Partial Pressure from Mole Fraction Equation, Derivation & Rearrangements

Core idea

Overview

This equation is a direct consequence of Dalton's Law of Partial Pressures and the Ideal Gas Law. It states that the partial pressure of a specific gas (P_i) in a mixture is equal to its mole fraction (X_i) multiplied by the total pressure of the gas mixture (P_total). This relationship is incredibly useful for determining the contribution of an individual gas to the overall pressure, especially when the number of moles of each gas is known.

When to use: Apply this formula when you know the total pressure of a gas mixture and the mole fraction of a specific component gas, and you need to find that component's partial pressure. It's also useful for calculating mole fraction if partial and total pressures are known.

Why it matters: Understanding this relationship is vital in fields like chemical engineering for designing separation processes, environmental science for analyzing atmospheric composition, and medicine for understanding gas exchange in the body. It allows for precise quantification of individual gas contributions in complex systems.

Symbols

Variables

$X_{i}$ = Mole Fraction of Gas i, $P_{t o t a l}$ = Total Pressure, $P_{i}$ = Partial Pressure of Gas i

X_{i}

Mole Fraction of Gas i

Variable

P_{t o t a l}

Total Pressure

atm

P_{i}

Partial Pressure of Gas i

atm

Walkthrough

Derivation

Formula: Partial Pressure from Mole Fraction

The partial pressure of a gas in a mixture is its mole fraction multiplied by the total pressure of the mixture.

The gases in the mixture behave ideally.
The gases do not chemically react with each other.

1

Start with Ideal Gas Law for component i:

The partial pressure ( $P_{i}$ ) of an individual gas (i) in a mixture can be expressed using the Ideal Gas Law, where $n_{i}$ is the number of moles of gas i, R is the ideal gas constant, T is temperature, and V is the total volume.

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

2

Start with Ideal Gas Law for total mixture:

Similarly, the total pressure ( $P_{t}$ otal) of the gas mixture can be expressed using the Ideal Gas Law for the total number of moles ( $n_{t}$ otal) in the same volume and temperature.

P_{t o t a l} V = n_{t o t a l} R T ⟹ P_{t o t a l} = n_{t o t a l} \frac{R T}{V}

3

Divide P_i by P_total:

Divide the expression for $P_{i}$ by the expression for $P_{t}$ otal. The terms (RT/V) cancel out, as they are common to both.

\frac{P _{i}}{P _{t o t a l}} = \frac{n _{i} ( R T / V )}{n _{t o t a l} ( R T / V )}

4

Simplify the Ratio:

After cancellation, the ratio of partial pressure to total pressure is equal to the ratio of moles of gas i to the total moles.

\frac{P _{i}}{P _{t o t a l}} = \frac{n _{i}}{n _{t o t a l}}

5

Define Mole Fraction:

The mole fraction ( $X_{i}$ ) of gas i is defined as the ratio of the number of moles of gas i ( $n_{i}$ ) to the total number of moles in the mixture ( $n_{t}$ otal).

X_{i} = \frac{n _{i}}{n _{t o t a l}}

6

Substitute Mole Fraction:

Substitute the definition of mole fraction into the simplified ratio.

\frac{P _{i}}{P _{t o t a l}} = X_{i}

7

Rearrange for P_i:

Multiply both sides by $P_{t}$ otal to isolate $P_{i}$ , yielding the final formula for partial pressure from mole fraction.

P_{i} = X_{i} P_{t o t a l}

Result

P_{i} = X_{i} P_{t o t a l}

Source: Chemistry: The Central Science, Brown, LeMay, Bursten — Chapter 10: Gases

Free formulas

Rearrangements

Solve for $X_{i}$

Partial Pressure: Make $X_{i}$ the subject

X_{i} = \frac{P _{i}}{P _{t o t a l}}

To make $X_{i}$ (Mole Fraction) the subject, divide the partial pressure ( $P_{i}$ ) by the total pressure ( $P_{t}$ otal).

Difficulty: 1/5

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

Partial Pressure: Make $P_{t}$ otal the subject

P_{t o t a l} = \frac{P _{i}}{X _{i}}

To make $P_{t}$ otal (Total Pressure) the subject, divide the partial pressure ( $P_{i}$ ) by the mole fraction ( $X_{i}$ ).

Difficulty: 1/5

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

Visual intuition

Graph

The graph displays a straight line starting at the origin where the slope represents the total pressure of the gas mixture. For a chemistry student, a large mole fraction indicates that a specific gas makes up a significant portion of the mixture, resulting in a higher contribution to the total pressure. The most important feature of this linear relationship is that doubling the mole fraction of a gas will exactly double its partial pressure.

