Non-Reactive Ideal Gas Mixtures: Dalton’s Law and Amagat’s Law

The thermodynamic behaviour of a mixture of gases depends upon the individual properties of its constituent gases. Because substances in the gaseous phase are so miscible with each other, there is no limit to the number of different gaseous mixtures. Furthermore, individual constituents of gaseous mixtures often react chemically with each other, and these reactions introduce another factor that exerts a strong influence on the properties of a gas.

A gas mixture is frequently treated as if it consisted of a single component rather than many properties of the individual constituents of an inert gas tend to be submerged, so that the gas behaves, in certain ways, as though it were a single, pure substance. The main constituents of substance are, for example, oxygen, nitrogen, argon and water vapour. But air is commonly said to have a molecular weight of 28.97 kg/K-mol, even though this figure represents a composite value based on the molecular weights and the proportions of the constituent species.

In the first case of mixtures of ideal or perfect gases, equations are derived which express the properties of mixtures in terms of properties of the constituents. Mixtures are treated in which no chemical reactions, condensa­tion or evaporation take place. The derived expressions apply, in general to gas mixtures over a wide range of temperatures and pressures. Although the study centers about perfect gases, mixtures containing non-perfect gases often show only negligible deviation from perfect gas behaviour.

Dalton’s Law of Partial Pressures:

It was John Dalton (1766 – 1844) who first stated that the total pressure p_m exerted by a mixture of gases is the sum of the pressures which each gas (or vapour), would exert were it to occupy the vessel alone at the volume V_m and temperature T_m of the mixture true only of ideal gases. Thus, if p_x, p_y, p_z…….. represent respectively the individual pressures of the mixed gases x, y, z.

Applying the ideal or perfect gas equation to a component x and to the mixture, we get,

Thus volume fraction is the ratio of the partial pressure to the total pressure the mixture. Figure 7.1 shows the mixtures and constituents.

Note:

Here that the individual pressure exerted by a real gas is not necessarily given by these equations.

Amagat’s Law and Volumetric Analysis:

Amagat’s Law states that the total volume of a mixture of gases is equal to the sum of the volumes that would he occupied by each component at the pressure p_m and temperature T_m of the mixture.

This law is found to be true for ideal gases only, though it is widely used for all kinds of gaseous components.

Figure 7.2 shows the mixture of gases following this Law.

It may be noted here that Dalton’s Law and Amagat’s Law are coincident with the kinetic theory of gases. These laws assume that no intermolecular forces exist in a mixture of gases and that each constituent acts as if no other constituents were present.

When this mixture is analyzed, the perfect gas law leads to –

 

Gravimetric Analysis:

Constituents or components of the mixture of gases are the individual gases and vapours in a mixture. Generally, the description of a mixture of any substances like solids, liquids or gases—can be given by a gravimetric analysis which is the percentage or fraction by weight or mass of each constituent.

Thus if a, b and c are the constituents in a mixture and they are having weights or masses as m_a, m_b and m_c respectively, then the total mass of the mixture is –

Conversion between Gravimetric and Volumetric Analysis:

For converting the gravimetric analysis of the mixture of gases into volumetric analysis and vice-versa, two tables are given below.

(a) Gravimetric to Volumetric Analysis-

(b) Volumetric to Gravimetric Analysis-

The use of these tables will be best understood when the numerical problems are solved.

Molecular Weight and Gas Constant for Mixture:

Consider the mixture of three gases x, y, and z having the molecular weights as M_x, M_y and M_z. Let the moles of each of these gases be -n_x, n_y, and n_z.

Internal Energy, Enthalpy, Specific Heats and Entropy of Mixture of Ideal Gases:

According to the Gibbs-Dalton Law, an extensive property of a mixture of perfect gases is the sum of the contribu­tions of the individual constituents. Properties of a mixture can be expressed on either a mass or a mole basis and are referred to as per kg or K mol, respectively. Expressions for some of the extensive properties are presented here on a mass basis. The internal energy per unit mass of a mixture is given by –

Where m_a, m_b and m_c are the masses of gases a, b and c in a mixture of 1 kg and U_a, U_b and U_c are the internal energies of gases a, b and c in a mixture of 1 kg.

We know that internal energy is a function of temperature and for constant volume specific heats of individual gas in mixture, you can calculate the specific heat at constant volume for the mixture is given by –

Processes of Gaseous Mixture:

There are two basic mixing processes for ideal gases:

1. Mixing Process when Temperature of Each Gas is Same:

Consider a system of gases as shown in Fig. 7.3 (a) and (b).

Let the system be ideally insulated so that the mixing of two gases is adiabatic mixing.

Let there are two gases a and b having the states as p_a, T_a, m_a and p_b, T_b, m_b as shown in Fig.7.3, Here, T_a = T_b. These states are shown on T-s diagram in Fig 7.4.

Let p_a > p_b so that the pressure of the mixture is p_m and p_a > p_m and p_m > p_b and t_m = t_a = t_b. After mixing, entropy of gas (a) increases and entropy of gas (b) decreases. Let the state of the mixture be given by the point m.

Then by Daltons-Gibbs law, we have the entropy of the mixture of gases is the sum of the partial entropies of the constituents and the partial entropy is calculated as if each constituent alone the occupied the total volume of the mixture at the temperature of the mixture.

Thus we observe from this equation that the increase in entropy for a mixture depends only on the number of moles of the gases in the mixture and does not depend on the gases themselves. Mixing of gases is an irreversible process and therefore the entropy increases according to second law of thermodynamics.

Again, we observe from these equations that even if the gases approach to similarity (same gases) but the pressure are different, the change of entropy does not become zero.

2. Different Gases at Different Initial Pressures and Temperatures:

The system considered for this problem may be represented by the two vessels, each containing an ideal gas, at different pressures and temperatures. Let these vessels be connected to each other by means of pipe containing a valve as shown in Fig. 7.5.

When the valve is open, then the gases will be allowed to diffuse and the pressure and temperatures of the mixture will be in between p₁ and p₂ and T₁ and T₂.

Let m = Mass of the mixture

= (m₁ + m₂)

Internal energy is a function of the temperature and the total internal energy is the sum of the total internal energies of the individual gases.

In this equation, T₁ and p₁ and T₁ and p₂ are the temperature and pressure before mixing and after mixing. T₂ represents the mixture temperature and p₂ represents the partial pressure of the constituent in the mixture after mixing.

After calculating the changes of entropy of each constituent, addition of all changes gives the change of entropy due to mixing.