Q. how to calculate vapor pressure of a solution

Goal. Explain how to calculate the vapor pressure of a solution and show a worked numerical example, with every step explained.

Basic principles. For an ideal solution made of volatile components, Raoult’s law gives the partial vapor pressure of component i as

\[ p_i = x_i \, p_i^\ast \]

where \( x_i \) is the mole fraction of component i in the liquid phase and \( p_i^\ast \) is the vapor pressure of the pure component i at the same temperature. The total vapor pressure of the solution is the sum of the partial pressures of all volatile components,

\[ p_{\text{total}} = \sum_i p_i = \sum_i x_i \, p_i^\ast . \]

For a solution in which one component is nonvolatile (for example a nonvolatile solute dissolved in a volatile solvent), only the volatile component contributes to the vapor pressure. If component 1 is the volatile solvent and component 2 is a nonvolatile solute,

\[ p = x_1 \, p_1^\ast . \]

For nonideal solutions, introduce the activity coefficient \( \gamma_i \) and replace \( x_i \) by the activity \( a_i = \gamma_i x_i \). Then

\[ p_i = a_i \, p_i^\ast = x_i \, \gamma_i \, p_i^\ast . \]

Step-by-step procedure to calculate vapor pressure (ideal solution or nonvolatile solute case).

Step 1. Identify which components are volatile. If only the solvent is volatile and the solute is nonvolatile, you will use \( p = x_{\text{solvent}} \, p_{\text{solvent}}^\ast \). If both are volatile, use \( p_{\text{total}} = \sum x_i p_i^\ast \).

Step 2. Obtain or compute the vapor pressures of the pure components at the temperature of interest. These are the \( p_i^\ast \) values from tables or experimental data.

Step 3. Compute the mole numbers of each species. If you are given masses, convert to moles by

\[ n_j = \frac{m_j}{M_j} \]

where \( m_j \) is the mass and \( M_j \) is the molar mass of species j.

Step 4. Compute mole fractions. For a two-component liquid,

\[ x_1 = \frac{n_1}{n_1 + n_2} , \qquad x_2 = \frac{n_2}{n_1 + n_2} . \]

Step 5. Apply Raoult’s law (or the activity-corrected form). For a volatile solvent 1 and nonvolatile solute 2, compute

\[ p = x_1 \, p_1^\ast . \]

If the solution is nonideal and you know the activity coefficient, use

\[ p = x_1 \, \gamma_1 \, p_1^\ast . \]

Step 6. If needed, compute the vapor-pressure lowering and relative lowering. Vapor-pressure lowering is

\[ \Delta p = p_1^\ast – p = p_1^\ast (1 – x_1) = p_1^\ast x_2 . \]

The relative lowering is

\[ \frac{\Delta p}{p_1^\ast} = x_2 . \]

Worked numerical example. Find the vapor pressure at 25 degrees Celsius of a solution made by dissolving 10.0 grams of a nonvolatile, non-electrolyte solute (molar mass 180.16 g mol^{-1}) in 100.0 grams of water. The vapor pressure of pure water at 25 degrees Celsius is \( p_{\text{H2O}}^\ast = 23.8 \) mmHg.

Step A. Compute moles of solute and solvent.

\[ n_{\text{solute}} = \frac{m_{\text{solute}}}{M_{\text{solute}}} = \frac{10.0}{180.16} = 0.0555\ \text{mol} . \]

\[ n_{\text{H2O}} = \frac{m_{\text{H2O}}}{M_{\text{H2O}}} = \frac{100.0}{18.015} = 5.553\ \text{mol} . \]

Step B. Compute mole fraction of the solvent (water).

\[ x_{\text{H2O}} = \frac{n_{\text{H2O}}}{n_{\text{H2O}} + n_{\text{solute}}} = \frac{5.553}{5.553 + 0.0555} = \frac{5.553}{5.6085} = 0.9911 . \]

Step C. Apply Raoult’s law for a nonvolatile solute.

\[ p = x_{\text{H2O}} \, p_{\text{H2O}}^\ast = 0.9911 \times 23.8\ \text{mmHg} = 23.6\ \text{mmHg} \text{ (rounded)}. \]

Step D. Vapor-pressure lowering.

\[ \Delta p = p_{\text{H2O}}^\ast – p = 23.8 – 23.6 = 0.18\ \text{mmHg} . \]

Notes and approximations.

1. For very dilute solutions, \( x_{\text{solute}} \ll 1 \) and \( x_{\text{solvent}} \approx 1 – x_{\text{solute}} \). Then \( \Delta p \approx p^\ast x_{\text{solute}} \).

2. If the solute is an electrolyte that dissociates into i particles per formula unit, you can approximate the effective solute mole fraction by multiplying by the van ‘t Hoff factor \( i \) for colligative properties. More rigorously, account for activity coefficients when ionic strength is significant.

3. If both components are volatile, calculate each partial pressure using \( p_i = x_i p_i^\ast \) and sum them for the total pressure. If nonideal, include activity coefficients \( \gamma_i \) so \( p_i = x_i \gamma_i p_i^\ast \).

Summary checklist you can follow for any problem.

1. Identify volatile species. 2. Convert masses to moles. 3. Compute mole fractions. 4. Use Raoult’s law \( p_i = x_i p_i^\ast \) for ideal behavior. 5. If nonideal, use \( p_i = x_i \gamma_i p_i^\ast \). 6. Sum partial pressures if more than one volatile component. 7. Compute vapor-pressure lowering if required.