Question

The volume fraction of a gas \(\mathrm{A}\) in a mixture is defined by the equation $$ \text { volume fraction } \mathrm{A}=\frac{V_{\mathrm{A}}}{V} $$ where \(V\) is the total volume and \(V_{A}\) is the volume that gas A would occupy alone at the same temperature and pressure. Assuming ideal gas behavior, show that the volume fraction is the same as the mole fraction. Explain why the volume fraction differs from the mass fraction.

Short answer

Answer: In an ideal gas, the volume fraction is the same as the mole fraction because the equations for both fractions in the Ideal Gas Law simplify to the same ratio \(V_A / V_{total} = n_A / n_{total}\). The mass fraction, however, differs from the volume and mole fractions because it is dependent on the molecular weight of the gases. The mass of each gas is affected by its molecular weight, whereas the volume and number of moles are not influenced by molecular weight.

Step-by-step solution

Write down the Ideal Gas Law for gas A and the mixture

Using the Ideal Gas Law, we will first write the equation for gas A and the mixture. Let's denote \(n_A\) as the number of moles of gas A, \(n_{total}\) as the total number of moles in the mixture, and \(V_{total}\) as the total volume of the mixture. For gas A: $$ P V_A = n_A RT $$ For the mixture: $$ P V_{total} = n_{total} RT $$

Rearrange and divide the Ideal Gas Law equations for gas A and the mixture

In this step, we want to rearrange the Ideal Gas Law equations to solve for \(V_A\) and \(V_{total}\). For gas A, divide by \(P\): $$ V_A = \frac{n_A RT}{P} $$ For the mixture, divide by \(P\): $$ V_{total} = \frac{n_{total} RT}{P} $$ Then, we will divide the equation for gas A by the equation for the mixture: $$ \frac{V_A}{V_{total}} = \frac{n_A RT / P}{n_{total} RT / P} $$

Simplify the equation and show the volume fraction is the same as the mole fraction

In this step, we will simplify the equation, canceling out R, T, and P on both sides and show that the volume fraction is the same as the mole fraction. Simplify the equation: $$ \frac{V_A}{V_{total}} = \frac{n_A}{n_{total}} $$ As the volume fraction A is defined as \(V_A / V_{total}\), we can see that it is equal to the mole fraction, which is \(n_A / n_{total}\).

Define mass fraction and compare with volume fraction

Lastly, we will define the mass fraction and compare it to the volume fraction. Mass fraction is defined as the ratio of the mass of a component to the total mass of the mixture: $$ \text{mass fraction A} = \frac{m_A}{m_{total}} $$ where \(m_A\) is the mass of gas A and \(m_{total}\) is the total mass of the mixture. It is important to note that mass fraction differs from volume or mole fraction because it is dependent on the molecular weight of the gases, whereas volume and mole fractions are not. In other words, the mass of each gas is affected by its molecular weight, but the volume and number of moles are not influenced by molecular weight. That is why the volume fraction differs from the mass fraction.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental principle that relates the pressure, volume, temperature, and number of moles of gas. Represented by the equation

\[ PV=nRT \]
where:\
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• \( P \) denotes the pressure of the gas,\
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• \( V \) represents the volume occupied by the gas,\
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• \( n \) is the number of moles of the gas,\
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• \( R \) is the gas constant, and\
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• \( T \) stands for the temperature of the gas in Kelvin.\
• \

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This law is pivotal in many calculations involving gases. In the context of gas mixtures, the Ideal Gas Law allows us to relate the properties of individual gases within the mixture to the properties of the entire mixture. By manipulating this equation, we can glean insights into the behaviour of gases and their interactions within a mixture. For instance, we deduce that if temperature and pressure are kept constant, the volume a gas occupies is directly proportional to the number of moles, which leads into our next concept, the mole fraction.

mole fraction

The mole fraction, a dimensionless quantity, provides a measure of the abundance of a particular substance within a mixture, expressed in terms of moles. It is calculated as the ratio of the number of moles of a component to the total number of moles of all components in the mixture. Concretely, for a component \( A \), the mole fraction \( x_A \) is given by

\[ x_A = \frac{n_A}{n_{total}} \]
where:
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• \( n_A \) is the number of moles of component \( A \), and\
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• \( n_{total} \) is the total number of moles in the mixture.\
• \

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Mole fraction is particularly useful when dealing with gas mixtures, especially under the assumption of ideal behavior, because it allows for the simplification of ratios, such as volume ratio, due to the direct relationship between the mole fraction and volume fraction in ideal gases. Understanding mole fraction is essential for correctly interpreting gas mixture compositions and for tasks like calculating partial pressures or concentration of gases.

mass fraction

The mass fraction takes into account the mass of each component in a mixture. Whereas the mole fraction was concerned with the number of moles, the mass fraction focuses on the mass percentage of each constituent. It is defined as the mass of a single component divided by the total mass of the mixture:

\[ \text{mass fraction A} = \frac{m_A}{m_{total}} \]
Here, \( m_A \) is the mass of component \( A \), and \( m_{total} \) is the total mass of all components in the mixture. Unlike mole fraction, the mass fraction is influenced by the molecular weight of the substances involved. This is because two gases with the same number of moles may have different masses if they have different molecular weights. Consequently, the mass fraction provides a different perspective than the mole fraction or volume fraction, especially in the context of gas mixtures where the components have varying densities and molecular weights. Therefore, understanding the distinction between these fractions is crucial for accurately analyzing the composition and properties of a gas mixture.