Studying the behavior of gases in physics, often problems arise in determining the energy stored in them, which theoretically can be used to perform some useful work. In this article, we consider the question by which formulas the internal energy of an ideal gas can be calculated.
The concept of perfect gas
A clear understanding of the ideal gas concept is important when solving problems with systems in this state of aggregation. Any gas takes the form and volume of a vessel into which it is placed; however, not every gas is ideal. For example, air can be considered a mixture of ideal gases, while water vapor is not. What is the fundamental difference between real gases and their ideal model?
The answer to this question will be two of the following features:
- the relationship between the kinetic and potential energy of the molecules and atoms that make up the gas;
- the ratio between the linear sizes of gas particles and the average distance between them.
A gas is considered ideal only if the average kinetic energy of its particles is incommensurably greater than the binding energy between them. The difference between these energies is such that we can assume that the interaction between particles is completely absent. Also, an ideal gas is characterized by a lack of particle size, or rather these sizes can be ignored, since they are much smaller than the average interparticle distances.
Good empirical criteria for determining the ideality of a gas system are its thermodynamic characteristics such as temperature and pressure. If the first is more than 300 K, and the second is less than 1 atmosphere, then any gas can be considered ideal.
What is the internal energy of the gas?
Before writing the formula of the internal energy of the ideal gas, it is necessary to get acquainted with this characteristic closer.
In thermodynamics, internal energy, as a rule, is denoted by the Latin letter U. It is determined in the general case by the following formula:
U = H - P * V
Where H is the enthalpy of the system, P and V are pressure and volume.
In its physical sense, internal energy consists of two components: kinetic and potential. The first is associated with various kinds of movement of the particles of the system, and the second - with the force interaction between them. If we apply this definition to the concept of an ideal gas, which has no potential energy, then the quantity U for any state of the system will be exactly equal to its kinetic energy, that is:
U = E k .
Derivation of the formula of internal energy
We established above that to determine it for a system with an ideal gas, it is necessary to calculate its kinetic energy. From the course of general physics it is known that the energy of a particle of mass m, which is translationally moving in a certain direction with a speed v, is determined by the formula:
E k1 = m * v 2/2.
It can also be applied to gas particles (atoms and molecules), however, some comments must be made.
Firstly, the speed v should be understood as some average value. The fact is that gas particles move at different speeds according to the Maxwell-Boltzmann distribution. The latter allows us to determine the average speed, which does not change over time if there are no external influences on the system.
Secondly, the formula for E k1 implies energy per degree of freedom. Gas particles can move in all three directions, as well as rotate depending on their structure. To take into account the value of the degree of freedom z, it should be multiplied by E k1 , that is:
E k1z = z / 2 * m * v 2 .
The kinetic energy of the entire system E k is N times greater than E k1z , where N is the total number of gas particles. Then for U we get:
U = z / 2 * N * m * v 2 .
According to this formula, a change in the internal energy of a gas is possible only if one changes the number of particles N in the system or their average velocity v.
Internal energy and temperature
Applying the principles of the molecular-kinetic theory of an ideal gas, we can obtain the following formula for the relationship between the average kinetic energy of one particle and the absolute temperature:
m * v 2/2 = 1/2 * k B * T.
Here k B is the Boltzmann constant. Substituting this equality into the formula for U obtained in the paragraph above, we arrive at the following expression:
U = z / 2 * N * k B * T.
This expression can be rewritten through the amount of substance n and the gas constant R in the following form:
U = z / 2 * n * R * T.
In accordance with this formula, a change in the internal energy of a gas is possible if its temperature is changed. The values โโof U and T depend on each other linearly, that is, the graph of the function U (T) is a straight line.
How does the structure of a gas particle affect the internal energy of a system?
By the structure of a gas particle (molecule) is meant the number of atoms that make it up. It plays a decisive role in substituting the corresponding degree of freedom z into the formula for U. If the gas is monatomic, the formula of the internal energy of the gas takes the following form:
U = 3/2 * n * R * T.
Where did z = 3 come from? Its appearance is associated with only three degrees of freedom that an atom possesses, since it can only move in one of three spatial directions.
If a diatomic gas molecule is considered, then the internal energy should be calculated by the following formula:
U = 5/2 * n * R * T.
As you can see, the diatomic molecule already has 5 degrees of freedom, 3 of which are translational and 2 rotational (in accordance with the geometry of the molecule, it can rotate around two mutually perpendicular axes).
Finally, if the gas is three or more atomic, then the following expression holds for U:
U = 3 * n * R * T.
Complex molecules have 3 translational and 3 rotational degrees of freedom.
Task example
Under the piston is a monatomic gas at a pressure of 1 atmosphere. As a result of heating, the gas expanded so that its volume increased from 2 liters to 3. How did the internal energy of the gas system change if the expansion process was isobaric?
To solve this problem, the formulas given in the article are not enough. It is necessary to recall the equation of state of an ideal gas. It has the form presented below.
Since the piston closes the cylinder with gas, during the expansion process the amount of substance n remains constant. During the isobaric process, the temperature changes in direct proportion to the volume of the system (Charles law). This means that the formula above is written like this:
P * ฮV = n * R * ฮT.
Then the expression for the internal energy of a monatomic gas will take the form:
ฮU = 3/2 * P * ฮV.
Substituting the pressure and volume changes in SI units into this equality, we get the answer: ฮU โ 152 J.