume is called isochoric. The isochores on the 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 give characteristics only to the three laws that are presented above and which are particular 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 remain, then its volume V is in direct proportion to the amount of substance n. It does not matter what chemical nature the substance is. Avogadro established the following relationship:
n / V = k 4,
where the constant k 4 is determined by the pressure and temperature in the system.
The Avogadro principle is sometimes formulated as follows: a volume that occupies 1 mole of an ideal gas at a given temperature and pressure is always the same, regardless of its nature. Recall that 1 mol of a substance is the number N A , which reflects the number of elementary units (atoms, molecules) that make up the substance (N A = 6.02 * 10 23 ).
Mendeleev-Clapeyron Law
Now it's time to get back to the main topic of the article. Any ideal gas in equilibrium can be described by the following equality:
P * V = n * R * T.
This expression is called the Mendeleev-Clapeyron law - according to the names of scientists who have made a huge contribution to its formulation. The law says that the product of pressure on the volume of gas is directly proportional to the product of the amount of the substance of this gas by its temperature.
Clapeyron first obtained this law, summarizing the results of the studies of Boyle-Mariotte, Charles, Gay-Lussac and Avogadro. Mendeleev’s merit is that he gave the basic ideal gas equation a modern form by introducing the constant R. Clapeyron in his mathematical formulation used a set of constants, which made it inconvenient to use this law to solve practical problems.
The quantity R introduced by Mendeleev is called the universal gas constant. It shows what work 1 mole of gas of any chemical nature does as a result of isobaric expansion with an increase in temperature by 1 kelvin. Through the Avogadro constant N A and the Boltzmann constant k B, this value is calculated as follows:
R = N A * k B = 8.314 J / (mol * K).
Derivation of the equation
The current state of thermodynamics and statistical physics makes it possible to obtain, in several different ways, the ideal gas equation written in the previous paragraph.
The first way is to generalize only two empirical laws: Boyle-Marriott and Charles. From this generalization follows the form:
P * V / T = const.
That is what Clapeyron did in the 30s of the XIX century.
The second way is to use the provisions of the ICB. If we consider the momentum that each particle transmits in a collision with the vessel wall, 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 from the kinetic theory the ideal gas equation in the following form:
P * V = N * k B * T.
Multiplying and dividing the right side of the equation by the number N A , 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 equation of state of an ideal gas - from statistical mechanics using the concept of Helmholtz free energy.
Writing the equation through the mass of gas and density
The above equation 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 is written in the following form:
P * V = m / M * R * T.
M is the molar mass for a given gas. For example, for oxygen O 2 it is equal to 32 g / mol.
Finally, transforming the last expression, we can rewrite it like this:
P = ρ / M * R * T
Where ρ is the density of the substance.
Gas mixture
A mixture of ideal gases is described by the so-called Dalton 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 record:
P = ∑ i P i = R * T / V * ∑ i n i .
That is, the total pressure in the mixture P is equal to the sum of the partial pressures P i of all components.