A certain indicator HIn has a \(\mathrm{p} K_{\mathrm{a}}\) of 3.00 and a color change becomes visible when \(7.00 \%\) of the indicator has been converted to \(\operatorname{In}^{-} .\) At what \(\mathrm{pH}\) is this color change visible?

Understanding the concept of pKa is crucial for students studying chemistry, as it represents the strength of an acid in solution. The pKa value is derived from the acid dissociation constant (Ka), which is a quantitative measure of how well an acid releases its protons into solution. A lower pKa value indicates a stronger acid, as it more readily gives up its protons.

The calculation of pKa is very straightforward: it is the negative base-10 logarithm of the Ka value. Mathematically, this is represented as \[ \text{pKa} = -\log(\text{Ka}) \]. When given a pKa, you can find Ka by taking the antilogarithm or the power of 10 of the negative pKa. Specifically, \[ \text{Ka} = 10^{-\text{pKa}} \].

For example, with a pKa of 3.00, the Ka can be calculated by \[ \text{Ka} = 10^{-3.00} = 1 \times 10^{-3} \]. Knowing how to interconvert these two quantities is essential when analyzing chemical equilibrium, especially in acid-base reactions.

Chemical equilibrium is a dynamic state where the rate of the forward chemical reaction equals the rate of the reverse reaction. At equilibrium, the concentrations of reactants and products remain constant but are not necessarily equal.

To quantify the position of equilibrium, we use the equilibrium constant (K). For acid-base reactions, we specifically talk about the acid dissociation constant (Ka), which is the equilibrium constant for the dissociation of an acid to its conjugate base and a proton. The general form of the equation is \[ \text{Ka} = \frac{[\text{Product 1}][\text{Product 2}]}{[\text{Reactant}]} \].

In the context of the exercise, the equilibrium expression for the dissociation of the indicator HIn is given by \[ \text{Ka} = \frac{[\text{In}^-][\text{H}^+]}{[\text{HIn}]} \]. By setting up this expression and knowing the Ka, we can solve for the unknown concentrations, assuming we have some initial concentration inputs, like the percentage of dissociation in this case.

Determining the pH of a solution is a fundamental aspect of chemistry, as it indicates the acidity or basicity of that solution. pH is a scale used to specify the hydrogen ion concentration (\( [\text{H}^+] \) ) in a solution. The pH scale ranges from 0 to 14, with 7 being neutral. A pH less than 7 indicates acidity, while a pH greater than 7 indicates alkalinity.

The pH can be calculated by taking the negative logarithm (base 10) of the hydrogen ion concentration: \[ \text{pH} = -\log([\text{H}^+]) \]. This formula shows the inverse logarithmic relationship between hydrogen ion concentration and pH.

For instance, if the concentration of \( [\text{H}^+] \) is determined to be \( \frac{0.93 \times 10^{-3}}{0.07} \) from the equilibrium calculation, then the pH is calculated as follows: \[ \text{pH} = -\log\left(\frac{0.93 \times 10^{-3}}{0.07}\right) \approx 1.66 \]. This shows that at the point where 7% of the indicator HIn has dissociated, the solution will have a pH of approximately 1.66, making it highly acidic. Thus, pH determination helps in understanding the extent of acid dissociation and visualizing the acid-base nature of a solution.