Raoult's Law describes the vapor pressure of a component in an ideal solution:
\[ p_A = x_A p_A^* \]This expression works especially well for the solvent in a dilute solution, because the solvent is nearly pure:
\[ x_A \rightarrow 1 \]But a solute in a dilute solution is in a very different environment. As
\[ x_B \rightarrow 0 \]each solute molecule is surrounded almost entirely by solvent molecules rather than by other solute molecules. Consequently, the interactions experienced by a dilute solute are very different from those in the pure liquid. Because of this, the pure solute is often not the most useful reference state for describing the behavior of the dissolved solute.
Experimentally, many dilute solutes show a different simple limiting behavior. Their vapor pressure is proportional to their mole fraction:
\[ p_B = K_H x_B \]This relationship is called Henry's Law. The constant \(K_H\) is the Henry's Law constant for the solute in a particular solvent at a particular temperature.
Henry's Law is most commonly applied to gases dissolved in liquids, such as oxygen dissolved in water or carbon dioxide dissolved in soft drinks.
The value of \(K_H\) measures how readily the solute escapes from the liquid phase into the vapor phase. A large value of \(K_H\) means that a small amount of dissolved solute produces a relatively large vapor pressure. A small value of \(K_H\) means that the solute remains more strongly in the liquid phase.
The important contrast is this: Raoult's Law describes the solvent limit, where a component approaches its pure-liquid state, while Henry's Law describes the dilute-solute limit, where a solute molecule is surrounded almost entirely by solvent.
Big picture: Henry's Law gives us the natural limiting behavior for dilute solutes. It also prepares the way for activity, because real solutions often require us to replace concentration with an effective concentration that reflects the molecular environment of the solute.
Henry's Law is often written in terms of mole fraction:
\[ p_A = K_H x_A \]However, in many practical applications, the concentration of the dissolved gas is more useful than its mole fraction. Environmental chemists, engineers, and biochemists are often interested in quantities such as the concentration of dissolved oxygen in a lake or the concentration of carbon dioxide in blood.
For this reason, Henry's Law is frequently written in the alternative form
\[ c_A = k_A p_A \]where \(c_A\) is the concentration of the dissolved gas, \(p_A\) is its partial pressure above the solution, and \(k_A\) is a Henry's Law constant appropriate for this form of the equation.
This expression states that the concentration of a dissolved gas is directly proportional to its partial pressure. Doubling the partial pressure doubles the equilibrium concentration of the gas in the liquid.
The value of \(k_A\) depends on the identity of the gas, the solvent, and the temperature. Different gases therefore dissolve to different extents even when exposed to the same pressure.
Big picture: The mole-fraction and concentration forms of Henry's Law describe the same physical phenomenon: the equilibrium distribution of a dilute solute between a gas phase and a liquid phase. The concentration form is often more convenient because concentrations can be measured directly and are commonly used in biological, environmental, and engineering applications.
The partial pressure of oxygen in air is approximately 0.19 atm. Assuming Henry's Law applies and that the Henry's Law constant for oxygen dissolved in water at 25 °C is
\[ K_H = 769\ \text{atm} \]calculate the mole fraction of dissolved oxygen in water.
Henry's Law relates the partial pressure of a dissolved gas to its mole fraction in solution:
\[ p_{O_2}=K_H x_{O_2} \]Solving for the mole fraction gives
\[ x_{O_2}=\frac{p_{O_2}}{K_H} \]Substituting the known values:
\[ x_{O_2} = \frac{0.19\ \text{atm}} {769\ \text{atm}} \] \[ x_{O_2} = 2.47 \times 10^{-4} \]Therefore, the equilibrium mole fraction of oxygen dissolved in water is
\[ \boxed{x_{O_2}=2.47\times10^{-4}} \]The mole fraction of oxygen dissolved in water at equilibrium with air was found to be
\[ x_{O_2}=2.47\times10^{-4} \]Because the solution is very dilute, the mole fraction can be interpreted as approximately 2.47 × 10-4 mol of oxygen per mole of water.
One liter of water has a mass of approximately 1000 g, corresponding to
\[ n_{H_2O} = \frac{1000\ \text{g}} {18.015\ \text{g mol}^{-1}} = 55.5\ \text{mol} \]Therefore, the number of moles of dissolved oxygen per liter is
\[ n_{O_2} = (2.47\times10^{-4})(55.5) = 1.37\times10^{-2}\ \text{mol} \]Since this amount is contained in approximately 1.00 L of solution,
\[ [O_2] = 1.37\times10^{-2}\ \text{mol L}^{-1} \]or
\[ \boxed{[O_2]=0.0137\ \text{M}} \]Physical interpretation: Even though the mole fraction of dissolved oxygen is very small, liquid water contains about 55.5 mol of water molecules per liter. As a result, a seemingly tiny mole fraction corresponds to a measurable concentration of dissolved oxygen.
Physical interpretation: Although oxygen is essential for aquatic life, only a very small fraction of the molecules in the solution are oxygen molecules. The relatively large value of the Henry's Law constant indicates that oxygen is only moderately soluble in water, so most oxygen remains in the gas phase rather than dissolving into the liquid.
Henry's Law relates the partial pressure of a dissolved solute to its mole fraction in solution:
\[ p_B = K_H x_B \]Two quantities are given below. Calculate the missing quantity.
| Quantity | Value |
|---|---|
| \(K_H\) | |
| \(p_B\) | |
| \(x_B\) | |
| Your answer | |
Henry's Law is sometimes written as
\[ c_A = k_A p_A \]where \(c_A\) is the concentration of the dissolved gas, \(k_A\) is the Henry's Law constant, and \(p_A\) is the partial pressure of the gas above the solution.
Two quantities are given below. Calculate the missing quantity.
| Quantity | Value |
|---|---|
| \(k_A\) | |
| \(p_A\) | |
| \(c_A\) | |
| Your answer | |
Big picture: Henry's Law provides the thermodynamic description of dilute solutes in solution, just as Raoult's Law provides the limiting behavior of solvents. Together these laws establish the reference states used to describe real solutions and motivate the introduction of activity and activity coefficients.