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Thermodynamics

Ideal Gas Law (Molecular Form)

The molecular ideal-gas law uses the number of particles rather than the amount in moles.

P V = N k_{B} T

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Source & review

Difficulty

Intermediate

Read time

5 min

Prerequisites

Basic algebra, consistent units, and the stated model assumptions

Review statusNeeds review

Source

OpenStax, College Physics 2e, §13.3 (Paul Peter Urone and Roger Hinrichs)

OpenFormulaGraph equation reference; FormuLab adaptation details and license are provided below.

Adapted from OpenFormulaGraph, under CC BY 4.0.

FormuLab adapted wording, notation, applicability conditions and topic placement, and expanded the numerical check into a worked example.

Definition

Understanding the core concept

The molecular ideal-gas law uses the number of particles rather than the amount in moles. It is equivalent to the molar form because the gas constant equals Avogadro’s constant times Boltzmann’s constant.

Variables & Units

Understanding each component

Symbol	Meaning	Units
$P$	Pressure	$Pa$
$V$	Volume	$m^{3}$
$N$	Number of Particles	$1$
$k_{B}$	Boltzmann Constant	$J/K$
$T$	Temperature	$K$

Real-World Applications

Where this formula is used in practice

Gas particle counts

Relate number of gas particles to a measured state.

Gas-state models

Estimate pressure using molecular rather than molar amount.

Worked Example

Step-by-step calculation with real numbers

Problem

Find the absolute pressure of 2.50 × 10²³ gas particles in 0.0100 m³ at 300 K.

Given

V = 0.01 m^{3}

N = 2.5 \times 1 0^{23}

k_{B} = 1.380649 \times 1 0^{- 23} J/K

T = 300 K

Solution

P V = N k_{B} T

P = \frac{N \cdot k _{B} \cdot T}{V}

P = \frac{( 2.5 \times 1 0 ^{23} ) \cdot ( 1.380649 \times 1 0 ^{- 23} J/K ) \cdot ( 300 K )}{( 0.01 m ^{3} )}

P \approx 103500 Pa

Final Answer

P \approx 103500 Pa

Formula Guide

Info

DifficultyIntermediate

Read Time5 min

PrerequisitesBasic algebra, consistent units, and the stated model assumptions

Tools

Source & review

StatusNeeds review

Last reviewedPending

Source

OpenStax, College Physics 2e, §13.3 (Paul Peter Urone and Roger Hinrichs)

OpenFormulaGraph equation reference; FormuLab adaptation details and license are provided below.

Adapted from OpenFormulaGraph, under CC BY 4.0.

FormuLab adapted wording, notation, applicability conditions and topic placement, and expanded the numerical check into a worked example.

Assumptions

• Assume a dilute ideal gas in an equilibrium state.
• Use absolute pressure, kelvin temperature, and the number of molecules or atoms represented by N.

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Ideal Gas Law (Molecular Form)

Gas particle counts

Gas-state models

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