Studying the properties and behavior of an ideal gas is the key to understanding the physics of this area as a whole. In this article, we will consider what the concept of an ideal monatomic gas includes, what equations describe its state and internal energy. We will also solve a couple of problems on this topic.
General concept
Every student knows that gas is one of the three aggregate states of matter, which, unlike solid and liquid, does not retain volume. In addition, it also does not retain its shape and always fills the volume provided to it completely. In fact, the last property applies to the so-called ideal gases.
The concept of an ideal gas is closely related to 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. The particles do not interact with each other, and the distancesbetween 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 media are close in their 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, due to the presence of strong hydrogen interactions between H2 molecules HO, strong hydrogen interactions are not considered an ideal gas, but air, consisting of non-polar molecules, is.
Clapeyron-Mendeleev law
During the analysis, from the point of view of the MKT, the behavior of a gas in equilibrium, the following equation can be obtained, which relates the main thermodynamic parameters of the system:
PV=nRT.
Here pressure, volume and temperature are denoted by 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 / (Kmol), 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 recorded equality is called the universal equation of state of Clapeyron-Mendeleev. Why? It is named so in honor of the French physicist Emile Clapeyron, who in the 30s of the 19th century, studying the experimental gas laws established before, wrote it down in general form. Subsequently, Dmitri Mendeleev led him to modernform by entering the constant R.
Internal energy of a monatomic medium
A monatomic ideal gas differs from a polyatomic one 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 for the average kinetic energy of one atom:
mv2 / 2=3 / 2kB T.
The speed v is called root mean square. The mass of an atom and the Boltzmann constant are denoted as m and kBrespectively.
According to the definition of internal energy, it is the sum of the kinetic and potential components. Let's consider in more detail. Since an ideal gas has no potential energy, its internal energy is kinetic energy. What is its 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 / 2nRT.
Related examples
Task 1. An ideal monatomic gas passes from state 1 to state 2. The mass of the 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 volume delta of the gas vessel was three liters.
Let's write out the formula for changing the internal energy U:
ΔU=3 / 2nRΔT.
Using the Clapeyron-Mendeleev equation,this expression can be rewritten as:
ΔU=3 / 2PΔV.
We know the pressure and change in volume from the condition of the problem, so it remains to translate their values into SI and substitute them into the formula:
ΔU=3 / 21013250.003 ≈ 456 J.
Thus, when a monatomic ideal gas passes from state 1 to state 2, its internal energy increases by 456 J.
Task 2. An ideal monatomic gas in an amount of 2 mol was in a vessel. After isochoric heating, its energy increased by 500 J. How did the temperature of the system change?
Let's write down the formula for changing the value of U again:
ΔU=3 / 2nRΔT.
From it it is easy to express the magnitude of the change in absolute temperature ΔT, we have:
ΔT=2ΔU / (3nR).
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 correct. The Clapeyron-Mendeleev law is valid for any ideal gas.
