The study of the properties and behavior of an ideal gas is a key point in understanding the physics of this field as a whole. Let us consider in this article what the concept of an ideal monatomic gas includes, what equations describe its state and internal energy. And also solve a couple of problems on this topic.
General concept
Every student knows that gas is one of the three aggregate states of a substance, which, unlike solid and liquid, does not preserve volume. In addition, he also does not retain his form and always fills out the volume fully provided to him. In fact, the latter property relates to the so-called ideal gases.
The concept of an ideal gas is closely related to the molecular kinetic theory (MKT). In accordance with it, the particles of the gas system move randomly in all directions. Their speeds obey the Maxwell distribution. Particles do not interact with each other, and the distances between them far exceed their size. If all of the above conditions are met with a certain accuracy, then the gas can be considered ideal.
Any real medium is close in its behavior to ideal if they have low densities and high absolute temperatures. In addition, they must be composed of chemically inactive molecules or atoms. So, water vapor due to the presence of strong hydrogen interactions between H 2 O molecules is not considered an ideal gas, and air consisting of non-polar molecules is it.
Clapeyron-Mendeleev Law
During the analysis, from the point of view of MKT, the behavior of the gas in equilibrium, the following equation can be obtained that relates the basic thermodynamic parameters of the system:
P * V = n * R * T.
Here pressure, volume and temperature are denoted by the Latin letters P, V and T, respectively. The value of n is the amount of substance that allows you to determine the number of particles in the system, R is the gas constant, independent of the chemical nature of the gas. It is equal to 8.314 J / (K * mol), that is, any ideal gas in the amount of 1 mol when it is heated by 1 K, expanding, does the work of 8.314 J.
The written equality is called the universal Clapeyron-Mendeleev equation of state . Why? It is named after the French physicist Emile Clapeyron, who in the 30s of the XIX century, studying the experimental gas laws established before, wrote it down in general form. Subsequently, Dmitry Mendeleev led him to modern form, introducing the constant R.
The internal energy of a monatomic medium
A monatomic ideal gas differs from a polyatomic gas in that its particles have only three degrees of freedom (translational motion along the three axes of space). This fact leads to the following formula of the average kinetic energy of one atom:
m * v 2/2 = 3/2 * k B * T.
The speed v is called the root mean square. The mass of the atom and the Boltzmann constant are denoted as m and k B, respectively.
According to the definition of internal energy, it is the sum of the kinetic and potential components. Let's consider in more detail. Since the ideal gas does not have potential energy, its internal energy is kinetic energy. What is her formula? Calculating the energy of all particles N in the system, we obtain the following expression for the internal energy U of a monatomic gas:
U = 3/2 * n * R * T.
Related examples
Task number 1. An ideal monatomic gas passes from state 1 to state 2. The mass of gas remains constant (closed system). It is necessary to determine the change in the internal energy of the medium if the transition is isobaric at a pressure equal to one atmosphere. The gas volume delta was three liters.
We write out the formula for changing the internal energy U:
ΔU = 3/2 * n * R * ΔT.
Using the Clapeyron-Mendeleev equation, this expression can be rewritten in the form:
ΔU = 3/2 * P * ΔV.
We know the pressure and the change in volume from the conditions of the problem; therefore, it remains to translate their values into SI and substitute in the formula:
ΔU = 3/2 * 101325 * 0.003 ≈ 456 J.
Thus, when a monatomic ideal gas passes from state 1 to state 2, its internal energy increases by 456 J.
Task number 2. The ideal 2-mole monatomic gas was in the vessel. After isochoric heating, its energy increased by 500 J. How did the temperature of the system change?
Again we write the formula for changing the value of U:
ΔU = 3/2 * n * R * ΔT.
From it, it is easy to express the magnitude of the change in the absolute temperature ΔT, we have:
ΔT = 2 * ΔU / (3 * n * R).
Substituting the data for ΔU and n from the condition, we get the answer: ΔT = +20 K.
It is important to understand that all the above calculations are valid only for a monatomic ideal gas. If the system is formed by polyatomic molecules, then the formula for U will no longer be true. The Clapeyron-Mendeleev law is valid for any ideal gas.