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. Each gas in the mixture exerts a partial pressure, which is the pressure it would exert if it alone occupied the entire volume. This law is fundamental to understanding gas behavior in various biological and chemical systems, particularly in respiratory physiology.

When to use: Apply this law when dealing with gas mixtures, such as atmospheric air, or gases within the lungs, to determine the overall pressure or the contribution of individual gases. It's crucial for understanding how gases move across membranes based on pressure gradients.

Why it matters: This law is vital in biology for understanding respiratory physiology, including how oxygen and carbon dioxide are exchanged in the lungs and tissues. It also explains phenomena like decompression sickness in diving and the effects of altitude on gas exchange, making it critical for medical and environmental sciences.

Symbols

Variables

P_1 = Partial Pressure of Gas 1, P_2 = Partial Pressure of Gas 2, P_3 = Partial Pressure of Gas 3, P_{\text{total}} = Total Pressure

P_{1}

Partial Pressure of Gas 1

k P a

P_{2}

Partial Pressure of Gas 2

k P a

P_{3}

Partial Pressure of Gas 3

k P a

P_{total}

Total Pressure

k P a

Walkthrough

Derivation

Formula: Dalton's Law of Partial Pressures

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, meaning their individual molecules occupy negligible volume and have no intermolecular forces.
All gases are at the same temperature and occupy the same volume.

1

Consider a Mixture of Ideal Gases:

Dalton's Law 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. Each gas contributes independently to the total pressure.

P_{total} = P_{1} + P_{2} + \dots + P_{n}

2

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 entire volume ( $V$ ) at the same temperature ( $T$ ), where $n_{i}$ is the number of moles of gas $i$ and $R$ is the ideal gas constant.

P_{i} = \frac{n _{i} R T}{V}

3

Summing the Partial Pressures:

By summing the partial pressures of all component gases, we obtain the total pressure of the gas mixture. This is a direct consequence of the ideal gas law and the assumption of non-reacting gases.

P_{total} = i = 1 \sum n P_{i}

Result

P_{total} = i = 1 \sum n P_{i}

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

Free formulas

Rearrangements

Solve for $P_{1}$

Dalton's Law: Make $P_{1}$ the subject

P_{1} = P_{total} - P_{2} - P_{3}

To make $P_{1}$ (Partial Pressure of Gas 1) the subject, subtract the partial pressures of all other gases from the total pressure.

Difficulty: 2/5

Solve for $P_{2}$

Dalton's Law: Make $P_{2}$ the subject

P_{2} = P_{total} - P_{1} - P_{3}

To make $P_{2}$ (Partial Pressure of Gas 2) the subject, subtract the partial pressures of the other gases ( $P_{1}$ and $P_{3}$ ) from the total pressure.

Difficulty: 2/5

Solve for $P_{3}$

Dalton's Law: Make $P_{3}$ the subject

P_{3} = P_{total} - P_{1} - P_{2}

To make $P_{3}$ (Partial Pressure of Gas 3) the subject, subtract the partial pressures of the other gases ( $P_{1}$ and $P_{2}$ ) from the total pressure.

Difficulty: 2/5

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

Visual intuition

Graph

The graph is a linear line with a slope of one that shifts vertically based on the sum of other partial pressures, meaning it only passes through the origin if those other pressures are zero. For a biology student, this shows that as the partial pressure of one gas increases, the total pressure of the mixture rises proportionally, reflecting how individual gas concentrations contribute to the overall pressure in gas exchange. The most important feature is that the line maintains a constant slope, which demonstrates

Graph type: linear

Why it behaves this way

Intuition

Imagine a container filled with different types of gas molecules, each type moving randomly and independently, and the total force (pressure)

P_{total}

The overall pressure exerted by the entire gas mixture on the container walls.

It's the combined 'push' or force per unit area resulting from all gas molecules hitting the walls.

