Question

Determine the \(\mathrm{H}_{3} \mathrm{O}^{+}\)concentration in solutions with each of the following \(\mathrm{pH}\) values. Identify each solution as acidic, basic, or neutral. (a) \(\mathrm{pH}=5.00\) (b) \(\mathrm{pH}=12.00\) (c) \(\mathrm{pH}=5.90\)

Short answer

For \(\mathrm{pH}=5.00\), \([\mathrm{H}_{3} \mathrm{O}^{+}] = 10^{-5.00}\) and is acidic. For \(\mathrm{pH}=12.00\), \([\mathrm{H}_{3} \mathrm{O}^{+}] = 10^{-12.00}\) and is basic. For \(\mathrm{pH}=5.90\), \([\mathrm{H}_{3} \mathrm{O}^{+}] = 10^{-5.90}\) and is acidic.

Step-by-step solution

Find concentration for \(\mathrm{pH}=5.00\).

Use the formula \(\mathrm{pH} = -\log[\mathrm{H}_{3} \mathrm{O}^{+}]\). So, rearranging, \([\mathrm{H}_{3} \mathrm{O}^{+}] = 10^{-\mathrm{pH}}\). Substitute \(\mathrm{pH} = 5.00\), get \([\mathrm{H}_{3} \mathrm{O}^{+}] = 10^{-5.00}\).

Identify the nature of the solution at \(\mathrm{pH}=5.00\).

Since \(\mathrm{pH} < 7\), the solution is acidic.

Find concentration for \(\mathrm{pH}=12.00\).

Use the formula from Step 1 with \(\mathrm{pH} = 12.00\), get \([\mathrm{H}_{3} \mathrm{O}^{+}] = 10^{-12.00}\).

Identify the nature of the solution at \(\mathrm{pH}=12.00\).

Since \(\mathrm{pH} > 7\), the solution is basic.

Find concentration for \(\mathrm{pH}=5.90\).

Use the formula from Step 1 with \(\mathrm{pH} = 5.90\), get \([\mathrm{H}_{3} \mathrm{O}^{+}] = 10^{-5.90}\).

Identify the nature of the solution at \(\mathrm{pH}=5.90\).

Since \(\mathrm{pH} < 7\), the solution is acidic.

Key concepts

pH Scale

The pH scale is a measure of how acidic or basic a solution is. It ranges from 0 to 14, with 7 being neutral. Solutions with a pH less than 7 are considered acidic, while those with a pH greater than 7 are deemed basic or alkaline. A key point to remember is that the pH scale is logarithmic, meaning each whole pH value below 7 is ten times more acidic than the next higher value, and each whole pH value above 7 is ten times less acidic, or more basic, than the one below it.

For example, a solution with a pH of 5 is ten times more acidic than a solution with a pH of 6 and a hundred times more acidic than a neutral pH of 7. This logarithmic nature is why pH calculations involve the exponentiation and logarithms of hydrogen ion concentration. Understanding the pH scale is essential when dealing with chemical solutions and interpreting their properties.

Acidic and Basic Solutions

Acidic solutions contain a higher concentration of hydrogen ions \( \left[ \mathrm{H}_{3}\mathrm{O}^{+} \right] \), also known as hydronium ions, than basic solutions. On the other hand, basic solutions have a higher concentration of hydroxide ions \( \left[ \mathrm{OH}^{-} \right] \). When \( \left[ \mathrm{H}_{3}\mathrm{O}^{+} \right] \ > 10^{-7}\) M, the solution is acidic, and when \( \left[ \mathrm{OH}^{-} \right] \ > 10^{-7}\) M, the solution is basic.

In the exercise, when the pH was 5.00 and 5.90, the nature of the solutions was acidic because their pH values were below 7. Conversely, the solution with a pH of 12.00 was basic because its pH was above 7. It's important to relate these concepts back to the pH scale to recognize that a lower pH means a higher acidity, and vice versa.

Hydrogen Ion Concentration

The concentration of hydrogen ions in a solution is a critical determinant of its pH value. This concentration is usually expressed in moles per liter (M). In the given exercise, the hydrogen ion concentration for each solution can be calculated using the formula \( \mathrm{pH} = -\log[\mathrm{H}_{3} \mathrm{O}^{+}] \). By rearranging the formula, you can find the hydronium ion concentration with \( [\mathrm{H}_{3} \mathrm{O}^{+}] = 10^{-\mathrm{pH}} \).

For instance, when the pH is 5.00, the hydronium ion concentration is \( 10^{-5}\) M, which implies there are \( 10^{-5}\) moles of \( \mathrm{H}_{3}\mathrm{O}^{+} \) ions in each liter of solution. This inverse relationship between pH and hydrogen ion concentration is fundamental to understanding the behavior of acids and bases. These calculations are vital in fields like chemistry, biology, environmental science, and various industrial applications.

Chemical Equilibrium

Chemical equilibrium refers to a state in a chemical reaction where the rate of the forward reaction equals the rate of the reverse reaction. This means that the concentrations of reactants and products remain constant over time. However, the concept of equilibrium extends beyond just the reactants and products of a single reaction; it also applies to the dissociation of water molecules into hydronium \( \mathrm{H}_{3}\mathrm{O}^{+} \) and hydroxide \( \mathrm{OH}^{-} \) ions. In pure water at 25°C, the product of these ion concentrations is always \( 10^{-14}\) M², which is reflected in the expression \( K_{w} = [\mathrm{H}_{3}\mathrm{O}^{+}][\mathrm{OH}^{-}] = 10^{-14}\).

This equilibrium constant for water determines the relationship between hydrogen ion and hydroxide ion concentrations and underpins the calculations for pH. For example, a change in \( \mathrm{H}_{3}\mathrm{O}^{+} \) concentration implies a complementary change in \( \mathrm{OH}^{-} \) concentration to maintain the equilibrium. Understanding chemical equilibrium is crucial for predicting the outcome of chemical reactions, especially in aqueous solutions.