Question

Solution \(\mathrm{X}\) has a \(\mathrm{pH}\) of \(4.35 .\) Solution \(\mathrm{Y}\) has \(\left[\mathrm{OH}^{-}\right]\) ten times as large as solution \(\mathrm{X}\). Solution \(\mathrm{Z}\) has a pH \(4.0\) units higher than that of solution \(\mathrm{X}\). (a) Calculate the ratio of \(\left[\mathrm{H}^{+}\right]\) in solutions \(\mathrm{X}\) and \(\mathrm{Y}\) and solutions \(\mathrm{X}\) and \(\mathrm{Z}\). (b) What is the \(\mathrm{pH}\) of solutions \(\mathrm{Y}\) and \(\mathrm{Z}\) ? (c) Classify each solution as acidic, basic, or neutral.

Short answer

Answer: The [H+] concentration ratios between solutions X and Y, and solutions X and Z can be found using the previously calculated H+ concentrations: \(\frac{[\mathrm{H}_\mathrm{X}^+]}{[\mathrm{H}_\mathrm{Y}^+]}\) and \(\frac{[\mathrm{H}_\mathrm{X}^+]}{[\mathrm{H}_\mathrm{Z}^+]}\) Regarding the classification, a solution is classified as acidic if its pH is less than 7, basic if its pH is greater than 7, and neutral if its pH is equal to 7. By calculating the pH values of solutions Y and Z from the provided information, we can classify each solution based on their pH values.

Step-by-step solution

Calculate the H+ concentration of solution X

The pH is defined as \(\mathrm{pH}=-\log [\mathrm{H}^+]\). Therefore, we can calculate the concentration of H+ ions in solution X as follows: \([\mathrm{H}^+]=10^{-\mathrm{pH}}=10^{-4.35}\).

Calculate the OH- concentration of solution X

Using the relationship between [H+] and [OH−] concentrations, we can calculate the concentration of OH− ions in solution X: \([\mathrm{H}^+][\mathrm{OH}^-]=1.0 \times 10^{-14}\). Therefore, \([\mathrm{OH}^-]=\frac{1.0 \times 10^{-14}}{[\mathrm{H}^+]}\)

Calculate the [OH-] and [H+] concentrations for solution Y

Solution Y’s [OH−] is 10 times larger than the [OH−] of solution X, so we have \([\mathrm{OH}_\mathrm{Y}^-]=10\times [\mathrm{OH}_\mathrm{X}^-]\). Then, we can calculate the concentration of H+ ions in solution Y: \([\mathrm{H}_\mathrm{Y}^+]=\frac{1.0 \times 10^{-14}}{[\mathrm{OH}_\mathrm{Y}^-]}\)

Calculate the pH value of solution Y

Now that we have the [H+] concentration of solution Y, we can calculate its pH: \(\mathrm{pH}=-\log [\mathrm{H}_\mathrm{Y}^+]\).

Calculate the pH value of solution Z

Solution Z has a pH 4.0 units higher than that of solution X, so we can write \(\mathrm{pH}_\mathrm{Z}=\mathrm{pH}_\mathrm{X}+4.0\).

Calculate the [H+] concentration for solution Z

To calculate the H+ concentration of solution Z, we use the formula \([\mathrm{H}_\mathrm{Z}^+]=10^{-\mathrm{pH}_\mathrm{Z}}\).

Find the ratios of [H+] concentrations

We are asked to find the ratio of H+ concentrations in solutions X and Y, and solutions X and Z. To do this, we perform the following calculations: \(\frac{[\mathrm{H}_\mathrm{X}^+]}{[\mathrm{H}_\mathrm{Y}^+]}\) and \(\frac{[\mathrm{H}_\mathrm{X}^+]}{[\mathrm{H}_\mathrm{Z}^+]}\).

