Question

A sample containing \(0.0500 \mathrm{~mol} \mathrm{Fe}_{2}\left(\mathrm{SO}_{4}\right)_{3}\) is dissolved in enough water to make \(1.00 \mathrm{~L}\) of solution. This solution contains hydrated \(\mathrm{SO}_{4}{ }^{2-}\) and \(\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}{ }^{3+}\) ions. The latter behaves as an acid: $$\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}^{3+}(a q) \rightleftharpoons \mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{5} \mathrm{OH}^{2+}(a q)+\mathrm{H}^{+}(a q)$$ a. Calculate the expected osmotic pressure of this solution at \(25^{\circ} \mathrm{C}\) if the above dissociation is negligible. b. The actual osmotic pressure of the solution is \(6.73\) atm at \(25^{\circ} \mathrm{C}\). Calculate \(K_{\mathrm{a}}\) for the dissociation reaction of \(\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}^{3+}\). (To do this calculation, you must assume that none of the ions goes through the semipermeable membrane. Actually, this is not a great assumption for the tiny \(\mathrm{H}^{+}\) ion.)

Short answer

The expected osmotic pressure of the solution at 25°C is 6.14 atm. The dissociation constant (Ka) for the dissociation reaction of \(\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}^{3+}\) is \(7.78 \times 10^{-3}\).

Step-by-step solution

Determine ion concentrations

First, let's find the initial concentrations of ions in the solution by taking into account the dissociation of \(\mathrm{Fe}_{2}\left(\mathrm{SO}_{4}\right)_{3}\) which forms 2 Fe³⁺ and 3 SO₄²⁻: For Fe³⁺: Initial concentration = 2 × 0.0500 mol/L = 0.100 mol/L For SO₄²⁻: Initial concentration = 3 × 0.0500 mol/L = 0.150 mol/L

Calculate the expected osmotic pressure

We should now calculate the expected osmotic pressure using the van 't Hoff equation: \(\Pi = iMRT\) where \(\Pi\) is the osmotic pressure, \(i\) is the van 't Hoff factor, \(M\) is the molar concentration, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin. Since the dissociation of \(\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}^{3+}\) is negligible, the van 't Hoff factor is the total number of dissociated ions, which is 2 + 3 = 5. Temperature: 25°C = 298.15 K Ideal gas constant (R) = 0.0821 L atm/mol K Expected osmotic pressure: \(\Pi = iMRT = (5)(0.0500\,\text{mol/L})(0.0821\,\text{L atm/mol K})(298.15\,\text{K})\) \(\Pi = 6.14\,\text{atm}\)

Calculate the dissociation constant (Ka)

We are given the actual osmotic pressure as 6.73 atm. To find the dissociation constant \(K_{a}\), we first determine the change in concentration caused by the dissociation of \(\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}^{3+}\): Difference in osmotic pressure: 6.73 atm - 6.14 atm = 0.59 atm Divide by RT to obtain the change in molar concentration: \(\Delta M = \frac{0.59\,\text{atm}}{(0.0821\,\text{L atm/mol K})(298.15\,\text{K})} = 0.0242\,\text{mol/L}\) Now we can set up an equilibrium expression for the dissociation reaction: \(K_{a} = \frac{[\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{5}\mathrm{OH}^{2+}][\mathrm{H}^{+}]}{[\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}^{3+}]}\) Initially: \([H^{+}] = \Delta M = 0.0242\,\text{mol/L}\) \([\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{5}\mathrm{OH}^{2+}] = \Delta M = 0.0242\,\text{mol/L}\) \([\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}^{3+}] = 0.100\,\text{mol/L} - \Delta M = 0.0758\,\text{mol/L}\) Substitute the values into the expression: \(K_{a} = \frac{(0.0242\,\text{mol/L})(0.0242\,\text{mol/L})}{(0.0758\,\text{mol/L})} = 7.78 \times 10^{-3}\) Thus, the dissociation constant (Ka) for the dissociation reaction of \(\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}^{3+}\) is \(7.78 \times 10^{-3}\).

Key concepts

Van 't Hoff Equation

To understand the behavior of solutions in terms of osmotic pressure, the van 't Hoff equation is fundamental. It relates osmotic pressure \(\Pi\) with the molar concentration of the particles in solution \(M\), the temperature \(T\) in Kelvin, and the ideal gas constant \(R\), expressed as \(\Pi = iMRT\). The factor \(i\) is known as the van 't Hoff factor and represents the number of particles into which a compound dissociates in solution. For example, a salt that dissociates into two ions would have an \(i\) of 2.

When dealing with non-ideal solutions or those that involve dissociation or association of particles, the accurate determination of \(i\) becomes more complex. In these scenarios, it's not just the initial amounts of substances that need to be considered, but also the degree to which they dissociate or associate. Therefore, using the van 't Hoff equation can give a predicted osmotic pressure useful for comparison with actual measured values, leading to insights about the nature of the solution's components.

Dissociation Constant (Ka)

The dissociation constant, denoted as \(Ka\), is a quantitative measure of the strength of an acid in solution. It refers to the equilibrium constant for the dissociation reaction of an acid into its ions in water. A larger \(Ka\) value corresponds to a stronger acid, indicating a greater tendency to lose its proton (\(H^+\)).

The calculation of \(Ka\) often involves setting up an equilibrium expression based on the concentrations of the reactants and products at equilibrium. In essence, \(Ka\) is defined by the ratio \(K_{a} = \frac{[A^{-}][H^{+}]}{[HA]}\), where \( [A^{-}] \) and \( [H^{+}] \) are the concentrations of the conjugate base and hydrogen ion, respectively, and \( [HA] \) is the concentration of the undissociated acid. Through \(Ka\) and its related \(pKa\), we can infer both the direction of the equilibrium and the relative strength of the acid.

Chemical Equilibrium

Chemical equilibrium is a state in which the rate of the forward reaction equals the rate of the reverse reaction, meaning the concentrations of reactants and products remain constant over time. It is represented in an equation by the equilibrium constant, \(K_{eq}\), which at a given temperature, specifies the ratio of product concentrations to reactant concentrations, each raised to their respective stoichiometric coefficients.

In the context of acid dissociation, the equilibrium will involve the concentrations of the acid, the conjugate base, and the hydrogen ions. Understanding equilibrium is crucial when interpreting the \(K_{a}\) of an acid, as it informs us of the relative amounts of species in solution at equilibrium. When solving problems related to equilibrium, it is essential to consider the initial concentrations of the species involved and how these concentrations change as equilibrium is established.

Acid Dissociation in Aqueous Solution

Acids dissolve in water by dissociating to some extent into ions. The degree to which this dissociation occurs is governed by the acid's \(K_{a}\) and the solution's pH. For a generic acid \(HA\), the dissociation in water can be represented as \(HA(aq) \rightleftharpoons A^{-}(aq) + H^{+}(aq)\). The pH of the solution provides information on the concentration of hydrogen ions (\(H^+\)) and consequently on the degree of acidity.

When an acid dissociates in an aqueous solution, it contributes to the solution's osmotic pressure because the dissociated ions (hydrogen ions and the conjugate base) affect the solution’s colligative properties. It's crucial to analyze the initial concentrations, the extent of dissociation, and the resulting equilibrium to accurately determine the osmotic pressure, as these factors play a direct role in the solution's behavior and characteristics.