P_{1}, P_{2}, P_{3}

The pressure that a single component gas (e.g., gas 1, gas 2, gas 3) would exert if it alone occupied the entire volume at the same temperature.

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

Signs and relationships

+: The addition sign indicates that the individual gas components contribute independently and additively to the total pressure, as they are assumed not to react or significantly interact with each other.

Free study cues

Insight

Canonical usage

This equation requires all pressure terms (partial pressures and total pressure) to be expressed in consistent units for the sum to be dimensionally correct.

Common confusion

A common mistake is to sum partial pressures expressed in different units (e.g., adding a pressure in Pa to one in mmHg) without first converting them to a common unit.

Unit systems

$P$ Pascal (Pa), millimeters of mercury (mmHg), atmosphere (atm), torr · All partial pressures (P_1, P_2, P_3, etc.) and the total pressure (P_total) must be expressed in the same unit for the sum to be valid. Conversion factors are necessary if different units are initially provided.

One free problem

Practice Problem

A gas mixture contains three non-reacting gases with partial pressures of 21.2 kPa, 79.8 kPa, and 0.1 kPa respectively. What is the total pressure of this gas mixture?

Partial Pressure of Gas 121.2 kPa

Partial Pressure of Gas 279.8 kPa

Partial Pressure of Gas 30.1 kPa

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

Calculating the partial pressure of oxygen in the air we breathe or in an oxygen tank.

Study smarter

Tips

Ensure all partial pressures are in the same units before summing them.
The law applies to ideal gases and non-reacting gas mixtures.
The partial pressure of a gas is proportional to its mole fraction in the mixture.
Remember that atmospheric pressure is the sum of partial pressures of N2, O2, Ar, CO2, etc.

Avoid these traps

Common Mistakes

Mixing pressure units (e.g., kPa and mmHg) without conversion.
Assuming the law applies to reacting gases.
Forgetting to account for all gases present in a mixture.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The total pressure of a gas mixture is the sum of the partial pressures of its individual component gases.

Apply this law when dealing with gas mixtures, such as atmospheric air, or gases within the lungs, to determine the overall pressure or the contribution of individual gases. It's crucial for understanding how gases move across membranes based on pressure gradients.

This law is vital in biology for understanding respiratory physiology, including how oxygen and carbon dioxide are exchanged in the lungs and tissues. It also explains phenomena like decompression sickness in diving and the effects of altitude on gas exchange, making it critical for medical and environmental sciences.

Mixing pressure units (e.g., kPa and mmHg) without conversion. Assuming the law applies to reacting gases. Forgetting to account for all gases present in a mixture.

Calculating the partial pressure of oxygen in the air we breathe or in an oxygen tank.

Ensure all partial pressures are in the same units before summing them. The law applies to ideal gases and non-reacting gas mixtures. The partial pressure of a gas is proportional to its mole fraction in the mixture. Remember that atmospheric pressure is the sum of partial pressures of N2, O2, Ar, CO2, etc.

References

Sources

Atkins' Physical Chemistry
Wikipedia: Dalton's law
NIST CODATA
Vander's Human Physiology: The Mechanisms of Body Function, 16th ed. (Eric P. Widmaier, Hershel Raff, Kevin T. Strang)
Atkins' Physical Chemistry, 11th ed. (Peter Atkins, Julio de Paula, James Keeler)
Atkins, P. W., & de Paula, J. (2014). Atkins' Physical Chemistry (10th ed.). Oxford University Press.
Bird, R. B., Stewart, W. E., & Lightfoot, E. N. (2007). Transport Phenomena (2nd ed.). John Wiley & Sons.
Callen, H. B. (1985). Thermodynamics and an Introduction to Thermostatistics (2nd ed.). John Wiley & Sons.

Dalton's Law of Partial Pressures

Overview

Variables

Derivation

Consider a Mixture of Ideal Gases:

Define Partial Pressure:

Summing the Partial Pressures:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Henry's Law

Ideal Gas Law

Frequently Asked Questions

Sources