Law of Combining Volumes (Gay-Lussac) — Theory
When gases react or are formed and all gas measurements are made at the
same temperature and pressure, their volumes are in the ratios of the
stoichiometric coefficients in the balanced chemical equation. This is
Gay-Lussac’s law of combining volumes. Formally, because an ideal gas obeys
\(PV=nRT\), at fixed \(T\) and \(P\) the volume of any gas is proportional to the
amount (in moles): \(V\propto n\).
Key relations
- Ideal Gas Equation: \(\displaystyle PV=nRT\)
- Same \(T\) & \(P\) (volume ratios):
\[
\frac{V_A}{V_B}=\frac{n_A}{n_B}=\frac{\nu_A}{\nu_B}
\]
where \(\nu_i\) is the stoichiometric coefficient of species \(i\) in the balanced equation.
- Stoichiometric link (any conditions):
\[
n_{\text{target}} = n_{\text{known}}\times
\frac{\nu_{\text{target}}}{\nu_{\text{known}}}
\]
then convert moles \(\leftrightarrow\) volume using \(PV=nRT\) if absolute volumes are required.
When to use what
- Use volume ratios (no \(R\), \(T\), \(P\) needed) only if both the known and the target are gases and all gas volumes are compared at the same \(T\) and \(P\).
- Use \(PV=nRT\) whenever you need an absolute gas volume at specified \(T\) and \(P\), or when the known amount is mass/moles rather than a gas volume ratio.
Constants & units
Choose a gas constant that matches the pressure–volume units:
- \(R=8.314462618\ \mathrm{J\,mol^{-1}\,K^{-1}}\) (with \(P\) in Pa, \(V\) in m³)
- \(R=0.082057\ \mathrm{L\,atm\,mol^{-1}\,K^{-1}}\)
- \(R=8.314\ \mathrm{L\,kPa\,mol^{-1}\,K^{-1}}\)
- \(R=0.08314\ \mathrm{L\,bar\,mol^{-1}\,K^{-1}}\)
Workflow used by the calculator
- Enter an unbalanced reaction. The tool balances it for you.
- Mark which species are gases. Only checked species can use volumes.
- Pick a known species and quantity (volume, moles, or mass). Mass is converted to moles using the molar mass.
- Apply the stoichiometric ratio to get moles of the target.
- If absolute gas volume is needed, convert with \(V=nRT/P\) at the specified \(T\) and \(P\). If “Same \(T\)&\(P\)” is on and both are gases, volume ratios may be used directly.
Worked examples
Example A — Volume ratio at the same \(T\) and \(P\)
Roast zinc sulfide: \(\mathrm{ZnS + O_2 \to ZnO + SO_2}\). Balanced:
\(\mathrm{2\,ZnS + 3\,O_2 \to 2\,ZnO + 2\,SO_2}\).
If \(1.00\ \mathrm{L}\) of \(\mathrm{O_2}\) reacts (same \(T,P\)), what volume of \(\mathrm{SO_2}\) forms?
Use volume ratios (coefficients of gases only):
\[
V_{\mathrm{SO_2}} = V_{\mathrm{O_2}}
\times \frac{\nu_{\mathrm{SO_2}}}{\nu_{\mathrm{O_2}}}
= 1.00\ \mathrm{L}\times \frac{2}{3}
= 0.667\ \mathrm{L}.
\]
Example B — Absolute volume via \(PV=nRT\)
Air-bag propellant: \(\mathrm{2\,NaN_3(s) \to 2\,Na(l) + 3\,N_2(g)}\).
What volume of \(\mathrm{N_2}\) is produced at \(P=98.0\ \mathrm{kPa}\) and \(T=299\ \mathrm{K}\) when \(75.0\ \mathrm{g}\) of \(\mathrm{NaN_3}\) decomposes?
- Convert mass to moles:
\[
n(\mathrm{NaN_3})=\frac{75.0\ \mathrm{g}}{65.01\ \mathrm{g\,mol^{-1}}}=1.154\ \mathrm{mol}.
\]
- Stoichiometry to moles of nitrogen:
\[
n(\mathrm{N_2})=1.154\ \mathrm{mol}\times \frac{3}{2}=1.731\ \mathrm{mol}.
\]
- Convert moles to volume with \(V=nRT/P\) (use \(R=8.314\ \mathrm{L\,kPa\,mol^{-1}\,K^{-1}}\)):
\[
V(\mathrm{N_2})=\frac{(1.731)(8.314)(299)}{98.0}\ \mathrm{L}
\approx 43.9\ \mathrm{L}.
\]
Assumptions & limits
- Gases behave ideally (low–moderate pressure, not near condensation).
- “Same \(T\)&\(P\)” must truly apply to all gas volumes compared when using ratios.
- Identify the limiting reagent when multiple reactants are provided. The tool’s single “known” amount is taken as the basis; add your own limiting-reagent check if needed.
Common pitfalls
- Using volume ratios when one of the compared species is not a gas.
- Mixing units for \(R\), \(P\) and \(V\) (e.g., kPa with L requires \(R=8.314\ \mathrm{L\,kPa\,mol^{-1}\,K^{-1}}\)).
- Forgetting to balance the equation—coefficients drive all ratios.
- Using °C in the gas law. Always convert to kelvins.
Quick checks
- At the same \(T,P\), if coefficients double from known to target, the gas volume should double.
- At STP, \(V_m \approx 22.7\ \mathrm{L\,mol^{-1}}\). Reasonable volumes should be in this ballpark times moles.
How this tool maps to the theory
- Balance & build creates the stoichiometric coefficients \(\nu_i\).
- Gas checkboxes decide which species can use volumes.
- Same T & P toggles pure volume-ratio mode. When off, the calculator applies \(PV=nRT\) using your \(P\) and \(T\).
- Known → Target: it converts the known amount to moles (if needed), applies the coefficient ratio, then converts to the requested output quantity.
