Problem 97 - Chemistry The Science in Context

Question

What is meant by the partial pressure of a gas?

Step-by-Step Solution

Answer

Answer: The partial pressure of a gas in a mixture is the pressure it would exert if it were the only gas present and occupied the same volume. This partial pressure is related to its mole fraction, which is the ratio of the moles of that specific gas to the total moles of all gases in the mixture. The partial pressure can be determined by multiplying the mole fraction of the gas by the total pressure of the gas mixture, as described by Dalton's Law.

Step 1: Define Partial Pressure

Partial pressure is the pressure that a gas in a mixture would exert if it were the only gas present and occupied the same volume, and it is related to its mole fraction. It is an important concept in understanding the behavior of a gas, as it helps to understand the behavior of gases in a mixture.

Step 2: The Ideal Gas Law

To understand partial pressure of a gas, we need to first understand the ideal gas law. It states that for an ideal gas, pressure (P), volume (V), and temperature (T) are related by the number of moles (n) and the ideal gas constant (R). It is represented by the equation: PV = nRT

Step 3: Dalton's Law

Dalton's Law of partial pressures states that the total pressure of a mixture of non-reacting gases is equal to the sum of the partial pressures of its individual components. Mathematically, it can be represented as: P_total = P₁ + P₂ + ... + P_n Where P_total is the total pressure, and P₁, P₂, ..., P_n are the partial pressures of the individual gases.

Step 4: Mole Fraction and Partial Pressure

Mole fraction is the ratio of the moles of a specific gas in a mixture to the total moles of all gases in the mixture. The mole fraction (χ) of a gas can be calculated by dividing the moles of that gas (n_i) by the total moles of all gases (n_total): χ_i = n_i / n_total The partial pressure of a gas can be found by multiplying its mole fraction by the total pressure of the gas mixture: P_i = χ_i * P_total

Step 5: Applying the Concepts

To calculate the partial pressure of a gas in a mixture, one needs to know the mole fraction of that gas and the total pressure of the mixture. Using Dalton's Law and the ideal gas law, the partial pressure can be determined, which helps in understanding the behavior of that specific gas in the mixture.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation that describes the behavior of gases under ideal conditions. This law connects four important variables: pressure (P), volume (V), temperature (T), and the number of moles (n) of the gas involved. The relationship is expressed with the equation \( PV = nRT \).
In this equation, \( R \) is the ideal gas constant, which is a known value that makes the units work in harmony across the various gases. The Ideal Gas Law allows us to predict how a gas will react under different scenarios of pressure, volume, and temperature.
For instance, if we have a fixed amount of gas and we increase the temperature, the pressure may rise unless the volume is allowed to expand. Understanding this law is crucial when calculating partial pressures, especially because it provides the framework for understanding how different conditions (like temperature) can influence gas behavior and interaction.

Dalton's Law

Dalton's Law, or the law of partial pressures, is a principle that explains how the pressure of a mixture of gases behaves. It states that the total pressure exerted by a mixture of non-reactive gases is equal to the sum of the partial pressures of each individual gas within the mixture.

• The equation representing Dalton's Law is: \( P_{\text{total}} = P_1 + P_2 + \ldots + P_n \).

The law is useful for understanding gas mixtures because it simplifies the calculation of the contribution of each gas to the overall pressure. Each gas in a mixture behaves independently and contributes to the total pressure based on its own properties, such as volume and temperature, as if it were alone in the container.
This is especially important in fields like chemistry and physics, where analyzing the behavior of gases in mixtures is common. Knowing each gas's partial pressure helps predict reactions, influence outcomes, and understand the physical chemistry that governs real-world scenarios.

Mole Fraction

The mole fraction is a way to express the concentration of a component in a mixture. It is especially important when discussing gas mixtures, as it provides a simple and clear way to determine how much of a particular substance is present in relation to all the substances in the mixture. The mole fraction \( \chi \) for a gas is calculated by taking the ratio of the moles of that specific gas to the total number of moles in the gas mixture:

• \( \chi_i = \frac{n_i}{n_{\text{total}}} \)

Once the mole fraction of a gas is known, its partial pressure can be easily found by multiplying the mole fraction by the total pressure of the gas mixture:

• \( P_i = \chi_i \times P_{\text{total}} \)

The mole fraction is unitless because it is simply a ratio. This concept is critical when using Dalton's Law to find partial pressures, as it provides the necessary link between the composition of the gas mixture and the pressures exerted by each individual gas. Understanding how to calculate the mole fraction and use it in other equations allows for greater insight into the behavior and interaction of gases in various mixtures.