In studying the behavior of gases in physics, much attention is paid to isoprocesses, that is, to such transitions between system states during which one thermodynamic parameter is preserved. However, there is a gas transition between states, which is not an isoprocess, but which plays an important role in nature and technology. This is an adiabatic process. In this article, we consider it in more detail, focusing on what is the gas adiabatic exponent.
Adiabatic process
According to the thermodynamic definition, an adiabatic process is understood to mean such a transition between the initial and final states of the system as a result of which there is no heat exchange between the external environment and the system under study. Such a process is possible under the following two conditions:
- thermal conductivity between the environment and the system for one reason or another is low;
- the speed of the process is high, so the heat exchange does not have time to occur.
In technology, the adiabatic transition is used both for heating a gas during its sharp compression, and for its cooling during rapid expansion. In nature, the thermodynamic transition under consideration manifests itself when the air mass rises or falls along the hillside. Such ascents and descents lead to a change in the dew point in the air and to precipitation.
Poisson equation for the ideal gas adiabat
An ideal gas is a system in which particles move randomly at high speeds, do not interact with each other and are dimensionless. Such a model is very simple in terms of its mathematical description.
According to the definition of the adiabatic process, we can write the following expression in accordance with the first law of thermodynamics:
dU = -P * dV.
In other words, the gas, expanding or contracting, performs the work P * dV due to a corresponding change in its internal energy dU.
In the case of an ideal gas, if we use the equation of state (Clapeyron-Mendeleev law), then we can obtain the following expression:
P * V γ = const.
This equality is called the Poisson equation. People who are familiar with gas physics will notice that if γ is 1, then the Poisson equation will go over to the Boyle-Mariotte law (isothermal process). However, such a transformation of the equations is impossible, since γ for any type of ideal gas is greater than unity. The quantity γ (gamma) is called the adiabatic exponent of an ideal gas. Let us consider in more detail its physical meaning.
What is the adiabatic exponent?
The exponent γ, which appears in the Poisson equation for an ideal gas, is the ratio of the specific heat at constant pressure to a similar value, but already at a constant volume. In physics, the heat capacity is the amount of heat that must be transferred to a given system or taken from it so that it changes its temperature by 1 Kelvin. We denote the isobaric heat capacity by the symbol C P , and the isochoric heat capacity by the symbol C V. Then for γ the equality
γ = C P / C V.
Since γ is always more than one, it shows how many times the isobaric heat capacity of the studied gas system exceeds the similar isochoric characteristic.
Heat capacities CP and CV
To determine the adiabatic exponent, the meaning of the values of C P and C V should be well understood. To do this, we will conduct the following thought experiment: imagine that the gas is in a closed system in a vessel with solid walls. If the vessel is heated, then all the reported heat will ideally pass into the internal energy of the gas. In such a situation, equality will be true:
dU = C V * dT.
The value of C V determines the amount of heat that should be transferred to the system in order to heat it isochorically by 1 K.
Now suppose that the gas is in a vessel with a movable piston. During the heating process of such a system, the piston will move, providing constant pressure. Since the enthalpy of the system in this case will be equal to the product of the isobaric heat capacity and the temperature change, the first law of thermodynamics will take the form:
C P * dT = C V * dT + P * dV.
This shows that C P > C V , since in the case of an isobaric state change, it is necessary to expend heat not only to increase the temperature of the system, and hence its internal energy, but also to perform gas work during its expansion.
The value of γ for an ideal monatomic gas
The simplest gas system is the monatomic ideal gas. Suppose we have 1 mole of such a gas. Recall that in the process of isobaric heating, 1 mole of gas is only 1 Kelvin, it performs work equal to the value of R. This symbol is used to denote the universal gas constant. It is equal to 8.314 J / (mol * K). Applying the last expression in the previous paragraph for this case, we obtain the following equality:
C P = C V + R.
Where can I determine the value of isochoric heat capacity C V :
γ = C P / C V ;
C V = R / (γ-1).
It is known that for one mole of a monatomic gas the value of isochoric heat capacity is:
C V = 3/2 * R.
From the last two equalities, the adiabatic exponent follows:
3/2 * R = R / (γ-1) =>
γ = 5/3 ≈ 1.67.
Note that the quantity γ depends solely on the internal properties of the gas itself (on the polyatomicity of its molecules) and does not depend on the amount of substance in the system.
The dependence of γ on the number of degrees of freedom
The equation for the isochoric heat capacity of a monatomic gas was written above. The coefficient 3/2 that appeared in it is related to the number of degrees of freedom for one atom. He has the ability to move only in one of the three directions of space, that is, there are only translational degrees of freedom.
If the system is formed by diatomic molecules, then two more rotational degrees are added to the three translational ones. Therefore, the expression for C V takes the form:
C V = 5/2 * R.
Then the value of γ will be equal to:
γ = 7/5 = 1.4.
Note that in fact there is another vibrational degree of freedom in a diatomic molecule, but at temperatures of several hundred Kelvin it is not involved and does not contribute to the specific heat.
If gas molecules are made up of more than two atoms, then they will have 6 degrees of freedom. The adiabatic index will be equal to:
γ = 4/3 ≈ 1.33.
Thus, with an increase in the number of atoms in a gas molecule, γ decreases. If we plot the adiabat in the PV axes, we can see that the curve for a monatomic gas will behave more sharply than for a polyatomic one.
Adiabatic exponent for gas mixture
We have shown above that the quantity γ does not depend on the chemical composition of the gas system. However, it depends on the number of atoms that make up its molecules. Assume that the system consists of N components. The atomic fraction of component i in the mixture is equal to a i . Then, to determine the adiabatic index of the mixture, you can use the following expression:
γ = ∑ i = 1 N (a i * γ i ).
Where γ i is the quantity γ for the ith component.
For example, this expression can be used to determine γ of air. Since it consists of 99% diatomic molecules of oxygen and nitrogen, its adiabatic index should be very close to 1.4, which is confirmed by the experimental determination of this value.