Total vapour pressure Equation, Derivation & Rearrangements

Core idea

Overview

Total vapor pressure represents the cumulative force per unit area exerted by all volatile components in a gas phase at equilibrium with their liquid counterparts. In physical chemistry, this is fundamentally described by Dalton’s Law of Partial Pressures, which states that the total pressure is the sum of the pressures each component would exert if it occupied the volume alone.

When to use: Use this formula when analyzing a mixture of non-reacting vapors or gases in a closed system to find the combined pressure. It is particularly relevant when working with Raoult's Law calculations for multi-component liquid solutions where each component contributes to the headspace pressure.

Why it matters: Understanding total vapor pressure is vital for the design of distillation columns in chemical engineering and for predicting the boiling points of mixtures. It also plays a key role in environmental science for determining the concentration of pollutants in the atmosphere above contaminated water sources.

Symbols

Variables

P_1 = Partial Pressure 1, P_2 = Partial Pressure 2, P_{total} = Total Pressure

P_{1}

Partial Pressure 1

k P a

P_{2}

Partial Pressure 2

k P a

P_{t o t a l}

Total Pressure

k P a

Walkthrough

Derivation

Understanding Total Vapour Pressure (Ideal Solutions)

Uses Dalton’s law and Raoult’s law to calculate total pressure above an ideal liquid mixture.

Mixture behaves ideally.

1

Apply Dalton’s Law:

Total pressure is the sum of partial pressures.

P_{total} = p_{A} + p_{B}

2

Substitute Raoult’s Law:

Replace each partial pressure using p=xp*.

P_{total} = x_{A} p_{A}^{⋆} + x_{B} p_{B}^{⋆}

Result

P_{total} = x_{A} p_{A}^{⋆} + x_{B} p_{B}^{⋆}

Source: Standard curriculum — A-Level Chemistry (Ideal solutions)

Free formulas

Rearrangements

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

Make Ptot the subject

P_{t o t a l} = \sum P_{i}

Ptot is already the subject of the formula.

Difficulty: 1/5

Solve for $P_{1}$

Make P1 the subject

P_{1} = P_{t o t a l} - P_{2}

To make $P_{1}$ the subject, first expand the sum of partial pressures, then isolate $P_{1}$ by subtracting other partial pressures from the total pressure.

Difficulty: 2/5

Solve for $P_{2}$

Make P2 the subject

P_{2} = P_{t o t a l} - P_{1}

To make P2 the subject from the total vapour pressure equation, first expand the sum of partial pressures, then subtract P1 from both sides and rearrange.

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 is a straight line with a positive slope of 1, where increasing P1 results in a direct, proportional increase in total pressure. The y-intercept represents the constant value of P2, meaning that even when P1 is at its smallest, the total pressure remains equal to the partial pressure of the second component. For a chemistry student, this linear relationship shows that P1 and Ptot change at the same rate, so a large x-value indicates a system dominated by the first component while a small x-value reflects

Graph type: linear

Why it behaves this way

Intuition

Imagine a container where different types of gas molecules are independently bouncing off the walls; the total force per unit area on the walls is the combined effect of all these individual collisions.

P_{t o t a l}

The total pressure exerted by a mixture of non-reacting gases or vapors in a closed system.

It's the overall 'push' on the container walls from all gas molecules combined.

P_{i}

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

Each type of gas molecule contributes its own 'share' to the total pressure, independent of the others.

\sum

The mathematical operator indicating the sum of all individual partial pressures P_i for 'i' from 1 to 'n' components.

Simply means to add up the pressures from each distinct gas in the mixture.

Free study cues

Insight

Canonical usage

The total pressure is calculated by summing individual partial pressures, requiring all terms to share the same pressure unit before addition.

Common confusion

Attempting to sum pressures expressed in different units, such as adding kilopascals (kPa) directly to atmospheres (atm) without conversion.

Dimension note

This equation is not dimensionless; it is a sum of quantities with dimensions of pressure.

Unit systems

$P_{t} o t a l$ Pa, bar, atm, or Torr · The resulting sum of all individual partial pressures.

$P_{i}$ Pa, bar, atm, or Torr · The partial pressure of component i; must match the units of P_total.

One free problem

Practice Problem

A container holds a mixture of two volatile liquids. At a specific temperature, the partial pressure of component 1 is 145 mmHg and the partial pressure of component 2 is 210 mmHg. Calculate the total vapour pressure of the system.

Partial Pressure 1145 kPa

Partial Pressure 2210 kPa

Solve for: $P t o t$

Hint: The total pressure is simply the sum of the individual partial pressures.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Calculating vapor pressure above a benzene⁻toluene mixture.

Study smarter

Tips

Always convert all individual pressure measurements to a common unit like atm, kPa, or mmHg before summing.
Recall that this law assumes ideal behavior where gas particles do not exert attractive forces on one another.
In liquid-vapor equilibrium, the total pressure is the sum of the product of each component's mole fraction and its pure vapor pressure.

Avoid these traps

Common Mistakes

Forgetting to include all volatile components.
Confusing with Raoult's law.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Uses Dalton’s law and Raoult’s law to calculate total pressure above an ideal liquid mixture.

Use this formula when analyzing a mixture of non-reacting vapors or gases in a closed system to find the combined pressure. It is particularly relevant when working with Raoult's Law calculations for multi-component liquid solutions where each component contributes to the headspace pressure.

Understanding total vapor pressure is vital for the design of distillation columns in chemical engineering and for predicting the boiling points of mixtures. It also plays a key role in environmental science for determining the concentration of pollutants in the atmosphere above contaminated water sources.

Forgetting to include all volatile components. Confusing with Raoult's law.

Calculating vapor pressure above a benzene⁻toluene mixture.

Always convert all individual pressure measurements to a common unit like atm, kPa, or mmHg before summing. Recall that this law assumes ideal behavior where gas particles do not exert attractive forces on one another. In liquid-vapor equilibrium, the total pressure is the sum of the product of each component's mole fraction and its pure vapor pressure.

References

Sources

Atkins, P. W., & de Paula, J. (2014). Atkins' Physical Chemistry (10th ed.). Oxford University Press.
Halliday, D., Resnick, R., & Walker, J. (2014). Fundamentals of Physics (10th ed.). Wiley.
Wikipedia: Dalton's law
IUPAC Gold Book
Atkins' Physical Chemistry
NIST Chemistry WebBook
Peter Atkins, Julio de Paula, James Keeler. Atkins' Physical Chemistry. 11th ed. Oxford University Press, 2018.
IUPAC Gold Book. 'Ideal gas'. DOI: 10.1351/goldbook.I02932.