Calculate the \(\mathrm{pH}\) of a solution that is \(1.00 \mathrm{M} \mathrm{HCN}\) and \(1.00 M\) HF. Compare the concentration (in molarity) of the \(\mathrm{CN}^{-}\) ion in this solution with that in a \(1.00 \mathrm{M}\) HCN solution. Comment on the difference.

The dissociation constant (Ka) is a measure of the strength of an acid in solution. It represents the equilibrium constant for the dissociation of an acid into its conjugate base and a hydrogen ion. Mathematically, for a generic acid HA, the Ka is expressed as follows:
\[ Ka = \frac{[A^-][H^+]}{[HA]} \]
where \([HA]\) is the concentration of the undissociated acid, \([A^-]\) is the concentration of its conjugate base, and \([H^+]\) is the concentration of hydrogen ions. In essence, a higher Ka value indicates a stronger acid that dissociates more completely in solution. Weak acids, such as HCN and HF, have relatively low Ka values, implying that they do not fully dissociate in solution. Understanding the Ka is essential to calculate the pH of a solution, especially when dealing with weak acids.

The ICE table method is an organized way to apply the stoichiometry of an equilibrium reaction to figure out the concentrations of reactants and products. ICE stands for Initial, Change, and Equilibrium. Initially, you list the molar concentrations of all species in the reaction. Then, you denote the changes that occur as the reaction proceeds towards equilibrium, often expressed as x. Lastly, you calculate the equilibrium concentrations of all species.
Used in conjunction with the equilibrium expression, the ICE table facilitates solving for unknowns, such as the concentration of hydrogen ions (\(H^+\)) from the dissociation of weak acids. This method is crucial for performing pH calculations because it allows for the systematic approach to determine the degree of dissociation of acids in solutions, which directly affects the pH.

Weak acids only partially dissociate in solution, establishing an equilibrium between the undissociated acid and its ions. This behavior contrasts with strong acids, which fully dissociate and release all their hydrogen ions. Weak acids like HCN and HF, as mentioned in the exercise, do not completely give up their protons (hydrogen ions). Instead, they set up a dynamic equilibrium where the forward reaction (the acid releasing H+ ions) and the reverse reaction (the conjugate base accepting H+ ions) occur at the same rate.
This partial dissociation results in lower hydrogen ion concentration and, thus, a higher pH when compared to strong acids. As weak acids react in a solution, the extent of their dissociation is determined by their Ka value, which influences pH calculations.

The concentration of ions in solution is directly tied to the dissociation of the substance in the solvent. When calculating the concentration of ions for weak acids, one must account for the partial dissociation, described by their dissociation constant. In the given exercise, for instance, the concentration of the \(CN^-\) ions is not simply the starting concentration of HCN. It's calculated based on how much HCN dissociates, which is influenced by the presence of other acids, such as HF.
Adding a different acid like HF to a HCN solution introduces additional H+ ions, which shift the equilibrium towards the undissociated form of HCN, thus decreasing the concentration of \(CN^-\) ions. This interplay and its effects on ion concentration must be explained correctly, as it impacts the resultant solution's characteristics, including its pH.