Dalton's Law of Partial Pressures Equation, Derivation & Rearrangements

Core idea

Overview

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 the individual gases. This law is fundamental in understanding the behavior of gas mixtures, particularly in atmospheric chemistry, diving, and industrial processes. It assumes ideal gas behavior and that the gases do not chemically react with each other.

When to use: Use this law when dealing with a mixture of gases in a container and you need to find the total pressure, or if you know the total pressure and all but one partial pressure. It's applicable for ideal gas mixtures where components don't react.

Why it matters: This law is crucial for fields like respiratory physiology (understanding gas exchange in lungs), meteorology (atmospheric pressure), and chemical engineering (designing gas separation processes). It helps predict gas behavior in complex systems, ensuring safety and efficiency.

Symbols

Variables

$P_{A}$ = Partial Pressure of Gas A, $P_{B}$ = Partial Pressure of Gas B, $P_{C}$ = Partial Pressure of Gas C, $P_{t o t a l}$ = Total Pressure

P_{A}

Partial Pressure of Gas A

atm

P_{B}

Partial Pressure of Gas B

atm

P_{C}

Partial Pressure of Gas C

atm

P_{t o t a l}

Total Pressure

atm

Walkthrough

Derivation

Formula: Dalton's Law of Partial Pressures

Dalton's Law states that the total pressure of a gas mixture is the sum of the partial pressures of its individual component gases.

The gases in the mixture do not chemically react with each other.
The gases behave ideally.
All component gases occupy the same volume and are at the same temperature.

1

Define Partial Pressure:

The partial pressure ( $P_{i}$ ) of an individual gas (i) in a mixture is the pressure it would exert if it alone occupied the same volume (V) at the same temperature (T), as described by the Ideal Gas Law, where $n_{i}$ is the moles of gas i and R is the ideal gas constant.

P_{i} = n_{i} \frac{R T}{V}

2

Total Pressure of a Mixture:

John Dalton proposed that the total pressure ( $P_{t}$ otal) of a mixture of non-reacting gases is simply the sum of the partial pressures of each individual gas component ( $P_{A}$ , $P_{B}$ , $P_{C}$ , etc.).

P_{t o t a l} = P_{A} + P_{B} + P_{C} + ...

3

Relating to Ideal Gas Law:

Since each gas contributes independently to the total pressure, the sum of their individual pressures is equivalent to the total number of moles ( $n_{t}$ otal) exerting pressure in the same volume and temperature, consistent with the Ideal Gas Law for the entire mixture.

P_{t o t a l} = (n_{A} + n_{B} + n_{C} + ...) \frac{R T}{V} = n_{t o t a l} \frac{R T}{V}

Result

P_{t o t a l} = (n_{A} + n_{B} + n_{C} + ...) \frac{R T}{V} = n_{t o t a l} \frac{R T}{V}

Source: Atkins' Physical Chemistry, 11th Edition — Chapter 1: The Properties of Gases

Free formulas

Rearrangements

Solve for $P_{A}$

Dalton's Law: Make $P_{A}$ the subject

P_{A} = P_{t o t a l} - P_{B} - P_{C}

To find the partial pressure of Gas A ( $P_{A}$ ), subtract the partial pressures of all other gases ( $P_{B}$ , $P_{C}$ , etc.) from the total pressure ( $P_{t}$ otal).

Difficulty: 2/5

Solve for $P_{B}$

Dalton's Law: Make $P_{B}$ the subject

P_{B} = P_{t o t a l} - P_{A} - P_{C}

To find the partial pressure of Gas B ( $P_{B}$ ), subtract the partial pressures of all other gases ( $P_{A}$ , $P_{C}$ , etc.) from the total pressure ( $P_{t}$ otal).

Difficulty: 2/5

Solve for $P_{C}$

Dalton's Law: Make $P_{C}$ the subject

P_{C} = P_{t o t a l} - P_{A} - P_{B}

To find the partial pressure of Gas C ( $P_{C}$ ), subtract the partial pressures of all other gases ( $P_{A}$ , $P_{B}$ , etc.) from the total pressure ( $P_{t}$ otal).

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

Graph unavailable for this formula.

The graph displays a linear line with a slope of one passing through the origin, representing the direct relationship where the total pressure is plotted against itself. For a chemistry student, this shape illustrates that as the sum of the individual partial pressures increases, the total pressure of the gas mixture rises at an identical rate. The most important feature of this curve is its constant linear slope, which demonstrates that doubling the combined pressure of the component gases results in an exact doubling of the total pressure.

