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.

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.

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.

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.