Dalton's Law Of Partial Pressure Example

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Dalton's Law of Partial Pressures: Understanding Gas Mixtures

Imagine a room filled with air. That air isn't just one type of gas, but a mixture of nitrogen, oxygen, and trace amounts of other gases like argon and carbon dioxide. How do we calculate the pressure exerted by each of these individual gases? This is where Dalton's Law of Partial Pressures comes into play. It's a fundamental concept in chemistry and physics that helps us understand and predict the behavior of gas mixtures.

This law, named after the British chemist John Dalton, provides a simple yet powerful way to determine the total pressure of a gas mixture based on the pressures of the individual gases. Let’s dive deeper into Dalton's Law, exploring its principles, applications, and practical examples.

What is Dalton's Law of Partial Pressures?

Dalton's Law of Partial Pressures states that the total pressure exerted by a mixture of non-reacting gases is equal to the sum of the partial pressures of the individual gases. In simpler terms, each gas in a mixture contributes to the total pressure, and the total pressure is just the sum of all these individual contributions.

Mathematically, Dalton's Law can be expressed as:

P_total = P_1 + P_2 + P_3 + ... + P_n

Where:

• P_total is the total pressure of the gas mixture.
• P_1, P_2, P_3, ..., P_n are the partial pressures of each individual gas in the mixture.

Understanding Partial Pressure

The partial pressure of a gas in a mixture is the pressure that the gas would exert if it occupied the same volume alone. In other words, it's the pressure that the gas "contributes" to the overall pressure of the mixture.

The partial pressure of a gas is proportional to its mole fraction in the mixture. The mole fraction (χ) of a gas is the ratio of the number of moles of that gas to the total number of moles of all gases in the mixture.

χ_i = n_i / n_total

Where:

• χ_i is the mole fraction of gas i.
• n_i is the number of moles of gas i.
• n_total is the total number of moles of all gases in the mixture.

The partial pressure of a gas can then be calculated using the mole fraction and the total pressure:

P_i = χ_i * P_total

Key Assumptions of Dalton's Law

Dalton's Law is based on a few key assumptions:

1. Non-reacting gases: The gases in the mixture must not react with each other. If they do, the law does not apply directly.
2. Ideal gas behavior: The gases are assumed to behave ideally, meaning that intermolecular forces between the gas molecules are negligible.
3. Uniform mixing: The gases are uniformly mixed throughout the volume.

Example 1: Calculating Total Pressure

Let's say we have a container that holds the following gases:

• Nitrogen (N₂) with a partial pressure of 2 atm
• Oxygen (O₂) with a partial pressure of 1 atm
• Carbon Dioxide (CO₂) with a partial pressure of 0.5 atm

To find the total pressure in the container, we simply add the partial pressures:

P_total = P_N₂ + P_O₂ + P_CO₂
P_total = 2 atm + 1 atm + 0.5 atm
P_total = 3.5 atm

Therefore, the total pressure in the container is 3.5 atm.

Example 2: Determining Partial Pressure from Mole Fraction

Consider a container with a total pressure of 5 atm. The container contains two gases:

• Gas A with a mole fraction of 0.6
• Gas B with a mole fraction of 0.4

To find the partial pressure of each gas, we use the formula:

P_i = χ_i * P_total

For Gas A:

P_A = 0.6 * 5 atm
P_A = 3 atm

For Gas B:

P_B = 0.4 * 5 atm
P_B = 2 atm

Thus, the partial pressure of Gas A is 3 atm, and the partial pressure of Gas B is 2 atm.

Example 3: Collecting Gas Over Water

A common laboratory technique involves collecting a gas produced in a reaction over water. When a gas is collected over water, it becomes saturated with water vapor. This means that the total pressure of the collected gas is the sum of the pressure of the gas itself and the vapor pressure of water.

For instance, suppose you collect hydrogen gas over water at 25°C. The total pressure of the gas collected is 760 torr. The vapor pressure of water at 25°C is 24 torr. To find the partial pressure of the hydrogen gas, you subtract the vapor pressure of water from the total pressure:

P_total = P_H₂ + P_H₂O
P_H₂ = P_total - P_H₂O
P_H₂ = 760 torr - 24 torr
P_H₂ = 736 torr

Therefore, the partial pressure of the hydrogen gas is 736 torr.

