Pressure is one of the three main thermodynamic macroscopic parameters of any gas system. In this article, we consider gas pressure formulas in the ideal gas approximation and in the framework of the molecular-kinetic theory.
Ideal gases
Every student knows that gas is one of four (including plasma) aggregate states of matter in which particles do not have certain positions and move randomly in all directions with the same probability. Proceeding from such a structure, gases retain neither volume nor shape at the slightest external force impact on them.
In any gas, the average kinetic energy of its particles (atoms, molecules) is greater than the energy of intermolecular interaction between them. In addition, the distances between particles far exceed their own sizes. If molecular interactions and particle sizes can be neglected, then such a gas is called ideal.
In an ideal gas, there is only one type of interaction — elastic collisions. Since the particle size is negligible in comparison with the distances between them, the probability of particle-particle collisions will be low. Therefore, in an ideal gas system, there are only collisions of particles with the walls of the vessel.
All real gases with good accuracy can be considered ideal if the temperature in them is higher than room temperature and the pressure does not greatly exceed atmospheric pressure.
Cause of gas pressure
Before writing down the formulas for calculating gas pressure, it is necessary to understand why it arises in the system under study.
According to the physical definition, pressure is a value equal to the ratio of the force that perpendicularly affects a certain area to the area of this area, that is:
P = F / S
We noted above that there is only one single type of interaction in an ideal gas system - these are absolutely elastic collisions. As a result of them, the particles transmit the momentum Δp to the walls of the vessel during the collision time Δt. For this case, Newton’s second law applies:
F * Δt = Δp
It is the force F that leads to the appearance of pressure on the walls of the vessel. The value of F itself from the collision of one particle is insignificant, however, the number of particles is huge (≈ 10 23 ), so they together create a significant effect, which manifests itself in the form of the presence of pressure in the vessel.
The ideal gas pressure formula from molecular kinetic theory
In explaining the concept of an ideal gas, the basic principles of molecular kinetic theory (MKT) were announced above. This theory is based on statistical mechanics. It was developed in the second half of the 19th century by such scientists as James Maxwell and Ludwig Boltzmann, although Bernoulli laid the foundation for it in the first half of the 18th century.
According to Maxwell-Boltzmann statistics, all particles in the system move at different speeds. In this case, there is a small fraction of particles whose velocity is practically zero, and the same proportion of particles having huge velocities. If we calculate the root mean square velocity, then it will take a certain value, which remains constant over time. The mean square velocity of the particles uniquely determines the temperature of the gas.
Using the MKT approximations (non-interacting dimensionless and randomly moving particles), we can obtain the following formula for the gas pressure in the vessel:
P = N * m * v 2 / (3 * V)
Here N is the number of particles in the system, V is the volume, v is the root mean square velocity, m is the mass of one particle. If all the indicated values are determined, then substituting them in SI units into this equality, we can calculate the gas pressure in the vessel.
Pressure formula from the equation of state
In the mid 30-ies of the XIX century, the French engineer Emile Clapeyron, generalizing the experimental experience accumulated before him on studying the behavior of gases during various isoprocesses, received an equation, which is now called the universal equation of state of an ideal gas. The corresponding formula is:
P * V = n * R * T
Here n is the amount of substance in moles, T is the temperature on an absolute scale (in kelvins). The value of R is called the universal gas constant, which was introduced into this equation by the Russian chemist D.I. Mendeleev, therefore, the written expression is also called the Clapeyron-Mendeleev law.
From the equation above it is easy to obtain the gas pressure formula:
P = n * R * T / V
Equality suggests that pressure increases linearly with temperature at a constant volume and increases along a hyperbole with a decrease in volume at a constant temperature. These dependencies are reflected in the laws of Gay-Lussac and Boyle-Marriott.
If we compare this expression with the formula written above, which follows from the provisions of the MCT, we can establish a relationship between the kinetic energy of one particle or the entire system and the absolute temperature.
Gas pressure
Answering the question of how to find the gas pressure and formulas, we did not say anything about whether the gas is pure or whether it is a gas mixture. In the case of the formula for P, which follows from the Clapeyron equation, there is no connection with the chemical composition of the gas; in the case of the expression for P from the MCT, this connection is present (parameter m). Therefore, when using the last formula for a gas mixture, it becomes unclear what mass of particles to choose.
When you need to calculate the pressure of a mixture of ideal gases, you should do one of two methods:
- Calculate the average particle mass m, or, preferably, the average molar mass M, based on the atomic percent of each gas in the mixture;
- Take advantage of Dalton's law. It states that the pressure in the system is equal to the sum of the partial pressures of all its components.
Task example
It is known that the average velocity of oxygen molecules is 500 m / s. It is necessary to determine the pressure in a vessel with a volume of 10 liters, in which there are 2 mol of molecules.
The answer to the problem can be obtained by using the formula for P from MKT:
P = N * m * v 2 / (3 * V)
It contains two inconvenient parameters for performing calculations - these are m and N. We transform the formula as follows:
m is M / N A ;
n = N / N A ;
m * N = M * n;
P = M * n * v 2 / (3 * V)
The volume of the vessel in cubic meters is 0.01 m 3 . The molar mass of the oxygen molecule M is 0.032 kg / mol. Substituting these values into the formula, as well as the velocity v and the amount of substance n from the conditions of the problem, we come to the answer: P = 533333 Pa, which corresponds to a pressure of 5.3 atmospheres.