Graph type: linear

Why it behaves this way

Intuition

Imagine a container filled with various types of tiny, fast-moving particles, where each type of particle independently collides with the container walls, and the total force per unit area on the walls is the sum of the

P_{t o t a l}

The total pressure exerted by the entire gas mixture on the walls of its container.

It's the overall 'push' from all the different types of gas molecules combined.

P_{A}, P_{B}, P_{C}, ...

The partial pressure of an individual gas component (A, B, C, etc.) in the mixture, defined as the pressure that gas would exert if it alone occupied the entire volume of the

It's the 'push' contributed by just one specific type of gas molecule, acting as if the other gases weren't present.

Signs and relationships

+: The addition operator signifies that each gas component contributes independently to the total pressure. In an ideal gas mixture, molecules of different gases do not interact with each other, so their individual pressure

Free study cues

Insight

Canonical usage

All pressure terms in the equation must be expressed in the same units for the summation to be valid.

Common confusion

A common mistake is to mix different units of pressure (e.g., adding a partial pressure in atm to another in kPa) before performing the summation, leading to incorrect results.

Unit systems

$P$ Pa · All partial pressures (P_A, P_B, etc.) and the total pressure (P_total) must be expressed in the same unit for the summation to be dimensionally consistent.

One free problem

Practice Problem

A gas mixture contains three non-reacting gases: Gas A, Gas B, and Gas C. Their partial pressures are 0.2 atm, 0.5 atm, and 0.1 atm, respectively. What is the total pressure of the mixture?

Partial Pressure of Gas A0.2 atm

Partial Pressure of Gas B0.5 atm

Partial Pressure of Gas C0.1 atm

Solve for: $P_{t} o t a l$

Hint: Sum the individual partial pressures.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In the total pressure inside a scuba diving tank containing a mixture of oxygen, Dalton's Law of Partial Pressures is used to calculate Total Pressure from Partial Pressure of Gas A, Partial Pressure of Gas B, and Partial Pressure of Gas C. The result matters because it helps check loads, margins, or component sizes before a design is treated as safe.

Study smarter

Tips

Ensure all partial pressures are in the same units before summing them.
The law applies only to non-reacting gases.
The volume and temperature of all component gases are the same as the total volume and temperature of the mixture.
Remember that '...' implies you sum all partial pressures present, not just three.

Avoid these traps

Common Mistakes

Forgetting to convert units of partial pressures before summing.
Applying the law to gases that chemically react with each other.
Confusing partial pressure with mole fraction.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Dalton's Law states that the total pressure of a gas mixture is the sum of the partial pressures of its individual component gases.

Use this law when dealing with a mixture of gases in a container and you need to find the total pressure, or if you know the total pressure and all but one partial pressure. It's applicable for ideal gas mixtures where components don't react.

This law is crucial for fields like respiratory physiology (understanding gas exchange in lungs), meteorology (atmospheric pressure), and chemical engineering (designing gas separation processes). It helps predict gas behavior in complex systems, ensuring safety and efficiency.

Forgetting to convert units of partial pressures before summing. Applying the law to gases that chemically react with each other. Confusing partial pressure with mole fraction.

In the total pressure inside a scuba diving tank containing a mixture of oxygen, Dalton's Law of Partial Pressures is used to calculate Total Pressure from Partial Pressure of Gas A, Partial Pressure of Gas B, and Partial Pressure of Gas C. The result matters because it helps check loads, margins, or component sizes before a design is treated as safe.

Ensure all partial pressures are in the same units before summing them. The law applies only to non-reacting gases. The volume and temperature of all component gases are the same as the total volume and temperature of the mixture. Remember that '...' implies you sum all partial pressures present, not just three.

References

Sources

Atkins' Physical Chemistry
Halliday, Resnick, Walker - Fundamentals of Physics
Wikipedia: Dalton's law
Atkins' Physical Chemistry, 11th Edition
IUPAC Gold Book (Compendium of Chemical Terminology)
Halliday, Resnick, and Walker, Fundamentals of Physics, 11th Edition
McQuarrie, Donald A. Physical Chemistry: A Molecular Approach
Brown, Theodore L., H. Eugene LeMay Jr., Bruce E. Bursten, Catherine J. Murphy, Patrick M. Woodward, and Matthew W. Stoltzfus.

Dalton's Law of Partial Pressures

Overview

Variables

Derivation

Define Partial Pressure:

Total Pressure of a Mixture:

Relating to Ideal Gas Law:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Partial Pressure from Mole Fraction

Graham's Law of Effusion

Frequently Asked Questions

Sources