When solving thermodynamic problems in physics, in which there are transitions between different states of an ideal gas, the Mendeleev-Clapeyron equation is an important reference point. In this article, we will consider what this equation is and how it can be used to solve practical problems.
Real and ideal gases
The gaseous state of matter is one of the existing four aggregate states 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 isthe momentum and kinetic energy before and after the collision is conserved.
- Particles have no volume, but 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 (105Pa) can be considered ideal.
Thermodynamic quantities describing the state of a gas
Thermodynamic quantities are macroscopic physical characteristics that uniquely determine the state of the system. There are three base values:
- Temperature T;
- volume V;
- pressure P.
Temperature reflects the intensity of movement of atoms and molecules in a gas, that is, it determines the kinetic energy of particles. This value is measured in Kelvin. To convert from degrees Celsius to Kelvin, use the equation:
T(K)=273, 15 + T(oC).
Volume - the ability of each real body or system to occupy part of the space. Expressed in SI in cubic meters (m3).
Pressure is a macroscopic characteristic that, on average, describes the intensity of collisions of gas particles with the vessel walls. The higher the temperature and the higher the particle concentration, the higher the pressure will be. It is expressed in pascals (Pa).
Further it will be shown that the Mendeleev-Clapeyron equation in physics contains one more macroscopic parameter - the amount of substance n. Under it is the number of elementary units (molecules, atoms), which is equal to the Avogadro number (NA=6,021023). The amount of a substance is expressed in moles.
Mendeleev-Clapeyron Equation of State
Let's write this equation right away, and then explain its meaning. This equation has the following general form:
PV=nRT.
The product of pressure and the volume of an ideal gas is proportional to the product of the amount of substance in the system and the absolute temperature. The proportionality factor R is called the universal gas constant. Its value is 8.314 J / (molK). The physical meaning of R is that it is equal to the work that 1 mol of gas does when expanding if it is heated by 1 K.
The written expression is also called the ideal gas equation of state. Its importance lies in the fact that it does not depend on the chemical type of gas particles. So, it can be oxygen molecules, helium atoms, or a gaseous air mixture in general, for all these substances the equation under consideration will be valid.
It can be written in other forms. Here's them:
PV=m / MRT;
P=ρ / MRT;
PV=NkB T.
Here m is the mass of the gas, ρ is its density, M is the molar mass, N is the number of particles in the system, kB is Boltzmann's constant. Depending on the condition of the problem, you can use any form of writing the equation.
A brief history of getting the equation
The Clapeyron-Mendeleev equation was firstobtained in 1834 by Emile Clapeyron as a result of a generalization of the laws of Boyle-Mariotte and Charles-Gay-Lussac. At the same time, 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 at a fixed one thermodynamic parameter (temperature or pressure).
D. Mendeleev's merit in writing the modern form of the ideal gas equation is that he first replaced a number of constants with a 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 provisions of molecular kinetic theory.
Special cases of the equation of state
There are 4 particular laws that follow from the equation of state for an ideal gas. Let's dwell briefly on each of them.
If a constant temperature is maintained in a closed system with gas, then any increase in pressure in it will cause a proportional decrease in volume. This fact can be written mathematically as follows:
PV=const at T, n=const.
This law bears the names of scientists Robert Boyle and Edme Mariotte. 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, thenyes:
V / T=const at P, n=const.
The process described by this equation is called isobaric. It 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 with V, n=const.
This equality is called Gay-Lussac's law.
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 a substance in the system will lead to a proportional increase in its volume:
n / V=const when P, T=const.
This equality is called the Avogadro principle. It underlies D alton's law for ideal gas mixtures.
Problem Solving
The Mendeleev-Clapeyron equation is convenient to use for solving various practical problems. Here is an example of one of them.
Oxygen with a mass of 0.3 kg is in a cylinder with a volume of 0.5 m3at a temperature of 300 K. How will the gas pressure change if the temperature is increased to 400 K?
Assuming the oxygen in the cylinder to be an ideal gas, we use the equation of state to calculate the initial pressure, we have:
P1 V=m / MRT1;
P1=mRT1 / (MV)=0, 38, 314300 / (3210-3 0.5)=46766.25Pa.
Now we calculate the pressure at which the gas will be in the cylinder, if we raise the temperature to 400 K, we get:
P2=mRT2 / (MV)=0, 38, 314400 / (3210-3 0, 5)=62355 Pa.
Change in pressure during heating will be:
ΔP=P2- P1=62355 - 46766, 25=15588, 75 Pa.
The resulting value of ΔP corresponds to 0.15 atmospheres.
