When studying the behavior of gases in physics, problems often arise to determine the energy stored in them, which theoretically can be used to perform some useful work. In this article, we will consider the question of what formulas can be used to calculate the internal energy of an ideal gas.
The concept of an ideal gas
A clear understanding of the concept of an ideal gas is important when solving problems with systems in this state of aggregation. Any gas takes the shape and volume of the vessel in 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 the question will be the following two features:
- the ratio between the kinetic and potential energy of the molecules and atoms that make up the gas;
- ratio between the linear sizes of particlesgas 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 the particles is completely absent. Also, an ideal gas is characterized by the absence of dimensions of its particles, or rather, these dimensions 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 greater than 300 K, and the second is less than 1 atmosphere, then any gas can be considered ideal.
What is the internal energy of a gas?
Before writing down the formula for the internal energy of an ideal gas, you need to get to know this characteristic more closely.
In thermodynamics, internal energy is usually denoted by the Latin letter U. In the general case, it is determined by the following formula:
U=H - PV
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 motion 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 value of U in any state of the system will be exactly equal to its kinetic energy, that is:
U=Ek.
Derivation of the internal energy formula
Above, we found 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 moving forward in a certain direction with a speed v, is determined by the formula:
Ek1=mv2/2.
It can also be applied to gas particles (atoms and molecules), however, some remarks need to 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 you to determine the average speed, which does not change over time if there are no external influences on the system.
Second, the formula for Ek1 assumes energy per degree of freedom. Gas particles can move in all three directions, and also rotate depending on their structure. To take into account the degree of freedom z, it should be multiplied by Ek1, i.e.:
Ek1z=z/2mv2.
The kinetic energy of the entire system Ek is N times greater than Ek1z, where N is the total number of gas particles. Then for U we get:
U=z/2Nmv2.
According to this formula, a change in the internal energy of a gas is possible only if the number of particles N is changed insystem, or their average speed v.
Internal energy and temperature
Applying the provisions 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:
mv2/2=1/2kBT.
Here kB 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/2NkBT.
This expression can be rewritten in terms of the amount of substance n and the gas constant R in the following form:
U=z/2nR T.
In accordance with this formula, a change in the internal energy of a gas is possible if its temperature is changed. The values 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?
The structure of a gas particle (molecule) refers to the number of atoms that make it up. It plays a decisive role when substituting the corresponding degree of freedom z in the formula for U. If the gas is monatomic, the formula for the internal energy of the gas becomes:
U=3/2nRT.
Where did the value z=3 come from? Its appearance is associated with only three degrees of freedom that an atom has, since it can only move in one of three spatial directions.
If a diatomicgas molecule, then the internal energy should be calculated using the following formula:
U=5/2nRT.
As you can see, a 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 for U is true:
U=3nRT.
Complex molecules have 3 translational and 3 rotational degrees of freedom.
Example problem
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 for an ideal gas. It looks like below.
Since the piston closes the cylinder with gas, the amount of substance n remains constant during the expansion process. During an isobaric process, the temperature changes in direct proportion to the volume of the system (Charles law). This means that the formula above would be:
PΔV=nRΔT.
Then the expression for the internal energy of a monatomic gas will take the form:
ΔU=3/2PΔV.
Substituting into this equation the values of pressure and volume change in SI units, we get the answer: ΔU ≈ 152 J.
