Question

Sodium tripolyphosphate \(\left(\mathrm{Na}_{5} \mathrm{P}_{3} \mathrm{O}_{10}\right)\) is used in many synthetic detergents. Its major effect is to soften the water by complexing \(\mathrm{Mg}^{2+}\) and \(\mathrm{Ca}^{2+}\) ions. It also increases the efficiency of surfactants, or wetting agents, that lower a liquid's surface tension. The pK value for the formation of \(\mathrm{MgP}_{3} \mathrm{O}_{10}^{3-}\) is \(-8.60 .\) The reaction is $$ \mathrm{Mg}^{2+}(a q)+\mathrm{P}_{3} \mathrm{O}_{10}^{5-}(a q) \rightleftharpoons \mathrm{MgP}_{3} \mathrm{O}_{10}^{3-}(a q) $$ Calculate the concentration of \(\mathrm{Mg}^{2+}\) in a solution that was originally \(50 . \mathrm{ppm} \mathrm{Mg}^{2+}(50 . \mathrm{mg} / \mathrm{L} \text { of solution) after } 40 . \mathrm{g} \text { of }\) \(\mathrm{Na}_{5} \mathrm{P}_{3} \mathrm{O}_{10}\) is added to 1.0 \(\mathrm{L}\) of the solution.

Short answer

The final concentration of Mg^2+ ions in the solution after adding 40 g of Na5P3O10 to 1.0 L of the solution is approximately 0 mol/L.

Step-by-step solution

Calculate the concentration of sodium tripolyphosphate

First, we need to calculate the concentration of sodium tripolyphosphate (Na5P3O10) added to the solution. We are given that 40 g of Na5P3O10 is added to 1.0 L of the solution. To find the concentration in mol/L, we will divide the mass by the molar mass and the volume: Molar mass of Na5P3O10 = (5 × 22.99) + (3 × 30.97) + (10 × 16.00) g/mol = 367.87 g/mol Concentration of Na5P3O10 = (40 g) / (367.87 g/mol × 1.0 L) = 0.1087 mol/L

Calculate the reaction quotient, Q

Now we will use the given pK value (-8.60) to find the equilibrium constant, K: K = 10^(-pK) = 10^8.60 We are also given the initial concentration of Mg^2+ ions as 50 ppm, which is equal to 50 mg/L. To convert this to molarity, we need to divide by the molar mass of Mg^2+ (24.31 g/mol): Initial concentration of Mg^2+ = (50 mg/L) × (1 g / 1000 mg) × (1 mol / 24.31 g) = 0.002055 mol/L Next, we will calculate the reaction quotient, Q, using the concentrations of Mg^2+ and Na5P3O10: Q = [MgP3O10^(3-)] / ([Mg^2+] × [P3O10^(5-)]) At the start, [MgP3O10^(3-)] = 0, so Q = 0.

Determine the direction of the reaction

As Q < K, the reaction will move towards the right side (forward direction).

Calculate the final concentration of Mg^2+ ions

Let x be the moles of Mg^2+ that reacted. Thus, the final concentrations at equilibrium will be: [Mg^2+] = 0.002055 - x [P3O10^(5-)] = 0.1087 - x [MgP3O10^(3-)] = x We can now set up an expression for K using these concentrations: K = [MgP3O10^(3-)] / ([Mg^2+] × [P3O10^(5-)]) = x / ((0.002055 - x) × (0.1087 - x)) Since K ≫ 1, the reaction proceeds almost to completion, and we can assume that x ≈ 0.002055 mol/L. Therefore, the final concentration of Mg^2+ ions in the solution is: [Mg^2+] ≈ 0.002055 - x ≈ 0.002055 - 0.002055 ≈ 0 mol/L However, this is an approximation, so the final concentration of Mg^2+ ions will be very close to 0 mol/L, but not exactly 0.

