An ideal gas is a successful model in physics, which allows one to study the behavior of real gases under various conditions. In this article, we will examine in more detail what an ideal gas is, with a formula which describes its state, and also how its energy is calculated.
Ideal gas concept
We are talking about gas, which is formed by particles that do not have size and do not interact with each other. Naturally, not a single gas system satisfies absolutely exactly the noted conditions. Nevertheless, many real fluid substances approach these conditions with an accuracy sufficient to solve many practical problems.
If in a gas system the distance between particles is much greater than their size, and the potential interaction energy is much less than the kinetic energy of translational and vibrational movements, then such a gas can rightly be considered ideal. For example, such is air, methane, noble gases at low pressures and high temperatures. On the other hand, water vapor, 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.
The 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 XVII 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, which now bears his last name.
All the achievements of scientists listed above led Emil Clapeyron in 1834 to write the equation of state of an ideal gas. We give this equation:
P × V = n × R × T.
The importance of the written equality is as follows:
- It is valid for any ideal gases regardless of their chemical composition.
- it connects three basic 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 put the value of P constant (isobaric process).
The universal law also allows us to obtain a formula for any thermodynamic parameter of the system. For example, the formula for the volume of an ideal gas is written as:
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 MKT. In accordance with them, each gas particle moves along a straight path until it meets the vessel wall. After an absolutely elastic collision with it, it moves along another straight path, preserving the kinetic energy that it had before the collision.
All gas particles have velocities in accordance with Maxwell-Boltzmann statistics. An important microscopic characteristic of the system is the average speed, which remains constant over time. Thanks to this fact, it is possible to calculate the temperature of the system. The corresponding ideal gas formula is:
m × v 2/2 = 3/2 × k B × T.
Where m is the particle mass, k B is the Boltzmann constant.
From MKT for ideal gas follows the formula for absolute pressure. It has the form:
P = N × m × v 2 / (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.
System internal energy
By 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 multiplying both sides of the equality for the kinetic energy of one particle by their number N in the system, that is:
N × m × v 2/2 = 3/2 × k B × T × N.
Then we get:
U = 3/2 × k B × T × N = 3/2 × n × R × T.
We got a logical conclusion: internal energy is directly proportional to the absolute temperature in the system. In fact, the expression obtained for U is valid only for a monoatomic 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:
U 2 = 5/2 × n × R × T.
If the system consists of polyatomic molecules, then the following expression is true:
U n> 2 = 3 × n × R × T.
The last two formulas also take into account rotational degrees of freedom.
Task example
Two moles of helium are in a vessel of 5 liters at a temperature of 20 o C. It is necessary to determine the pressure and internal energy of the gas.
First of all, we will transfer all known quantities to SI:
n = 2 mol;
V = 0.005 m 3 ;
T = 293.15 K.
Helium pressure is calculated by 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, it can be considered ideal at this pressure.
For a monatomic ideal gas, the formula for U has the form:
U = 3/2 × n × R × T.
Substituting into it the values of temperature and the amount of substance, we obtain the helium energy: U = 7311.7 J.