An ideal gas is a successful model in physics that allows you to study the behavior of real gases under various conditions. In this article, we will take a closer look at what an ideal gas is, what formula describes its state, and also how its energy is calculated.
Ideal gas concept
This is a gas, which is formed by particles that do not have a size and do not interact with each other. Naturally, not a single gas system satisfies the absolutely precisely noted conditions. However, many real fluid substances approach these conditions with sufficient accuracy to solve many practical problems.
If in a gas system the distance between particles is much greater than their size, and the potential energy of interaction is much less than the kinetic energy of translational and oscillatory motions, then such a gas is rightly considered ideal. For example, such is air, methane, noble gases at low pressures and high temperatures. On the other hand, watersteam, even at low pressures, does not satisfy the concept of an ideal gas, since the behavior of its molecules is greatly influenced by hydrogen intermolecular interactions.
Equation of state of an ideal gas (formula)
Humanity has been studying the behavior of gases using a scientific approach for several centuries. The first breakthrough in this area was the Boyle-Mariotte law, obtained experimentally at the end of the 17th century. A century later, two more laws were discovered: Charles and Gay Lussac. Finally, at the beginning of the 19th century, Amedeo Avogadro, studying various pure gases, formulated the principle that now bears his last name.
All the achievements of scientists listed above led Emile Clapeyron in 1834 to write the equation of state for an ideal gas. Here is the equation:
P × V=n × R × T.
The importance of the recorded equality is as follows:
- it is true for any ideal gases, regardless of their chemical composition.
- it links three main thermodynamic characteristics: temperature T, volume V and pressure P.
All of the above gas laws are easy to obtain from the equation of state. For example, Charles's law automatically follows from Clapeyron's law if we set the value of P constant (isobaric process).
The universal law also allows you to obtain a formula for any thermodynamic parameter of the system. For example, the formula for the volume of an ideal gas is:
V=n × R × T / P.
Molecular Kinetic Theory (MKT)
Although the universal gas law was obtained purely experimentally, there are currently several theoretical approaches leading to the Clapeyron equation. One of them is to use the postulates of the MKT. In accordance with them, each particle of gas moves along a straight path until it meets the wall of the vessel. After a perfectly elastic collision with it, it moves along a different straight trajectory, retaining the kinetic energy it had before the collision.
All gas particles have velocities according to Maxwell-Boltzmann statistics. An important microscopic characteristic of the system is the average velocity, which remains constant in time. Thanks to this fact, it is possible to calculate the temperature of the system. The corresponding formula for an ideal gas is:
m × v2 / 2=3 / 2 × kB × T.
Where m is the mass of the particle, kB is the Boltzmann constant.
From the MKT for an ideal gas follows the formula for absolute pressure. It looks like:
P=N × m × v2 / (3 × V).
Where N is the number of particles in the system. Given the previous expression, it is not difficult to translate the formula for absolute pressure into the universal Clapeyron equation.
Internal energy of the system
According to the definition, an ideal gas has only kinetic energy. It is also its internal energy U. For an ideal gas, the energy formula U can be obtained by multiplyingboth parts of the equality for the kinetic energy of one particle per their number N in the system, that is:
N × m × v2 / 2=3 / 2 × kB × T × N.
Then we get:
U=3 / 2 × kB × T × N=3 / 2 × n × R × T.
We got a logical conclusion: the internal energy is directly proportional to the absolute temperature in the system. In fact, the resulting expression for U is valid only for a monatomic gas, since its atoms have only three translational degrees of freedom (three-dimensional space). If the gas is diatomic, then the formula for U will take the form:
U2=5 / 2 × n × R × T.
If the system consists of polyatomic molecules, then the following expression is true:
Un>2=3 × n × R × T.
The last two formulas also take rotational degrees of freedom into account.
Example problem
Two moles of helium are in a 5 liter vessel at a temperature of 20 oC. It is necessary to determine the pressure and internal energy of the gas.
First of all, let's convert all known quantities to SI:
n=2 mol;
V=0.005 m3;
T=293.15 K.
Helium pressure is calculated using the formula from Clapeyron's law:
P=n × R × T/V=2 × 8.314 × 293.15 / 0.005=974,899.64 Pa.
The calculated pressure is 9.6 atmospheres. Since helium is a noble and monatomic gas, at this pressure it can beconsidered ideal.
For a monatomic ideal gas, the formula for U is:
U=3 / 2 × n × R × T.
Substituting the values of temperature and amount of substance into it, we get the energy of helium: U=7311.7 J.