Example 4: Mixture of Gases in a Fixed Volume

A 10.0 L flask contains 0.200 mol of methane (CH₄) and 0.300 mol of ethane (C₂H₆) at a temperature of 27°C. Calculate the partial pressure of each gas and the total pressure in the flask.

First, convert the temperature from Celsius to Kelvin:

T(K) = T(°C) + 273.15
T(K) = 27°C + 273.15 = 300.15 K

Use the ideal gas law to calculate the partial pressure of each gas:

PV = nRT

Where:

• P is the pressure (in atm)
• V is the volume (in L)
• n is the number of moles
• R is the ideal gas constant (0.0821 L·atm/mol·K)
• T is the temperature (in K)

For methane (CH₄):

P_CH₄ = (n_CH₄ * R * T) / V
P_CH₄ = (0.200 mol * 0.0821 L·atm/mol·K * 300.15 K) / 10.0 L
P_CH₄ = 0.492 atm

For ethane (C₂H₆):

P_C₂H₆ = (n_C₂H₆ * R * T) / V
P_C₂H₆ = (0.300 mol * 0.0821 L·atm/mol·K * 300.15 K) / 10.0 L
P_C₂H₆ = 0.739 atm

The total pressure is the sum of the partial pressures:

P_total = P_CH₄ + P_C₂H₆
P_total = 0.492 atm + 0.739 atm
P_total = 1.231 atm

Thus, the partial pressure of methane is 0.492 atm, the partial pressure of ethane is 0.739 atm, and the total pressure in the flask is 1.231 atm.

Real-World Applications of Dalton's Law

Dalton's Law has many practical applications in various fields:

• Diving: Divers use Dalton's Law to calculate the partial pressures of gases in their breathing mixtures at different depths. This is crucial for preventing nitrogen narcosis and oxygen toxicity.
• Anesthesia: Anesthesiologists use Dalton's Law to control the concentrations of anesthetic gases delivered to patients.
• Respiratory Physiology: Understanding partial pressures is essential for studying gas exchange in the lungs and blood.
• Industrial Processes: Many industrial processes involve gas mixtures, and Dalton's Law is used to design and optimize these processes.
• Meteorology: Meteorologists use Dalton's Law to analyze the composition of the atmosphere and predict weather patterns.

Limitations of Dalton's Law

While Dalton's Law is a useful tool, it has some limitations:

• Non-ideal gases: Dalton's Law is based on the assumption that gases behave ideally. However, real gases deviate from ideal behavior, especially at high pressures and low temperatures.
• Reacting gases: If the gases in the mixture react with each other, Dalton's Law does not apply directly. In such cases, you need to consider the stoichiometry of the reaction and account for the changes in the number of moles of each gas.

Dalton's Law and Vapor Pressure

As demonstrated in Example 3, Dalton's Law is often used when dealing with gases collected over water or other liquids. The total pressure of the gas collected is the sum of the partial pressure of the gas and the vapor pressure of the liquid.

The vapor pressure of a liquid is the pressure exerted by its vapor when the liquid is in equilibrium with its vapor. Vapor pressure depends on temperature; as temperature increases, vapor pressure also increases.

FAQ about Dalton's Law of Partial Pressures

• Q: Does Dalton's Law apply to mixtures of liquids?

  • A: No, Dalton's Law specifically applies to mixtures of non-reacting gases.

• Q: Can Dalton's Law be used to calculate the density of a gas mixture?

  • A: Yes, you can use Dalton's Law to find the average molar mass of the mixture, and then use that to calculate the density.

• Q: What happens if the gases in the mixture react with each other?

  • A: If the gases react, Dalton's Law does not apply directly. You need to account for the reaction stoichiometry and the changes in the number of moles of each gas.

In Conclusion

Dalton's Law of Partial Pressures is a fundamental principle in chemistry and physics that provides a simple yet powerful way to understand the behavior of gas mixtures. By knowing the partial pressures of the individual gases in a mixture, we can determine the total pressure of the mixture. This law has numerous applications in various fields, from diving and anesthesia to respiratory physiology and industrial processes. While Dalton's Law has some limitations, it remains an essential tool for studying and working with gas mixtures.

How might Dalton's Law affect the design of a scuba diving tank? What other scenarios can you think of where understanding partial pressures would be crucial?