Amagat's law explanation, examples, exercises

The Amagat's Law establishes that the total volume of a gas mixture is equal to the sum of the partial volumes that each gas that composes it would have, if it were alone and at 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 by Dalton.

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..

Article index

• 1 Formula
• 2 Examples
  • 2.1 Ideal gases and component volumes
• 3 Exercises
  • 3.1 Exercise 1
  • 3.2 Exercise 2
• 4 References

Formula

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

V_T = V₁ + V_two + V₃ +…. = ∑ V_i (T_m, P_m)

Where the letter V represents the volume, where V_T the total volume. The summation symbol serves as a compact notation. T_m And p_m are respectively the temperature and pressure of the mixture.

The volume of each gas is V_i and is called 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, the mixture of which 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.

-Various 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 such as 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..

Ideal gases and component volumes

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

P.V = nRT

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

V_i = n_iRT_m / P_m

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 it. The number of moles ni is:

n_i = P_m V_i / RT_m

While for the complete mix, n Is given by:

n = P_mV / RT_m

Dividing the expression for ni by the latter:

n_i / n = V_i / V

Solving for V_i:

V_i = (n_i / n) V

Therefore:

V_i = x_i V

Where x_i it is called Molar fraction y 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 ideal gas equation of state must be multiplied by this factor:

P.V = Z.nRT

Training

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 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:

n = 11 + 8 + 1 moles = 20 moles

The mole fraction of each gas is:

-Nitrogen: x _Nitrogen = 11/20

-Oxygen: x _Oxygen = 8/20

-Carbon dioxide: x _{Carbon dioxide} = 1/20

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

-Nitrogen: P_N = 760mm Hg. (11/20) = 418mm Hg; V_N = 10 liters. (11/20) = 5.5 liters.

-Oxygen: P_OR = 760mm Hg. (8/20) = 304mm Hg; V_N = 10 liters. (8/20) = 4.0 liters.

-Carbon dioxide: P_A-C = 760mm Hg. (1/20) = 38mm 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:

10 liters = 5.5 + 4.0 + 0.5 liters.

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:

V _Oxygen = n _Oxygen. RT_m / P_m = 50 mol × 8.314472 J / mol. K × 298.15 K / 101 325 Pa = 1.22 m³.

V _Nitrogen = n _Nitrogen. RT_m / P_m = 190 × 8.314472 J / mol. K × 298.15 K / 101 325 Pa = 4.66 m³.

In conclusion, the volume of the mixture is:

V_T = 1.22 + 4.66 m³ = 5.88 m³.

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.