Question

Solutions of sodium thiosulfate are used to dissolve unexposed \(\operatorname{AgBr}\left(K_{\text {sp }}=5.0 \times 10^{-13}\right.\) ) in the developing process for blackand-white film. What mass of \(\mathrm{AgBr}\) can dissolve in \(1.00 \mathrm{~L}\) of \(0.500 \mathrm{M} \mathrm{Na}_{2} \mathrm{~S}_{2} \mathrm{O}_{3} ? \mathrm{Ag}^{+}\) reacts with \(\mathrm{S}_{2} \mathrm{O}_{3}{ }^{2-}\) to form a complex ion: \(\mathrm{Ag}^{+}(a q)+2 \mathrm{~S}_{2} \mathrm{O}_{3}{ }^{2-}(a q) \rightleftharpoons \mathrm{Ag}\left(\mathrm{S}_{2} \mathrm{O}_{3}\right)_{2}{ }^{3-}(a q)\)

Short answer

Approximately \(4.21 \times 10^{-5} \: g\) of AgBr can dissolve in 1.00 L of 0.500 M Na2S2O3 solution.

Step-by-step solution

Write down the balanced equation for the given reaction

The balanced equation for the reaction of Ag+ with S2O3^2- is given as: \[ Ag^+(aq) + 2 S2O3^{2-}(aq) \rightleftharpoons Ag(S2O3)_2^{3-}(aq) \]

Determine the reaction quotient (Q) for the reaction of Ag+ with S2O3^2-

To determine the reaction quotient (Q) for the reaction of Ag+ with S2O3^2-, we will create an ICE table (Initial, Change, and Equilibrium) for the equation: | | Ag^+ | S2O3^2- | Ag(S2O3)_2^{3-} | |--------|------|---------|-----------------| |Initial | x | 0.500 | 0 | |Change | -x | -2x | x | |Equil. | 0 | 0.500 - 2x | x | Since Ag+ reacts completely with S2O3^2-: Q = [Ag(S2O3)_2^{3-}] = x

Find the Ksp expression for the dissolution of AgBr

The dissolution of AgBr is as follows: \[ AgBr(s) \rightleftharpoons Ag^+(aq) + Br^-(aq) \] The Ksp expression for the dissolution of AgBr is given by: \[ K_{sp} = [Ag^+][Br^-] \] We will use the provided Ksp value of 5.0 x 10^{-13} to solve for [Ag^+] and [Br^-].

Use the Ksp expression to solve for [Ag^+] and [Br^-]

Note that, at equilibrium, [Ag^+] = [Br^-]. Let's use the Ksp value: \[ 5.0 \times 10^{-13} = [Ag^+][Br^-] = [Ag^+]^2 \] Solving for [Ag^+]: \[ [Ag^+] = \sqrt{5.0 \times 10^{-13}} \approx 2.24 \times 10^{-7} \: M \] Since [Ag^+] = [Br^-], we also have: \[ [Br^-] = 2.24 \times 10^{-7} \: M \]

Calculate the mass of AgBr that can dissolve in the solution

Now that we have the concentration of Ag^+ in equilibrium, we can find the mass of AgBr that can dissolve in 1.00 L of 0.500 M Na2S2O3 solution. The molar mass of AgBr is approximately 187.77 g/mol: \[ \text{Mass of AgBr} = \text{(conc. of Ag^+)} \times \text{(volume of solution)} \times \text{(molar mass of AgBr)} \] \[ \text{Mass of AgBr} = (2.24 \times 10^{-7} \: M) \times (1.00 \: L) \times (187.77 \: g/mol) \] \[ \text{Mass of AgBr} \approx 4.21 \times 10^{-5} \: g \] So, approximately \(4.21 \times 10^{-5} \: g\) of AgBr can dissolve in 1.00 L of 0.500 M Na2S2O3 solution.

Key concepts

Silver Bromide (AgBr)

Silver bromide, denoted as AgBr, is a compound that is commonly utilized in photography, particularly in black-and-white film. It plays an important role during the film developing process. In certain solutions, AgBr can undergo dissolution that is limited by its low solubility.

AgBr is an ionic compound composed of silver ions (Ag⁺) and bromide ions (Br⁻). It is known for its extremely low solubility in water, characterized by a very small Ksp value. This means that only a tiny amount of AgBr can dissolve in water to produce Ag⁺ and Br⁻ ions. However, adding certain substances can help "pull" AgBr into solution, such as through the formation of complex ions, which we will explore later to help more AgBr dissolve.

Understanding the dissolution of AgBr is crucial for applications where fine control over the amount of silver ions in a solution is necessary, such as in the development of photographs.

Complex Ion Formation

Complex ion formation refers to the process in which central metal ions like Ag⁺ form a complex, a type of coordination compound, with other ions or molecules. In our scenario, the presence of excess sodium thiosulfate (Na₂S₂O₃) in solution can lead to complex ion formation.

When Ag⁺ ions react with thiosulfate ions (S₂O₃²⁻), they form a complex ion, \[ \text{Ag}^+_{(aq)} + 2 \text{S}_2\text{O}_3^{2-}_{(aq)} \rightleftharpoons \text{Ag(S}_2\text{O}_3)_2^{3-}_{(aq)} \]

This reaction greatly increases the solubility of AgBr compared to when only water is present. This is because the formation of Ag(S₂O₃)₂³⁻ reduces the concentration of free Ag⁺ ions, encouraging more AgBr to dissolve to replenish the Ag⁺ ions.

The ability to form complex ions is extensively used to control solubility and reactions in various chemical processes, particularly in industrial applications and laboratory settings.

Reaction Quotient

The reaction quotient, denoted as Q, is a measure of the relative quantities of products and reactants present in a reaction at any given point in time. Unlike equilibrium constants, Q can be calculated at any moment whether the system is at equilibrium or not. It gives insight into which direction a reaction will proceed to achieve equilibrium.

In the dissolution of silver bromide, Q often helps predict whether more of the solid will dissolve or if ions will precipitate out from solution. As Ag⁺ reacts with S₂O₃²⁻ to form Ag(S₂O₃)₂³⁻, calculating a reaction quotient for this newly formed complex ion assists in understanding the process.

\[ Q = [\text{Ag(S}_2\text{O}_3)_2^{3-}] \]

If Q is less than the equilibrium constant, K, for the complex ion formation reaction, the system will shift to form more products, aiding dissolution. Monitoring Q assists in controlling the extent of reaction and optimizing the solubility of AgBr under developing conditions.

Ksp (Solubility Product Constant)

The solubility product constant, Ksp, is an important concept when dealing with sparingly soluble salts such as silver bromide (AgBr). Ksp is a measure of the solubility of a compound in solution at a given temperature. It is unique for each compound and provides insight into how much of the compound can dissolve in water.

The Ksp expression for AgBr is: \[ K_{sp} = [\text{Ag}^+][\text{Br}^-] \]

For AgBr, the Ksp value is given as \(5.0 \times 10^{-13}\). This low value indicates that AgBr dissolves sparingly in water, producing only minute concentrations of Ag⁺ and Br⁻ ions.

In practical applications, calculating Ksp allows chemists and students to determine the maximum concentrations of ions before precipitation occurs or the residual concentration of a compound in solution. By using the Ksp value, one can predict solubility under different conditions, crucial for applications like photographic film development where precise control is needed.