Classify the solutions

A solution is considered acidic if its pH is less than 7, basic if its pH is greater than 7, and neutral if its pH is equal to 7. We can classify each solution based on their pH values calculated in the previous steps.

Key concepts

Acid and Base Concentration

Understanding the concentration of acids and bases in a solution is crucial for many aspects of chemistry, including neutralization reactions, buffer systems, and solubility equilibria. In the context of our exercise, the concentration of hydrogen ions (\textsf{[H+]}) and hydroxide ions (\textsf{[OH−]}) determine the acidity or basicity of solutions X, Y, and Z.

When we refer to the concentration of an acid, we're usually talking about the concentration of \textsf{[H+]} ions it contributes to a solution. Conversely, the concentration of a base corresponds to the \textsf{[OH−]} ions it contributes. Using the concept of molarity (moles per liter), we express these concentrations and make use of the ion-product constant for water (\textsf{Kw}), which is \(1.0 \times 10^{-14}\) at 25°C, to relate \textsf{[H+]} and \textsf{[OH−]}.

For example, the \textsf{[H+]} concentration of solution X calculated in step 1 was \(10^{-4.35}\). Using this, we determined the \textsf{[OH−]} in solution X in step 2 by applying the ion-product constant of water. In acidic solutions, \textsf{[H+]} > \textsf{[OH−]}, while in basic solutions, the reverse is true. One can also calculate the \textsf{[H+]} or \textsf{[OH−]} concentration for a given solution when one is known, by using the value of \textsf{Kw}.

pOH and pH Relation

In chemistry, the pH and pOH are both measures of the acidic and basic properties of a solution but are inversely related. The pH measures the concentration of hydrogen ions (\textsf{[H+]}), while the pOH measures the concentration of hydroxide ions (\textsf{[OH-]}). The pH scale typically ranges from 0 to 14, where a pH of 7 is neutral, a pH less than 7 is acidic, and a pH greater than 7 is basic.

The relationship between pH and pOH can be described by the equation: \(\text{pH} + \text{pOH} = 14\), which is derived from the ion-product constant of water (\textsf{Kw}). Using this relationship, if one knows either the pH or the pOH of a solution, the other can be quickly determined. This is illustrated in our exercise when calculating the \textsf{[OH−]} concentration for solution Y after finding its pH.

Hydrogen Ion Concentration

Hydrogen ion concentration, expressed as \textsf{[H+]}, is a measure of the number of moles of hydrogen ions per liter of solution. It is an essential indicator of the solution's acidity. The more hydrogen ions present, the more acidic the solution is. The pH is a logarithmic scale used to simplify the expression of \textsf{[H+]}, calculated using the formula \(\mathrm{pH} = -\log [\mathrm{H}^+]\).

In the given exercise, the pH of solution X was initially given, allowing for the calculation of \textsf{[H+]} in step 1, and consequently, the calculation of the pH of solutions Y and Z in steps 4 and 5. It is important to recognize that a small change in pH corresponds to a significant change in \textsf{[H+]} due to the logarithmic nature of the pH scale. For instance, a pH difference of 1 unit reflects a tenfold difference in \textsf{[H+]} concentration.

Hydroxide Ion Concentration

Hydroxide ion concentration, noted as \textsf{[OH−]}, represents the number of moles of hydroxide ions in one liter of solution. The hydroxide ions are indicative of the solution's basicity—the higher their concentration, the more basic the solution. To calculate the concentration of hydroxide ions, we use the ion-product constant of water (\textsf{Kw}) and the known concentration of hydrogen ions, with the formula \(\text{[OH−]} = \frac{\text{Kw}}{\text{[H+]}}\).

This calculation enables us to find the pOH using the expression \(\text{pOH} = -\log \text{[OH−]}\), and it plays a crucial role in steps 2 and 3 of our exercise when determining [OH−] for solutions X and Y. Remember that just like with pH, a small change in pOH indicates a significant change in [OH−], and this is vital when comparing the basicity of different solutions.