Problem 33 - A Complete Resource Book in Chemistry for JEE Main

Question

The \(\mathrm{pH}\) of a \(10^{-8}\) molar solution of \(\mathrm{HCl}\) in water is (a) \(-8\) (b) 8 (c) between 6 and 7 (d) between 7 and 8

Step-by-Step Solution

Answer

The pH is between 6 and 7.

Step 1: Understanding the pH Scale

The \(B\mathrm{pH} scale\) is a measure of the acidity or alkalinity of a solution. \(\mathrm{pH} = -\log_{10}[\mathrm{H^+}]\), where \(\left[\mathrm{H^+}\right]\) is the concentration of hydrogen ions. Pure water at 25°C has a \(\mathrm{pH}\) of 7 which is considered neutral.

Step 2: Calculate the pH of HCL Solution

For a \(10^{-8} \mathrm{M}\) \(\mathrm{HCl}\) solution, \(\left[\mathrm{H^+}\right]\) from \(\mathrm{HCl}\) alone would be \(10^{-8}\). However, contributions from autoionization of water must also be considered since \(K_w = [\mathrm{H^+}][\mathrm{OH^-}] = 10^{-14}\). In pure water, \(\left[\mathrm{H^+}\right]\) and \(\left[\mathrm{OH^-}\right]\) are each \(10^{-7} \mathrm{M}\). This means that actual \(\mathrm{H^+}\) concentration in the solution is \(10^{-8} + 10^{-7} = 1.1 \times 10^{-7}\).

Step 3: Calculate the pH Using Total Hydrogen Ion Concentration

With the total \(\mathrm{H^+}\) concentration being \(1.1 \times 10^{-7}\:\\mathrm{pH} = -\log_{10}(1.1 \times 10^{-7}) \approx 6.96\).

Step 4: Match the pH Value to the Given Options

The calculated \(\mathrm{pH}\) of approximately \(6.96\) fits the option that states between \(6\) and \(7\). Therefore, the correct choice is \(\mathrm{(c)}\) between \(6\) and \(7\).

Key Concepts

pH scale

The pH scale is a crucial tool for scientists, particularly when assessing the acidity or basicity of solutions. The scale typically ranges from 0 to 14. A pH level of 7 is considered neutral, such as pure water. Below 7 indicates an acidic solution, whereas values above 7 signify a basic (or alkaline) solution.

Calculating pH involves using the formula:

• \( \mathrm{pH} = -\log_{10}[\mathrm{H^+} ] \)

This means that pH is inversely related to the hydrogen ion concentration, \( [\mathrm{H^+}] \). The higher the \([\mathrm{H^+} ]\), the lower the pH, and thus, the more acidic the solution becomes. This logarithmic nature of the scale means even small changes in \([\mathrm{H^+} ]\) can lead to significant pH changes.

In practical terms, the pH scale helps determine the chemical behavior of substances, how they might react with others, and their suitability for various uses, like in chemical experiments or biological environments.

autoionization of water

Water has a unique property known as autoionization, where water molecules interact to form hydrogen ions \( \mathrm{H^+} \) and hydroxide ions \( \mathrm{OH^-} \). This even happens in absolutely pure water! It’s a natural equilibrium:

• \( \mathrm{H_2O} \rightleftharpoons \mathrm{H^+} + \mathrm{OH^-} \)

This equilibrium is responsible for water's neutral pH of 7 at 25°C, as both ions are present at concentrations of \( 10^{-7} \) M. This balanced concentration is disrupted when acids or bases are introduced.

Understanding this intrinsic quality of water is essential when calculating the pH of weakly acidic or basic solutions, especially when their hydrogen ion concentrations are close to the base level created by water's autoionization itself. This is crucial for determining accurate pH levels as seen in the original exercise.

hydrogen ion concentration

The concentration of hydrogen ions, or \( [\mathrm{H^+} ] \), is a pivotal determinant of a solution's acidity. An increased \( [\mathrm{H^+} ] \) leads to a lower pH, indicating an acidic environment. Conversely, a lower \( [\mathrm{H^+} ] \) suggests a higher pH and a more basic solution.

In the exercise, even though \( [\mathrm{H^+} ] \) from a \( 10^{-8} \) M \( \mathrm{HCl} \) solution seems minimal, the natural \( [\mathrm{H^+} ] \) contributed by water through autoionization results in a total \( [\mathrm{H^+} ] \) of \( 1.1 \times 10^{-7} \) M. This reinforces how crucial accounting for all sources of hydrogen ions is in obtaining an accurate measure of pH.

Students must pay attention to both external hydrogen ions introduced to the solution and those already present due to water's autoionization. This dual-source understanding is crucial, particularly in low-concentration solutions.

Kw equilibrium

The term \( K_w \) refers to the ion-product constant for water at a specific temperature (commonly 25°C). It’s a measure of the extent of the autoionization of water:

• \( K_w = [\mathrm{H^+}][\mathrm{OH^-}] = 10^{-14} \)

This equilibrium constant is fundamental when analyzing acidic or basic solutions, because it always holds true for watery solutions.

In a neutral solution at 25°C, \([\mathrm{H^+}] = [\mathrm{OH^-}] = 10^{-7} \) M, resulting in a pH of 7. Deviations in \( [\mathrm{H^+}] \) or \( [\mathrm{OH^-}] \) balance due to additional solutes affecting pH can be calculated while respecting this constant, as was needed in the explanation of the problem.

Understanding \( K_w \) is critical in pH calculation, particularly for those solutions falling near neutral pH values, like the exercise's solution of \( \mathrm{HCl} \). It reminds us how water inherently maintains a balance, even when acids and bases are present.

HCl solution

Hydrochloric acid, or \( \mathrm{HCl} \), is a strong acid commonly found in both laboratories and industries. When dissolved in water, it dissociates completely into \( \mathrm{H^+} \) and \( \mathrm{Cl^-} \) ions, significantly increasing the hydrogen ion concentration, \( [\mathrm{H^+} ] \).

When determining the pH of a weak \( \mathrm{HCl} \) solution, such as \( 10^{-8} \) M, the natural \( [\mathrm{H^+} ] \) from the acid might seem negligible compared to the intrinsic \( [\mathrm{H^+} ] \) from water's autoionization. That's why one must consider both contributions to \( [\mathrm{H^+} ] \) in total when calculating the pH.

Thus, even tiny concentrations of \( \mathrm{HCl} \) can alter the solution's overall acidity slightly from neutrality due to their additive effect with hydrogen ions from water, as emphasized in the given problem resolution. This complexity highlights the importance of taking all ion contributions into account equally in chemistry.