Graph type: linear

Why it behaves this way

Intuition

Imagine a container where different types of gas molecules move randomly and independently, each type contributing to the total pressure on the walls in direct proportion to its relative number (mole fraction)

P_{i}

The pressure that gas 'i' would exert if it alone occupied the entire volume of the mixture at the same temperature.

This is the 'individual share' of the total pressure contributed by gas 'i', reflecting its molecular collisions with the container walls.

X_{i}

The ratio of the number of moles of component 'i' to the total number of moles of all components in the mixture.

This represents the proportion or relative abundance of gas 'i' in the mixture. A higher mole fraction means gas 'i' makes up a larger part of the total gas molecules.

P_{t o t a l}

The sum of the partial pressures of all individual gases in the mixture; the overall pressure exerted by the gas mixture on the container walls.

This is the combined effect of all gas molecules colliding with the walls, representing the total force per unit area.

Free study cues

Insight

Canonical usage

This equation requires the partial pressure and total pressure to be expressed in the same units, as the mole fraction is dimensionless.

Common confusion

A common mistake is using different units for $P_{i}$ and $P_{t}$ otal (e.g., Pa for $P_{i}$ and atm for $P_{t}$ otal) without proper conversion, leading to incorrect results.

Dimension note

The mole fraction ( $X_{i}$ ) is a dimensionless quantity, representing the ratio of the number of moles of a component to the total number of moles in the mixture. This makes it a pure number that scales the total pressure.

Unit systems

$P_{i}$ Pa, atm, bar, mmHg · The partial pressure of component 'i'. Must be in the same unit as P_total.

$P_{t o t a l}$ Pa, atm, bar, mmHg · The total pressure of the gas mixture. Must be in the same unit as P_i.

$X_{i}$ dimensionless (e.g., mol/mol) · The mole fraction of component 'i'. It is a ratio of moles and therefore has no units.

One free problem

Practice Problem

A gas mixture has a total pressure of 1.5 atm. If the mole fraction of Gas A is 0.2, what is the partial pressure of Gas A?

Mole Fraction of Gas i0.2

Total Pressure1.5 atm

Solve for: $P_{i}$

Hint: Multiply the mole fraction by the total pressure.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Determining the partial pressure of oxygen in the air at a given atmospheric pressure to understand its availability for respiration.

Study smarter

Tips

Ensure the mole fraction is a dimensionless value between 0 and 1.
The units of partial pressure ( $P_{i}$ ) will be the same as the units of total pressure ( $P_{t}$ otal).
Remember that the sum of all mole fractions in a mixture must equal 1.
Mole fraction ( $X_{i}$ ) is defined as $n_{i}$ / $n_{t}$ otal.

Avoid these traps

Common Mistakes

Using percentage composition instead of mole fraction (convert percentage to decimal first).
Confusing partial pressure with total pressure.
Incorrectly calculating mole fraction (e.g., using mass fraction instead of mole fraction).

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The partial pressure of a gas in a mixture is its mole fraction multiplied by the total pressure of the mixture.

Apply this formula when you know the total pressure of a gas mixture and the mole fraction of a specific component gas, and you need to find that component's partial pressure. It's also useful for calculating mole fraction if partial and total pressures are known.

Understanding this relationship is vital in fields like chemical engineering for designing separation processes, environmental science for analyzing atmospheric composition, and medicine for understanding gas exchange in the body. It allows for precise quantification of individual gas contributions in complex systems.

Using percentage composition instead of mole fraction (convert percentage to decimal first). Confusing partial pressure with total pressure. Incorrectly calculating mole fraction (e.g., using mass fraction instead of mole fraction).

Determining the partial pressure of oxygen in the air at a given atmospheric pressure to understand its availability for respiration.

Ensure the mole fraction is a dimensionless value between 0 and 1. The units of partial pressure (P_i) will be the same as the units of total pressure (P_total). Remember that the sum of all mole fractions in a mixture must equal 1. Mole fraction (X_i) is defined as n_i / n_total.

References

Sources

Atkins' Physical Chemistry (P. W. Atkins, J. de Paula)
Wikipedia: Dalton's law
IUPAC Gold Book: Partial pressure
IUPAC Gold Book: Mole fraction
Atkins' Physical Chemistry
Atkins' Physical Chemistry, 11th Edition, Chapter 1: The properties of gases
Wikipedia: Ideal Gas Law
Chemistry: The Central Science, Brown, LeMay, Bursten — Chapter 10: Gases