Key concepts

Complex Ion Formation

Complex ion formation is a fascinating part of chemical equilibrium where a central metal ion binds with one or more ligands, forming a complex ion. Metals like magnesium (\(\text{Mg}^{2+}\)) and calcium (\(\text{Ca}^{2+}\)) often form these complexes. In the context of our example, when magnesium ions bind with sodium tripolyphosphate (\(\text{Na}_5\text{P}_3\text{O}_{10}\)), it results in the complex \(\text{MgP}_3\text{O}_{10}^{3-}\).

• This complex ion formation is crucial in processes like water softening.
• The equilibrium involved means that the reaction can proceed forward or reverse depending on conditions.
• The formation constant, represented by \(K\), tells us how strongly the ions interact to form the complex.

This interaction is a dynamic equilibrium, meaning both forward and backward reactions occur simultaneously, though the extent varies with conditions such as concentration and temperature.

Water Softening

Water softening is an essential process that removes hardness from water, typically caused by minerals like calcium and magnesium. This is particularly important in preventing scale formation in pipes and improving detergent efficiency.
In our scenario, sodium tripolyphosphate acts as a chelating agent by binding with \(\text{Mg}^{2+}\) and \(\text{Ca}^{2+}\) ions to form soluble complexes. This prevents the ions from reacting with soap or surfactants, which would usually form scum.

• By removing these hard ions, water softening improves water quality and prevents appliance degradation.
• Sodium tripolyphosphate is not the only compound used; others like EDTA can also sequester metal ions.

The effectiveness of a water softener is often maintained by replenishing or regenerating agents, ensuring continued removal of hardness ions from the water.

Equilibrium Constant

The equilibrium constant, \(K\), is a vital concept in understanding chemical reactions. It represents the ratio of concentrations of products to reactants at equilibrium. For the complex ion formation reaction, the equilibrium constant tells us how far the reaction will proceed.
When we have a high \(K\) like in the sodium tripolyphosphate and magnesium reaction, it indicates the reaction strongly favors product formation. For the equilibrium: \[ \mathrm{Mg}^{2+}(aq) + \mathrm{P}_{3} \mathrm{O}_{10}^{5-}(aq) \rightleftharpoons \mathrm{MgP}_{3} \mathrm{O}_{10}^{3-}(aq) \]
The equilibrium constant expression would be:
\[ K = \frac{[\mathrm{MgP}_{3}\mathrm{O}_{10}^{3-}]}{[\mathrm{Mg}^{2+}] [\mathrm{P}_{3}\mathrm{O}_{10}^{5-}]} \]

• The large \(K\) value means that at equilibrium, the concentration of \(\text{MgP}_3\text{O}_{10}^{3-}\) is much higher than that of \(\text{Mg}^{2+}\) and \(\text{P}_3\text{O}_{10}^{5-}\).
• This ensures most \(\text{Mg}^{2+}\) ions are removed from the water.

Understanding \(K\) helps predict how a reaction will respond to changes in conditions, such as concentration or temperature shifts.

Surfactants

Surfactants, or surface-active agents, are compounds that significantly lower the surface tension of liquids. This property makes them essential in cleaning products and detergents. Surfactants work by orienting themselves at the boundary between water and dirt or grease, facilitating the removal of these substances.
Sodium tripolyphosphate plays a role in enhancing the effectiveness of surfactants. By forming complexes with \(\text{Mg}^{2+}\) and \(\text{Ca}^{2+}\), which are known to hinder surfactants by forming insoluble compounds, the detergents can work more efficiently.

• Softened water allows surfactants to effectively spread and penetrate surfaces.
• Without interference from hardness ions, surfactants maintain their full cleansing power.

Typically, surfactants are a blend of different chemical structures, each contributing to a specific aspect of the cleaning process. Whether it's foaming, emulsifying, or dispersing actions, surfactants and chelating agents like sodium tripolyphosphate ensure that cleaning processes are more effective.