When solving thermodynamic problems in physics, in which transitions between different states of an ideal gas arise, the Mendeleev-Clapeyron equation is an important reference point. In this article, we consider what this equation is and how it can be used to solve practical problems.
Real and perfect gases
The gas state of matter is one of the four existing state of aggregation of matter. Examples of pure gases are hydrogen and oxygen. Gases can mix with each other in arbitrary proportions. A well-known example of a mixture is air. These gases are real, but under certain conditions they can be considered ideal. An ideal gas is one that meets the following characteristics:
- The particles that form it do not interact with each other.
- Collisions between individual particles and between particles and vessel walls are absolutely elastic, that is, the momentum and kinetic energy are saved before and after the collision.
- Particles do not have volume, but have some mass.
All real gases at temperatures of the order of and above room temperature (more than 300 K) and at pressures of the order of and below one atmosphere (10 5 Pa) can be considered ideal.
Thermodynamic quantities describing the state of a gas
Thermodynamic quantities are understood to mean macroscopic physical characteristics that uniquely determine the state of the system. There are three basic values:
- temperature T;
- volume V;
- P. pressure
The temperature reflects the intensity of movement of atoms and molecules in the gas, that is, it determines the kinetic energy of the particles. This value is measured in Kelvin. To translate from degrees Celsius to Kelvin, you should use the equality:
T (K) = 273.15 + T ( o C).
Volume - the ability of each real body or system to occupy part of the space. It is expressed in SI in cubic meters (m 3 ).
Pressure is a macroscopic characteristic that on average describes the intensity of collisions of gas particles with the walls of a vessel. The higher the temperature and the higher the concentration of particles, the greater the pressure. It is expressed in pascals (Pa).
It will be shown below that the Mendeleev-Clapeyron equation in physics contains another macroscopic parameter - the amount of substance n. Under it is supposed the number of elementary units (molecules, atoms), which is equal to the Avogadro number (N A = 6.02 * 10 23 ). The amount of substance in moles is expressed.
Mendeleev-Clapeyron equation of state
We immediately write this equation, and then explain its meaning. This equation has the following general form:
P * V = n * R * T.
The product of pressure on the volume of an ideal gas is proportional to the product of the amount of substance in the system by the absolute temperature. The proportionality coefficient R is called the universal gas constant. Its value is 8.314 J / (mol * K). The physical meaning of the value of R lies in the fact that it is equal to the work that it does during the expansion of 1 mole of gas if it is heated by 1 K.
The recorded expression is also called the ideal gas equation of state. Its importance is that it does not depend on the chemical type of gas particles. So, it can be oxygen molecules, helium atoms or a gas-air mixture in general; for all these substances the equation under consideration will be valid.
It can be written in other forms. We give them:
P * V = m / M * R * T;
P = Ο / M * R * T;
P * V = N * k B * T.
Here m is the mass of the gas, Ο is its density, M is the molar mass, N is the number of particles in the system, k B is the Boltzmann constant. Depending on the conditions of the problem, you can use any form of equation writing.
A brief history of the equation
The Clapeyron-Mendeleev equation was first obtained in 1834 by Emil Clapeyron as a result of a generalization of the laws of Boyle-Mariotte and Charles-Gay-Lussac. Moreover, the Boyle-Mariotte law was already known in the second half of the 17th century, and the Charles-Gay-Lussac law was first published at the beginning of the 19th century. Both laws describe the behavior of a closed system with a fixed single thermodynamic parameter (temperature or pressure).
The merit of D. Mendeleev in writing the modern form of the ideal gas equation is that for the first time he replaced a series of constants with one single value R.
Note that at present, the Clapeyron-Mendeleev equation can be obtained theoretically if we consider the system from the point of view of statistical mechanics and apply the principles of molecular kinetic theory.
Special cases of the equation of state
There are 4 particular laws that follow from the equation of state of an ideal gas. Let us dwell briefly on each of them.
If in a closed system with gas to maintain a constant temperature, then any increase in pressure in it will cause a proportional decrease in volume. This fact can be written mathematically in the following form:
P * V = const at T, n = const.
This law bears the names of scientists Robert Boyle and Edm Marriott. The graph of the function P (V) is a hyperbola.
If the pressure is fixed in a closed system, then any increase in temperature in it will lead to a proportional increase in volume, that is:
V / T = const at P, n = const.
The process described by this equation is called isobaric. He bears the names of the French scientists Charles and Gay-Lussac.
If the volume does not change in a closed system, then the process of transition between the states of the system is called isochoric. During it, any increase in pressure leads to a similar increase in temperature:
P / T = const at V, n = const.
This equality is called the Gay-Lussac Act.
The graphs of isobaric and isochoric processes are straight lines.
Finally, if macroscopic parameters (temperature and pressure) are fixed, then any increase in the amount of substance in the system will lead to a proportional increase in its volume:
n / V = ββconst at P, T = const.
This equality is called the Avogadro principle. It underlies Dalton's law for ideal gas mixtures.
The solution of the problem
The Mendeleev-Clapeyron equation is conveniently used to solve various practical problems. We give an example of one of them.
Oxygen weighing 0.3 kg is in a 0.5 m 3 cylinder at a temperature of 300 K. How will the gas pressure change if the temperature is increased to 400 K?
Assuming that the oxygen in the cylinder is ideal gas, we use the equation of state to calculate the initial pressure, we have:
P 1 * V = m / M * R * T 1 ;
P 1 = m * R * T 1 / (M * V) = 0.3 * 8.314 * 300 / (32 * 10 -3 * 0.5) = 46766.25 Pa.
Now we calculate the pressure at which the gas will be in the cylinder, if you raise the temperature to 400 K, we get:
P 2 = m * R * T 2 / (M * V) = 0.3 * 8.314 * 400 / (32 * 10 -3 * 0.5) = 62355 Pa.
The change in pressure during heating will be:
ΞP = P 2 - P 1 = 62355 - 46766.25 = 15588.75 Pa.
The obtained ΞP value corresponds to 0.15 atmosphere.