AMAGAT'S LAW: EXPLANATION, EXAMPLES, EXERCISES - CHEMISTRY - 2025

The Amagat's Law states that the total volume of a mixture of gases is equal to the sum of the partial volumes each gas that would include, if alone and the pressure and temperature of the mixture.

It is also known as the law of partial or additive volumes and its name is due to the French physicist and chemist Emile Hilaire Amagat (1841-1915), who formulated it for the first time in 1880. It is analogous in volume to the law of partial pressures of Dalton.

Air in the atmosphere and in balloons can be treated as an ideal gas mixture, to which Amagat's law can be applied. Source: PxHere.

Both laws hold exactly in ideal gas mixtures, but they are approximate when applied to real gases, in which the forces between molecules play a prominent role. On the other hand, when it comes to ideal gases, the molecular attractive forces are negligible.

Formula

In mathematical form, Amagat's law takes the form:

V _T = V ₁ + V ₂ + V ₃ +…. = ∑ V _i (T _m, P _m)

Where the letter V represents the volume, where V _{T is} the total volume. The summation symbol serves as a compact notation. T _m and P _m are the temperature and pressure of the mixture, respectively.

The volume of each gas is V _i and is called the component volume. It is important to note that these partial volumes are mathematical abstractions and do not correspond to the real volume.

In fact, if we left only one of the gases in the mixture in the container, it would immediately expand to occupy the total volume. However, Amagat's law is very useful, because it facilitates some calculations in gas mixtures, giving good results especially at high pressures.

Examples

Gas mixtures abound in nature, to begin with, living beings breathe a mixture of nitrogen, oxygen and other gases in a lower proportion, so this is a very interesting gas mixture to characterize.

Here are some examples of gas mixtures:

-The air in the Earth's atmosphere, whose mixture can be modeled in various ways, either as an ideal gas or with one of the models for real gases.

-Gas engines, which are internal combustion, but instead of using gasoline they use a natural gas-air mixture.

-The carbon monoxide-dioxide mixture that gasoline engines expel through the exhaust pipe.

-The hydrogen-methane combination that abounds in the gas giant planets.

Interstellar gas, a mixture consisting mostly of hydrogen and helium that fills the space between stars.

-Diverse mixtures of gases at an industrial level.

Of course, these gaseous mixtures generally do not behave as ideal gases, since the pressure and temperature conditions are far from those established in that model.

Astrophysical systems like the Sun are far from ideal, since variations in temperature and pressure appear in the layers of the star and the properties of matter change as it evolves over time.

Gas mixtures are determined experimentally with different devices, such as the Orsat analyzer. For exhaust gases there are special portable analyzers that work with infrared sensors.

There are also devices that detect gas leaks or are designed to detect certain gases in particular, used mainly in industrial processes.

Figure 2. Old-fashioned gas analyzer for detecting vehicle emissions, specifically carbon monoxide and hydrocarbon emissions. Source: Wikimedia Commons.

Ideal gases and component volumes

Important relationships between the variables in the mixture can be derived by using Amagat's law. Starting from the ideal gas equation of state:

Next, the volume of a component i of the mixture is solved, which can then be written as follows:

Where n _i represents the number of moles of gas present in the mixture, R is the gas constant, T _m is the temperature of the mixture and P _m the pressure of the mixture. The number of moles ni is:

While for the complete mix, n is given by:

Dividing the expression for nor by the latter:

Solving for V _i:

Thus:

Where x _i is called the mole fraction and is a dimensionless quantity.

The mole fraction is equivalent to the volume fraction V _i / V and it can be shown that it is also equivalent to the pressure fraction P _i / P.

For real gases, another appropriate equation of state must be used or the compressibility factor or compression factor Z must be used. In this case, the equation of state for ideal gases must be multiplied by this factor:

Exercises

Exercise 1

The following gas mixture is prepared for a medical application: 11 moles of nitrogen, 8 moles of oxygen and 1 mole of carbon dioxide. Calculate the partial volumes and the partial pressures of each gas present in the mixture, if it must have a pressure of 1 atmosphere in 10 liters.

1 atmosphere = 760 mm Hg.

Solution

The mixture is considered to conform to the ideal gas model. The total number of moles is:

The mole fraction of each gas is:

-Nitrogen: x _Nitrogen = 11/20

-Oxygen: x _Oxygen = 8/20

-Carbonic anhydride: x _{Carbonic anhydride} = 1/20

The pressure and the partial volume of each gas are calculated respectively as follows:

-Nitrogen: P _N = 760 mm Hg. (11/20) = 418 mm Hg; V _N = 10 liters. (11/20) = 5.5 liters.

-Oxygen: P _O = 760 mm Hg (8/20) = 304 mm Hg;. V _N = 10 liters. (8/20) = 4.0 liters.

-Carbonic anhydride: P _A-C = 760 mm of Hg. (1/20) = 38 mm of Hg; V _N = 10 liters. (1/20) = 0.5 liters.

Indeed, it can be seen that what was said at the beginning is true: that the volume of the mixture is the sum of the partial volumes:

Exercise 2

50 moles of oxygen are mixed with 190 moles of nitrogen at 25 ° C and one atmosphere of pressure.

Apply Amagat's law to calculate the total volume of the mixture, using the ideal gas equation.

Solution

Knowing that 25 ºC = 298.15 K, 1 atmosphere of pressure is equivalent to 101325 Pa and the gas constant in the International System is R = 8.314472 J / mol. K, the partial volumes are:

In conclusion, the volume of the mixture is:

References

1. Borgnakke. 2009. Fundamentals of Thermodynamics. 7th Edition. Wiley and Sons.
2. Cengel, Y. 2012. Thermodynamics. 7th Edition. McGraw Hill.
3. Chemistry LibreTexts. Amagat's Law. Recovered from: chem.libretexts.org.
4. Engel, T. 2007. Introduction to Physicochemistry: Thermodynamics. Pearson.
5. Pérez, S. Real gases. Recovered from: depa.fquim.unam.mx.