Vapour Pressure of Solutions — Raoult’s Law (Ideal Solutions)
For an ideal liquid solution composed of volatile components A, B, … at fixed temperature,
the partial vapour pressure of each component is proportional to its liquid-phase mole fraction.
\[
P_i = x_i\,P_i^{\ast} \qquad (i=A,B,\ldots)
\]
Here \(x_i\) is the mole fraction of component \(i\) in the liquid, and \(P_i^{\ast}\) is the
vapour pressure of the pure component \(i\) at the same temperature. The total
pressure is the sum of partial pressures (Dalton’s law):
\[
P_{\text{tot}}=\sum_i P_i, \qquad
y_i=\frac{P_i}{P_{\text{tot}}}\ \text{is the vapour-phase mole fraction.}
\]
From composition to mole fractions
Let the user supply either moles or masses. The calculator handles both:
- If moles are given: \(n_i\) known ⇒ \(x_i = n_i/\sum_j n_j\).
- If masses are given: compute \(n_i = m_i/M_i\) from the chemical formula (molar mass \(M_i\) in \(\mathrm{g\,mol^{-1}}\)), then
\(x_i = n_i/\sum_j n_j\).
Two-component (A–B) quick relations
- \(x_A + x_B = 1\).
- \(P_A = x_A P_A^{\ast},\quad P_B = x_B P_B^{\ast}\).
- \(P_{\text{tot}} = x_A P_A^{\ast} + x_B P_B^{\ast}\).
- \(y_A = P_A/P_{\text{tot}},\quad y_B = P_B/P_{\text{tot}}\).
Pressure units handled by the calculator
Inputs for \(P_i^{\ast}\) may be in any common unit, and results can be displayed in your chosen unit.
Internally the tool converts via \(\mathrm{Pa}\) using:
- \(1\ \mathrm{atm}=101.325\ \mathrm{kPa}=1.01325\ \mathrm{bar}=760\ \mathrm{mmHg}=760\ \mathrm{Torr}=14.6959\ \mathrm{psi}\)
- \(1\ \mathrm{bar}=10^5\ \mathrm{Pa}\), \(1\ \mathrm{kPa}=10^3\ \mathrm{Pa}\), \(1\ \mathrm{mbar}=100\ \mathrm{Pa}\)
- \(1\ \mathrm{mmHg} = 133.322\ \mathrm{Pa}\) (exact to four sig figs in this tool).
Worked example (matches textbook data)
At \(25^\circ\mathrm{C}\), \(P^{\ast}_{\mathrm{benz}}=95.1\ \mathrm{mmHg}\) and \(P^{\ast}_{\mathrm{tol}}=28.4\ \mathrm{mmHg}\).
A solution has \(x_{\mathrm{benz}}=x_{\mathrm{tol}}=0.500\). Find \(P_{\mathrm{benz}}, P_{\mathrm{tol}}, P_{\text{tot}}\) and the vapour composition.
\[
P_{\mathrm{benz}} = x_{\mathrm{benz}}P_{\mathrm{benz}}^{\ast}
= 0.500\times 95.1 = 47.6\ \mathrm{mmHg}
\]
\[
P_{\mathrm{tol}} = x_{\mathrm{tol}}P_{\mathrm{tol}}^{\ast}
= 0.500\times 28.4 = 14.2\ \mathrm{mmHg}
\]
\[
P_{\text{tot}} = 47.6 + 14.2 = 61.8\ \mathrm{mmHg}
\]
\[
y_{\mathrm{benz}}=\frac{47.6}{61.8}=0.770,\qquad
y_{\mathrm{tol}} =\frac{14.2}{61.8}=0.230.
\]
Including a non-volatile solute
To model a non-volatile solute B, set \(P_B^{\ast}=0\). Then
\(P_{\text{tot}}=x_A P_A^{\ast}\) and the vapour is pure A (\(y_A=1\)).
Assumptions and when Raoult’s law fails
- Ideal solution: intermolecular interactions A–A, B–B, and A–B are effectively the same.
- Same temperature for all \(P_i^{\ast}\): \(P_i^{\ast}\) is strongly temperature-dependent.
- Deviations from ideality: positive or negative deviations occur when A–B interactions differ from like–like.
Activity coefficients \(\gamma_i\neq 1\) are then needed; the simple linear law no longer holds.
- Azeotropes: some mixtures have extrema in \(P_{\text{tot}}\) and constant-boiling compositions; Raoult’s law is only approximate near these points.
What the calculator shows
- Parses each formula to determine \(M_i\) (g·mol⁻¹).
- Converts masses to moles (if needed) and computes \(x_i\).
- Converts your input \(P_i^{\ast}\) to \(\mathrm{Pa}\), applies \(P_i=x_i P_i^{\ast}\), and sums to \(P_{\text{tot}}\).
- Computes vapour composition \(y_i=P_i/P_{\text{tot}}\) and reports everything in your selected pressure unit.
Related concept: at very low liquid-phase mole fractions of a volatile solute in a solvent, the solute often follows
Henry’s law \(P_{\text{solute}}=k_x x_{\text{solute}}\) rather than Raoult’s law.
