Question

Consider the reaction $$ \mathrm{AgCl}(s) \longrightarrow \mathrm{Ag}^{+}(a q)+\mathrm{Cl}^{-}(a q) $$ (a) Calculate \(\Delta G^{\circ}\) at \(25^{\circ} \mathrm{C}\). (b) What should the concentration of \(\mathrm{Ag}^{+}\) and \(\mathrm{Cl}^{-}\) be so that \(\Delta G=-1.0 \mathrm{~kJ}\) (just spontaneous)? Take \(\left[\mathrm{Ag}^{+}\right]=\) \(\left[\mathrm{Cl}^{-}\right]\) (c) The \(K_{\mathrm{sp}}\) for \(\mathrm{AgCl}\) is \(1.8 \times 10^{-10} .\) Is the answer to \((\mathrm{b})\) reasonable?

Short answer

Based on the calculations above, calculate the values for \(\Delta G^\circ\), the concentration of \(\mathrm{Ag}^{+}\) and \(\mathrm{Cl}^{-}\) (labeled as \(x\)), and determine if the answer to part (b) is reasonable by comparing \(x^2\) to the given value of \(K_{\mathrm{sp}}=1.8 \times 10^{-10}\).

Step-by-step solution

Calculate \(\Delta G^\circ\) at \(25^{\circ} \mathrm{C}\)

First, find the standard Gibbs free energy change, \(\Delta G^\circ\). We are given the \(K_{\mathrm{sp}}\) for \(\mathrm{AgCl}\), which is \(1.8 \times 10^{-10}\). We know that \(\Delta G^\circ=-RT \ln K_p\). Since we have \(K_{\mathrm{sp}}\), we can use the relationship \(K_p = K_{\mathrm{sp}}\) in this case. Given \(T=25^{\circ} \mathrm{C}\), we need to convert it to Kelvin: \(T = 25 + 273.15 = 298.15\,\mathrm{K}\). The gas constant, \(R\), is \(8.314 \,\mathrm{\frac{J}{mol\cdot K}}\). Now, we calculate \(\Delta G^\circ\): $$ \Delta G^\circ=-R T \ln K_{\mathrm{sp}}=-8.314 \,\frac{\mathrm{J}}{\mathrm{mol \cdot K}}(298.15 \,\mathrm{K}) \ln(1.8 \times 10^{-10}) $$

Find the concentrations of \(\mathrm{Ag}^{+}\) and \(\mathrm{Cl}^{-}\)

Now, we need to find the concentration of \(\mathrm{Ag}^{+}\) and \(\mathrm{Cl}^{-}\) so that \(\Delta G = -1.0 \mathrm{~kJ}\). We are given \(\Delta G\) and want to find the corresponding \(Q\). Using the formula \(\Delta G = \Delta G^\circ + RT\ln{Q}\), we need to isolate \(Q\): $$ \begin{aligned} Q =& e^{(\Delta G - \Delta G^\circ) / (RT)} \\ =& e^{((-1.0 \times 10^3 \,\mathrm{J}) - \Delta G^\circ) / (8.314 \,\mathrm{J/mol \cdot K} \cdot 298.15 \,\mathrm{K})} \\ \end{aligned} $$ Now, we are given that \(\left[\mathrm{Ag}^{+}\right]=\) \(\left[\mathrm{Cl}^{-}\right]\), which we will label as \(x\) for simplicity. The reaction quotient, \(Q\), will depend on the concentrations of \(\mathrm{Ag}^{+}\) and \(\mathrm{Cl}^{-}\): $$ Q = x^2 $$ Substitute the value of \(Q\) above into the equation we derived previously, and solve for \(x\), which represents the concentrations of \(\mathrm{Ag}^{+}\) and \(\mathrm{Cl}^{-}\): $$ x^2 = e^{((-1.0 \times 10^3 \,\mathrm{J}) - \Delta G^\circ) / (8.314 \,\mathrm{J/mol \cdot K} \cdot 298.15 \,\mathrm{K})} $$

Check if the answer to part (b) is reasonable

Lastly, we need to check if our answer in part (b), \(x\), is reasonable using the provided value of \(K_{\mathrm{sp}} = 1.8 \times 10^{-10}\). Using the relationship \(K_p = K_{\mathrm{sp}}\), we know that \(Q\) must be equal to \(K_{\mathrm{sp}}\) at equilibrium. We found \(Q = x^2\). Therefore, to determine if our calculated value of \(x\) is reasonable, we should compare \(x^2\) to \(K_{\mathrm{sp}}=1.8 \times 10^{-10}\). If the values are close to one another, then our answer to part (b) is reasonable.

Key concepts

Chemical Equilibrium

At the heart of many chemical processes is the concept of chemical equilibrium, a state in which the rate of the forward reaction equals the rate of the reverse reaction, resulting in no net change in the concentrations of reactants and products over time. Equilibrium does not mean that the reactants and products are equal in concentration, but rather that they have reached a constant ratio that remains unchanged.

When a system is at equilibrium, the Gibbs free energy change (abla G) is equal to zero. This concept is crucial in predicting the direction of a chemical reaction and understanding when a reaction will cease to undergo a change without external influences. In the exercise given, the dissolution of silver chloride (mathrm{AgCl}) into silver (mathrm{Ag^+}) and chloride ions (mathrm{Cl^-}) reaches equilibrium when the system's Gibbs free energy is minimized and does not change with time.

Reaction Quotient (Q)

The reaction quotient (Q) helps to determine the direction a reaction will proceed to reach equilibrium. It is defined as the ratio of the products' concentrations to the reactants' concentrations at any point in time, raised to the power of their respective coefficients in the balanced equation.

To calculate Q, you use the same expression as the equilibrium constant (K), but Q can be determined at any stage of the reaction, not just at equilibrium. If QK, the reaction will shift to produce more reactants. In the exercise, by calculating Q for the dissolution of mathrm{AgCl}, students can predict whether the reaction will proceed forward or shift backward at any given moment.

Solubility Product Constant (Ksp)

The solubility product constant (Ksp) is a special type of equilibrium constant that applies to the dissolution of sparingly soluble salts. It is a measure of the saturated solution's ability to maintain equilibrium by signifying the maximum concentration of ions that can exist before the salt starts to precipitate.

In the exercise, mathrm{AgCl} has a known Ksp, which indicates the extent to which it can dissolve in water to form mathrm{Ag^+} and mathrm{Cl^-} ions. The smaller the value of Ksp, the lower the solubility of the compound. Students are tasked with relating Ksp to the Gibbs free energy to assess the feasibility of the dissolution process at a given concentration level.

Thermodynamics

Thermodynamics plays an essential role in understanding chemical reactions, particularly in the context of Gibbs free energy, equilibrium, and solubility. It provides a quantitative way to determine whether a process can occur spontaneously, as all spontaneous processes are characterized by a decrease in the system's free energy (ΔG).

The equation ΔG = ΔGº + RTlnQ encapsulates the thermodynamic principles that link the standard free energy change (ΔGº), temperature (T), the universal gas constant (R), and the reaction quotient (Q). In the silver chloride exercise, students assess the spontaneity of ion formation at varying concentrations by applying these thermodynamic concepts to calculate ΔG based on the reaction's current state. Understanding the thermodynamic underpinnings helps students predict the direction and extent of chemical reactions.