Question

State Dalton's law of partial pressures and explain what mole fraction is. Does mole fraction have units?

Short answer

Mole fraction is dimensionless, and Dalton's Law states total pressure is the sum of partial pressures.

Step-by-step solution

Understanding Dalton's Law

Dalton's Law of Partial Pressures states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of individual gases. Mathematically, this can be expressed as: \( P_{total} = P_1 + P_2 + P_3 + ext{...} \), where \( P_1, P_2, \) and \( P_3 \) are the partial pressures of the individual gases.

Defining Mole Fraction

The mole fraction of a component in a mixture is the ratio of the moles of that component to the total moles of all components in the mixture. It is commonly denoted by \( x_i \). Mathematically, it is expressed as: \( x_i = \frac{n_i}{n_{total}} \), where \( n_i \) is the number of moles of the component \( i \) and \( n_{total} \) is the total number of moles in the mixture.

Examining the Units of Mole Fraction

Mole fraction is a dimensionless quantity, meaning it has no units. This is because it is a ratio of moles to moles, which cancels out any units.

Key concepts

Mole Fraction

The concept of mole fraction is crucial in understanding how different components behave in a chemical mixture. Simply put, the mole fraction is the ratio of the number of moles of a particular component to the total number of moles in the entire mixture. This is denoted by the symbol \( x_i \), and mathematically it is expressed as:\[x_i = \frac{n_i}{n_{\text{total}}}\]where \( n_i \) is the number of moles of a specific component, and \( n_{\text{total}} \) is the total moles of all components. The beauty of the mole fraction lies in its straightforwardness, as it provides a simple way to describe the composition of a mixture.

Partial Pressure

In any mixture of gases, each gas contributes to the total pressure through what is known as partial pressure. The partial pressure of a gas is the pressure that gas would exert if it were alone in the container. According to Dalton's Law of Partial Pressures, the total pressure of a mixture of non-reacting gases is simply the sum of each individual gas's partial pressure:\[P_{\text{total}} = P_1 + P_2 + P_3 + \ldots\]where \( P_1, P_2, \) and \( P_3 \) are the partial pressures of the individual gases. Understanding partial pressures is essential, particularly in fields like chemistry and environmental science, as it helps predict gas behavior and perform calculations involving gas mixtures.

Non-reacting Gases

Non-reacting gases in a mixture are gases that do not chemically interact with each other. This allows them to be studied using Dalton's Law because their behaviors are independent of one another. For example, if you have a container with oxygen and nitrogen, these gases do not typically react at room temperature. As a result, each gas contributes its partial pressure to the total pressure, and the properties of one do not affect the properties of the other. This concept is important because it simplifies the analysis of gas mixtures, allowing scientists and engineers to predict how different gases will behave when they are combined.

Dimensionless Quantity

Some measurements are referred to as dimensionless quantities, meaning they do not have any physical units. The mole fraction is one such dimensionless quantity. Because mole fraction is the ratio of moles of one component to the total moles, the units cancel out. This results in a pure number that represents a proportion. Being dimensionless makes mole fractions incredibly useful for simplifying calculations and comparisons, as it provides a direct and unit-independent way to express concentrations in mixtures. This characteristic is particularly beneficial when dealing with mixtures that vary widely in temperature and pressure.