Mixtures of Gases — Ideal Gas & Dalton’s Law
For a nonreactive mixture of ideal gases in one container, the gases share the same
temperature \(T\), volume \(V\), and total pressure \(P_\text{tot}\).
Two ideas tie everything together:
Ideal-gas (mixture):
\[
P_\text{tot}=\frac{n_\text{tot}RT}{V},\qquad n_\text{tot}=\sum_i n_i
\]
Dalton’s law of partial pressures:
\[
P_\text{tot}=\sum_i P_i,\qquad P_i=x_i\,P_\text{tot},\qquad x_i=\frac{n_i}{n_\text{tot}}
\]
At the same \(T\) and \(P\) for all components:
\(\displaystyle \frac{P_i}{P_\text{tot}}=\frac{n_i}{n_\text{tot}}=\frac{V_i}{V_\text{tot}}=x_i\).
Thus, “percent by volume” equals “mole percent” for ideal gases measured together.
What you can compute
- Container mode (PV = nRT): Give \(T\), \(V\), and each \(n_i\). The tool finds \(n_\text{tot}\), \(P_\text{tot}\), all \(x_i\), and each \(P_i\).
- Dalton mode (given \(P_\text{tot}\)): Give \(P_\text{tot}\) and either all \(x_i\) (they are normalized if needed) or all \(n_i\). The tool returns all \(x_i\) and \(P_i=x_iP_\text{tot}\).
Units & constants
Use absolute temperature (kelvin) and keep \(R\) consistent with your pressure–volume units:
- \(R=8.314462618\ \mathrm{Pa\,m^3\,mol^{-1}\,K^{-1}}\)
- \(R=0.08314\ \mathrm{bar\,L\,mol^{-1}\,K^{-1}}\)
- \(R=0.082057\ \mathrm{atm\,L\,mol^{-1}\,K^{-1}}\)
- \(R=62.364\ \mathrm{L\,torr\,mol^{-1}\,K^{-1}}\)
How to solve (by hand or to follow the tool)
- Convert \(T\to\mathrm{K}\) (add 273.15 if °C) and \(V\) to a unit matching your \(R\).
- If \(n_i\) are known: sum to \(n_\text{tot}\), get \(P_\text{tot}\) from \(PV=nRT\).
- If \(x_i\) are known: ensure \(\sum x_i=1\) (normalize if not), then \(P_i=x_iP_\text{tot}\).
- Report each \(x_i\) and \(P_i\). Check that \(\sum P_i=P_\text{tot}\).
Worked mini-examples
Example A (container mode): Mixture has \(0.50\ \mathrm{mol}\ \mathrm{H_2}\) and \(1.25\ \mathrm{mol}\ \mathrm{He}\) in \(V=5.0\ \mathrm{L}\) at \(T=293\ \mathrm{K}\). Find \(P_\text{tot}\), \(P_{\mathrm{H_2}}\), \(P_{\mathrm{He}}\) in bar.
\[
n_\text{tot}=0.50+1.25=1.75\ \mathrm{mol},\qquad
P_\text{tot}=\frac{n_\text{tot}RT}{V}=
\frac{(1.75)(0.08314)(293)}{5.0}\ \text{bar}\approx 8.5\ \text{bar}
\]
\[
x_{\mathrm{H_2}}=\frac{0.50}{1.75}=0.286,\quad
x_{\mathrm{He}}=\frac{1.25}{1.75}=0.714
\]
\[
P_{\mathrm{H_2}}=0.286\times 8.5\approx 2.4\ \text{bar},\qquad
P_{\mathrm{He}}=0.714\times 8.5\approx 6.1\ \text{bar}
\]
Example B (Dalton mode): A gas mixture at \(P_\text{tot}=1.00\ \text{atm}\) has composition by moles \(x_{\mathrm{CO_2}}=0.10\), \(x_{\mathrm{O_2}}=0.20\), \(x_{\mathrm{N_2}}=0.70\). Find each partial pressure.
\[
P_i=x_iP_\text{tot}\ \Rightarrow\
P_{\mathrm{CO_2}}=0.10\ \text{atm},\;
P_{\mathrm{O_2}}=0.20\ \text{atm},\;
P_{\mathrm{N_2}}=0.70\ \text{atm}
\]
\[
\sum P_i=1.00\ \text{atm}=P_\text{tot}\quad \checkmark
\]
Quality checks & common pitfalls
- Checks: \(\sum x_i=1\) and \(\sum P_i=P_\text{tot}\). Increasing \(n_\text{tot}\) or \(T\) at fixed \(V\) raises \(P_\text{tot}\); increasing \(V\) lowers it.
- Absolute temperature: never use °C in \(PV=nRT\); convert to K.
- Right \(R\): choose \(R\) consistent with the pressure & volume units you use.
- Nonreactive assumption: Dalton’s law is applied to mixtures that do not react during measurement.
- Fractions vs. percents: if the inputs are in %, convert to fractions by dividing by 100.
How this tool applies the theory
- Container mode: converts \(T, V\) to SI/selected units, sums \(n_i\) to get \(n_\text{tot}\), computes \(P_\text{tot}\) from \(PV=nRT\), then finds all \(x_i\) and \(P_i=x_iP_\text{tot}\).
- Dalton mode: takes \(P_\text{tot}\) and either:
- all \(x_i\) (normalizes if \(\sum x_i\neq 1\)) and returns \(P_i=x_iP_\text{tot}\); or
- all \(n_i\), computes \(x_i=n_i/n_\text{tot}\), then \(P_i\).
