One of the important questions in the study of thermodynamic systems in physics is the question of the possibility of this system performing some useful work. The concept of work is closely related to the concept of internal energy. In this article, we will consider what is the internal energy of an ideal gas and give formulas for its calculation.
Perfect gas
Every student knows about gas as an aggregate state that does not have any elastic force when externally exposed to it and, as a result, does not preserve volume and shape. The concept of ideal gas for many remains incomprehensible and unclear. Explain her.
Ideal is any gas that satisfies the following two important conditions:
- Its constituent particles have no size. In fact, they have a size, but it is so small in comparison with the distances between them that it can be ignored in all mathematical calculations.
- Particles do not interact with each other with the help of Van der Waals forces or forces of a different nature. In fact, in all real gases, such an interaction is present, but its energy is negligible compared to the average energy of kinetic particles.
The described conditions are satisfied by almost all real gases whose temperatures lie above 300 K and the pressures do not exceed one atmosphere. For too high pressures and low temperatures, a deviation of gases from ideal behavior is observed. In this case, they talk about real gases. They are described by the van der Waals equation.
The concept of the internal energy of an ideal gas
In accordance with the definition, the internal energy of a system is understood to mean the sum of the kinetic and potential energies contained within this system. If you apply this concept to an ideal gas, you should discard the potential component. Indeed, since particles of an ideal gas do not interact with each other, they can be considered as moving freely in an absolute vacuum. To extract one particle from the studied system, it is not necessary to do work against the internal forces of interaction, since these forces do not exist.
Thus, the internal energy of an ideal gas always coincides with its kinetic energy. The latter, in turn, is uniquely determined by the molar mass of the particles of the system, their quantity, as well as the average speed of translational and rotational motion. Speed depends on temperature. An increase in temperature leads to an increase in internal energy, and vice versa.
Formula for internal energy
We denote the internal energy of an ideal gas system by the letter U. According to thermodynamics, it is defined as the difference between the enthalpy H of the system and the product of pressure by volume, that is:
U = H - p * V.
In the paragraph above, we found out that the value U corresponds to the total kinetic energy E k of all gas particles:
U = E k .
From statistical mechanics, within the framework of the principles of molecular kinetic theory (MKT) of an ideal gas, it follows that the average kinetic energy of one particle E k1 is equal to the following value:
E k1 = z / 2 * k B * T.
Here k B and T are the Boltzmann constant and temperature, z is the number of degrees of freedom. The total kinetic energy of the system E k can be obtained by multiplying E k1 by the number of particles N in the system:
E k = N * E k1 = z / 2 * N * k B * T.
Thus, we obtained a formula for the internal energy of an ideal gas, written in general terms through the absolute temperature and the number of particles in a closed system:
U = z / 2 * N * k B * T.
Monatomic and polyatomic gas
The formula for U written in the previous paragraph of the article is inconvenient for its practical use, since the number of particles N is difficult to determine. Nevertheless, if we take into account the determination of the amount of substance n, then this expression can be rewritten in a more convenient form:
n = N / N A ; R = N A * k B = 8.314 J / (mol * K);
U = z / 2 * n * R * T.
The number of degrees of freedom z depends on the geometry of the particles making up the gas. So, for a monatomic gas, z = 3, since an atom can independently move only in three directions of space. If the gas is diatomic, then z = 5, since two more rotational degrees of freedom are added to the three translational degrees of freedom. Finally, for any other polyatomic gas z = 6 (3 translational and 3 rotational degrees of freedom). With this in mind, we can write in the following form the formulas of the internal energy of an ideal gas of a monatomic, diatomic and polyatomic:
U 1 = 3/2 * n * R * T;
U 2 = 5/2 * n * R * T;
U ≥3 = 3 * n * R * T.
Example of the task of determining the internal energy
In a cylinder of 100 liters there is pure hydrogen under a pressure of 3 atmospheres. Assuming that hydrogen is an ideal gas under given conditions, it is necessary to determine what its internal energy is equal to.
In the above formulas for U, the amount of substance and the temperature of the gas are present. In the condition of the problem, absolutely nothing is said about these quantities. To solve the problem, it is necessary to recall the universal Clapeyron-Mendeleev equation. It has the view shown in the figure.
Since hydrogen H 2 is a diatomic molecule, the formula for internal energy is written in the form:
U H2 = 5/2 * n * R * T.
Comparing both expressions, we arrive at the final formula for solving the problem:
U H2 = 5/2 * P * V.
It remains to convert the pressure and volume units from the condition to the SI unit system, substitute the corresponding values in the formula for U H2 and get the answer: U H2 ≈ 76 kJ.