The General Gas Equation

General Chemistry • Gases

Written by STEM Calculators Team Published October 19, 2025 Updated February 24, 2026

Combined Gas Law

For the same amount of gas between two states, the combined gas law is \( \dfrac{P_iV_i}{T_i}=\dfrac{P_fV_f}{T_f} \). Solve for any single unknown; units may differ between states.

Solve for Significant figures

Initial state (i)

Pressure \(P_i\)

Volume \(V_i\)

Temperature \(T_i\)

Final state (f)

Pressure \(P_f\)

Volume \(V_f\)

Temperature \(T_f\)

Temperatures are converted to Kelvin internally. Units can differ between states; the solver converts to SI (Pa, m³, K).

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General Gas Equation — Theory & Guide

For the same amount of gas (constant \(n\)) changing from an initial state (i) to a final state (f), applying \(PV=nRT\) to both states and dividing gives the general gas equation (combined gas law):

\[ \frac{P_i V_i}{T_i}=\frac{P_f V_f}{T_f} \]

Derivation (brief)

Write \(PV=nRT\) for each state with the same \(n\) and \(R\); then divide:

\[ \begin{aligned} P_iV_i &= nRT_i \\ P_fV_f &= nRT_f \\ \frac{P_iV_i}{nRT_i} &= \frac{P_fV_f}{nRT_f} \\ \Rightarrow\quad \frac{P_iV_i}{T_i} &= \frac{P_fV_f}{T_f}. \end{aligned} \]

Symbols

\(P\): absolute pressure (Pa, kPa, bar, atm, mmHg/torr).
\(V\): volume (m³, L, mL, cm³).
\(T\): temperature in Kelvin. Convert with \(T_\text{K}=t_{^\circ\!C}+273.15\) or \(T_\text{K}=(t_{^\circ\!F}-32)\tfrac{5}{9}+273.15\).

Use absolute pressure and \(T>0\ \mathrm{K}\).

Rearrangements (solve for any one)

Final pressure \(P_f\)

\[ \begin{aligned} \frac{P_iV_i}{T_i} &= \frac{P_fV_f}{T_f} \\ P_f &= \frac{P_iV_i}{T_i}\,\frac{T_f}{V_f} \end{aligned} \]

Final volume \(V_f\)

\[ \begin{aligned} \frac{P_iV_i}{T_i} &= \frac{P_fV_f}{T_f} \\ V_f &= \frac{P_iV_i}{T_i}\,\frac{T_f}{P_f} \end{aligned} \]

Final temperature \(T_f\)

\[ \begin{aligned} \frac{P_iV_i}{T_i} &= \frac{P_fV_f}{T_f} \\ T_f &= \frac{P_fV_f}{P_iV_i}\,T_i \end{aligned} \]

Initial pressure \(P_i\)

\[ \begin{aligned} \frac{P_iV_i}{T_i} &= \frac{P_fV_f}{T_f} \\ P_i &= \frac{P_fV_f}{T_f}\,\frac{T_i}{V_i} \end{aligned} \]

Initial volume \(V_i\)

\[ \begin{aligned} \frac{P_iV_i}{T_i} &= \frac{P_fV_f}{T_f} \\ V_i &= \frac{P_fV_f}{T_f}\,\frac{T_i}{P_i} \end{aligned} \]

Initial temperature \(T_i\)

\[ \begin{aligned} \frac{P_iV_i}{T_i} &= \frac{P_fV_f}{T_f} \\ T_i &= \frac{P_iV_i}{P_fV_f}\,T_f \end{aligned} \]

Special cases

Boyle’s law (isothermal \(T\) constant): \(P_1V_1=P_2V_2 \Rightarrow P\propto \tfrac1V\).
Charles’s law (isobaric \(P\) constant): \(\dfrac{V_1}{T_1}=\dfrac{V_2}{T_2} \Rightarrow V\propto T\).
Amontons/Gay-Lussac (isochoric \(V\) constant): \(\dfrac{P_1}{T_1}=\dfrac{P_2}{T_2} \Rightarrow P\propto T\).

Worked example

A 2.50 L sample at \(98.4\ \text{kPa}\) and \(24^\circ\text{C}\) is cooled to \(-25^\circ\text{C}\) at the same pressure. Find \(V_f\).

Plan. Pressure is constant, so \(\dfrac{V_i}{T_i}=\dfrac{V_f}{T_f}\Rightarrow V_f=V_i\,\dfrac{T_f}{T_i}\).

\[ \begin{aligned} T_i &= 24 + 273.15 = 297.15\ \text{K} \\ T_f &= -25 + 273.15 = 248.15\ \text{K} \\ V_f &= 2.50\ \text{L}\times \frac{248.15}{297.15} = 2.09\ \text{L}\ \text{(3 s.f.)} \end{aligned} \]

Frequently Asked Questions

What is the general gas equation (combined gas law) formula?

For a fixed amount of gas between two states, the combined gas law is Pi x Vi / Ti = Pf x Vf / Tf. It combines the effects of changing pressure, volume, and temperature into one relationship.

Why must temperature be in Kelvin for the combined gas law?

The gas-law relationships use absolute temperature, so Celsius and Fahrenheit must be converted to Kelvin before substituting into Pi x Vi / Ti = Pf x Vf / Tf. Using non-absolute temperature breaks the proportional relationships.

When can I use the combined gas law instead of PV = nRT?

Use the combined gas law when the amount of gas stays constant and you are comparing two states. If the amount changes, PV = nRT (or another model) is needed.

How does Boyle's law or Charles's law relate to the general gas equation?

Boyle's law is the special case when temperature is constant, giving P1 x V1 = P2 x V2. Charles's law is the special case when pressure is constant, giving V1/T1 = V2/T2.

Combined Gas Law

Initial state (i)

Final state (f)

Calculation steps

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Frequently Asked Questions

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