Section 3: The Gaseous State - Dalton’s Law of Partial Pressures

Hi there! Welcome to one of the most practical parts of the "Gaseous State" chapter. So far, you have probably spent a lot of time looking at single gases using the Ideal Gas Equation \( (pV = nRT) \). But in the real world—and in the chemistry lab—gases are rarely alone. Think about the air you are breathing right now; it’s a mixture of nitrogen, oxygen, argon, and carbon dioxide.

In this section, we will learn how to calculate the pressure contribution of each individual gas within a mixture. Don't worry if it seems like a lot of variables at first; once you see the relationship between moles and pressure, it becomes much simpler!


1. What is Partial Pressure?

Before we jump into the law itself, we need to understand the term partial pressure. Imagine you have a container filled with a mixture of Gas A and Gas B.

The partial pressure of Gas A is the pressure that Gas A would exert if it were alone in that same container at the same temperature.

The "Party Analogy":
Imagine a room full of people. The "Total Noise" in the room comes from everyone talking. If only the students were in the room, they would make a certain amount of noise (Partial Pressure of Students). If only the teachers were there, they would make their own amount of noise (Partial Pressure of Teachers). The total noise is just the sum of the noise from both groups!


2. Dalton’s Law of Partial Pressures

Dalton’s Law states that for a mixture of non-reacting gases, the total pressure exerted is the sum of the partial pressures of the individual gases.

Mathematically, it looks like this:
\( P_{total} = P_A + P_B + P_C + \dots \)

Where:
\( P_{total} \) = Total pressure of the mixture
\( P_A, P_B, P_C \) = Partial pressures of the individual gases

Important Condition:

This law only applies to gases that do not react with each other. If Gas A and Gas B reacted to form a solid or a new gas, the number of particles would change, and the law wouldn't work the same way!

Quick Review:
Total Pressure = Sum of all individual "Partial" Pressures.


3. Mole Fraction: The "Slice of the Pie"

To find out exactly how much pressure one gas contributes, we need to know what fraction of the total "crowd" it makes up. We call this the mole fraction.

The mole fraction of a gas (represented by \( x \)) is the number of moles of that specific gas divided by the total number of moles in the mixture.

For Gas A:
\( x_A = \frac{n_A}{n_{total}} \)

Where:
\( n_A \) = Moles of Gas A
\( n_{total} \) = \( n_A + n_B + n_C + \dots \)

Key Facts about Mole Fraction:
  • It is always a value between 0 and 1.
  • It has no units (because it is a ratio).
  • The sum of all mole fractions in a mixture always equals 1.

4. Calculating Partial Pressure from Mole Fraction

Here is the "Golden Rule" for your H2 Chemistry calculations: The partial pressure of a gas is directly proportional to its mole fraction. If a gas makes up 50% of the molecules, it will exert 50% of the total pressure.

The Formula:
\( P_A = x_A \times P_{total} \)

This is incredibly useful because it links the amount of substance (moles) to the physical property (pressure).


5. Step-by-Step Calculation Guide

If you are faced with a problem involving gas mixtures, follow these steps:

Step 1: Calculate the number of moles of each individual gas. (You might need to use \( n = \frac{mass}{M_r} \) or \( n = \frac{pV}{RT} \)).
Step 2: Find the total number of moles by adding them all together.
Step 3: Calculate the mole fraction for the gas you are interested in.
Step 4: Multiply the mole fraction by the total pressure to find the partial pressure.

Example:
A container holds 2.0 moles of Nitrogen and 3.0 moles of Oxygen. The total pressure is 10 atm. What is the partial pressure of Nitrogen?
1. \( n_{total} = 2.0 + 3.0 = 5.0 \) moles.
2. \( x_{nitrogen} = \frac{2.0}{5.0} = 0.4 \).
3. \( P_{nitrogen} = 0.4 \times 10 \) atm = 4 atm.


6. Common Pitfalls to Avoid

Don't worry if you get stuck at first; many students make these common mistakes:

  • Mixing Units: Ensure all pressures are in the same units (e.g., all in Pa or all in atm) before adding them.
  • Reacting Gases: Always check if the gases react. For example, you cannot simply use Dalton's Law for a mixture of \( NH_3 \) and \( HCl \) because they react to form solid \( NH_4Cl \).
  • Temperature Changes: Remember that if the temperature changes, the total pressure changes (Amontons' Law), but the mole fractions stay the same as long as no gas is added or removed.

7. Real-World Connection: Deep Sea Diving

Did you know? Scuba divers have to worry about partial pressures! At high pressures underwater, the partial pressure of oxygen in regular air can become toxic. Divers use special gas mixtures (like Nitrox) where they carefully control the mole fraction of oxygen to keep its partial pressure within a safe range for the depth they are diving.


Key Takeaways

1. Total Pressure is the sum of all individual Partial Pressures.
2. Mole Fraction \( (x) \) is the ratio of moles of one gas to the total moles.
3. Partial Pressure Formula: \( P_i = x_i \times P_{total} \).
4. Dalton's Law only applies to non-reacting ideal gases.