At the end of the XVIII and in the first half of the XIX century, scientists from different countries actively studied the behavior of gaseous, liquid and solid matter under various external conditions, relying in their studies on the ideas about the atomic and molecular structure of matter. One of these scientists was the British John Dalton. The law for a gas mixture that currently bears his last name is considered in this article.
Special conditions
Before formulating the Dalton law for a mixture of gases, you should deal with one of the concepts. This is very important, since only for such a substance is this law valid. It's about perfect gas. What is it?
An ideal gas is gas for which the following requirements are true:
- the sizes of molecules and atoms in it are so small that they can be considered material points having zero volume;
- molecules and atoms do not interact with each other.
Thus, an ideal gas is a combination of material points moving randomly. Their speed and mass uniquely determine the temperature of the entire mixture. The pressure that the test substance exerts on the walls of the vessel depends on such macroscopic parameters as temperature, volume of the vessel and the number of molecules.
For such a gas model, the equality
P * V = n * R * T
It is called the equation of state and combines pressure (P), temperature (T), volume (V) and the amount of substance in moles (n). The value of R is the coefficient of proportionality, which is equal to 8.314 J / (K * mol).
The amazing thing about this formula is that it does not include a single parameter that would depend on the chemical nature of molecules and atoms.
Almost any gas and its mixture can be considered ideal if the temperature is not too low and the pressure is not too high. Note! Room temperature and atmospheric pressure fall within these limits.
Partial pressure
Dalton's law for a mixture of ideal gases implies knowledge of yet another macroscopic parameter - partial pressure.
Suppose that there is some mixture consisting of 2 components, for example, H 2 and He. This mixture is in a vessel of a specific volume and creates a certain pressure on its walls. Since hydrogen molecules and helium atoms do not interact with each other, then for any calculations of macroscopic characteristics, both components can be considered independently of each other.
The partial pressure of a component is the pressure that it creates independently of the other components of the mixture, occupying the volume provided to it. In this example, we can talk about the partial pressure of H 2 and the same characteristics for He. This value is expressed in pascals and is designated for the i-th component as P i .
Gas mixtures and Dalton's law
John Dalton, studying various volatiles, including water vapor, at different temperatures and pressures, came to the following conclusion: the pressure of a mixture of absolutely any such substances in any proportions is equal to the sum of the partial pressures of all its components. This formulation is called the Dalton law for the pressure of a mixture of gases and is written by the following mathematical equality:
P tot = ∑ i (P i )
Here P tot is the total pressure of the mixture.
This fairly simple law holds true only for ideal gas mixtures whose components do not chemically interact with each other.
Another wording of Dalton's law
Dalton's law for a mixture of gases can be expressed not only through partial pressures, but also through the mole fractions of each component. We get the corresponding formula.
Since each component behaves independently of the others in the gas mixture, then the equation of state can be written for it:
P i * V = n i * R * T
This equation is valid for each ith component, since for all of them the temperature T and volume V are the same. The value of n i is the number of moles of component i in the mixture.
We now express the partial pressure, and divide it by the total pressure of the whole mixture, then we get:
P i / P tot = n i * R * T / V / (n * R * T / V) = n i / n
Here n is the total amount of substance in the whole mixture. It can be obtained by summing all n i . The ratio n i / n is called the mole fraction of component i in the mixture. It is usually denoted by x i . Through mole fractions, Dalton's law is written as follows:
P i = P tot * x i
The mole fraction is often represented as atomic percent components in the mixture. For example, 21% O 2 in air indicates that its molar fraction is 0.21, that is, every fifth molecule of air is oxygen.
Application of the considered law to solve the problem
It is known that a gas mixture of oxygen and nitrogen is under a pressure of 5 atmospheres in a cylinder. Knowing that it contains 10 mol of nitrogen and 3 mol of oxygen, it is necessary to determine the partial pressure of each substance.
To answer the question of the problem, we first find the total amount of substance:
n = n N2 + n O2 = 10 + 3 = 13 mol
Now you can calculate the molar fraction of each component in the mixture. We have:
x N2 = n N2 / n = 10/13 = 0.7692
x O2 = n O2 / n = 3/13 = 0.2308
Using the formula of Dalton’s law through the mole fraction of the component, we calculate the partial pressure of each gas in the cylinder:
P N2 = 5 * 0.7692 = 3.846 atm.
P O2 = 5 * 0.2308 = 1.154 atm.
As can be seen from the figures obtained, the sum of these pressures will give 5 atmospheres. The partial pressure of each gas is directly proportional to its molar fraction in the mixture.